Daily Papers

Segmenting medical images is critical to facilitating both patient diagnoses and quantitative research. A major limiting factor is the lack of labeled data, as obtaining expert annotations for each new set of imaging data and task can be labor intensive and inconsistent among annotators. We present CUTS, an unsupervised deep learning framework for medical image segmentation. CUTS operates in two stages. For each image, it produces an embedding map via intra-image contrastive learning and local patch reconstruction. Then, these embeddings are partitioned at dynamic granularity levels that correspond to the data topology. CUTS yields a series of coarse-to-fine-grained segmentations that highlight features at various granularities. We applied CUTS to retinal fundus images and two types of brain MRI images to delineate structures and patterns at different scales. When evaluated against predefined anatomical masks, CUTS improved the dice coefficient and Hausdorff distance by at least 10% compared to existing unsupervised methods. Finally, CUTS showed performance on par with Segment Anything Models (SAM, MedSAM, SAM-Med2D) pre-trained on gigantic labeled datasets.

The manifold hypothesis, which assumes that data lies on or close to an unknown manifold of low intrinsic dimension, is a staple of modern machine learning research. However, recent work has shown that real-world data exhibits distinct non-manifold structures, i.e. singularities, that can lead to erroneous findings. Detecting such singularities is therefore crucial as a precursor to interpolation and inference tasks. We address this issue by developing a topological framework that (i) quantifies the local intrinsic dimension, and (ii) yields a Euclidicity score for assessing the 'manifoldness' of a point along multiple scales. Our approach identifies singularities of complex spaces, while also capturing singular structures and local geometric complexity in image data.

Real-world phenomena do not generate arbitrary variability: their signals concentrate on compact, low-variability subsets of functional space, enabling rapid generalization from few examples. A small child can recognize a dog after extremely limited exposure because the perceptual manifold of "dog" is compact, structured, and low-dimensional. We formalize this principle through a deterministic functional-topological framework in which the set of valid realizations produced by a physical process forms a compact subset of a Banach space, endowed with stable invariants, a finite Hausdorff radius, and an induced continuous perceptual functional. This geometry provides explicit limits on knowledge, conditions for identifiability, and guarantees for generalization from sparse evidence -- properties fundamental to both natural and artificial intelligence. Across electromechanical, electrochemical, and physiological domains, we show that real-world processes consistently generate compact perceptual manifolds with the same geometric characteristics. Their boundaries can be discovered in a fully self-supervised manner as the empirical radius saturates with increasing sampling, even when the governing equations are unknown. These results demonstrate that deterministic functional topology offers a unified mathematical foundation for perception, representation, and world-model construction. It provides a geometric explanation for why biological learners and self-supervised AI systems can generalize from few observations, and establishes compact perceptual manifolds as a fundamental building block for future AI architectures. Finally, this work unifies biological perception and modern self-supervised models under a single geometric principle: both derive their generalization ability from the compactness and invariants of real-world perceptual manifolds.

Topological deep learning has emerged for modeling higher-order relational structures beyond pairwise interactions that standard graph neural networks fail to capture. Although combinatorial complexes offer a unified topological framework, most existing topological deep learning methods rely on local message passing via attention mechanisms, which incur quadratic complexity and remain low-dimensional, limiting scalability and rank-aware information aggregation in higher-order complexes.We propose Combinatorial Complex Mamba (CCMamba), the first unified mamba-based neural framework for learning on combinatorial complexes. CCMamba reformulates message passing as a selective state-space modeling problem by organizing multi-rank incidence relations into structured sequences processed by rank-aware state-space models. This enables adaptive, directional, and long range information propagation in linear time without self attention. We further establish the theoretical analysis that the expressive power upper-bound of CCMamba message passing is the 1-Weisfeiler-Lehman test. Experiments on graph, hypergraph, and simplicial benchmarks demonstrate that CCMamba consistently outperforms existing methods while exhibiting improved scalability and robustness to depth.

We focus on the utilisation of reactive trajectory imitation controllers for goal-directed mobile robot navigation. We propose a topological navigation graph (TNG) - an imitation-learning-based framework for navigating through environments with intersecting trajectories. The TNG framework represents the environment as a directed graph composed of deep neural networks. Each vertex of the graph corresponds to a trajectory and is represented by a trajectory identification classifier and a trajectory imitation controller. For trajectory following, we propose the novel use of neural object detection architectures. The edges of TNG correspond to intersections between trajectories and are all represented by a classifier. We provide empirical evaluation of the proposed navigation framework and its components in simulated and real-world environments, demonstrating that TNG allows us to utilise non-goal-directed, imitation-learning methods for goal-directed autonomous navigation.

Topological Neural Networks (TNNs) incorporate higher-order relational information beyond pairwise interactions, enabling richer representations than Graph Neural Networks (GNNs). Concurrently, topological descriptors based on persistent homology (PH) are being increasingly employed to augment the GNNs. We investigate the benefits of integrating these two paradigms. Specifically, we introduce TopNets as a broad framework that subsumes and unifies various methods in the intersection of GNNs/TNNs and PH such as (generalizations of) RePHINE and TOGL. TopNets can also be readily adapted to handle (symmetries in) geometric complexes, extending the scope of TNNs and PH to spatial settings. Theoretically, we show that PH descriptors can provably enhance the expressivity of simplicial message-passing networks. Empirically, (continuous and E(n)-equivariant extensions of) TopNets achieve strong performance across diverse tasks, including antibody design, molecular dynamics simulation, and drug property prediction.

This work introduces TopoBenchmarkX, a modular open-source library designed to standardize benchmarking and accelerate research in Topological Deep Learning (TDL). TopoBenchmarkX maps the TDL pipeline into a sequence of independent and modular components for data loading and processing, as well as model training, optimization, and evaluation. This modular organization provides flexibility for modifications and facilitates the adaptation and optimization of various TDL pipelines. A key feature of TopoBenchmarkX is that it allows for the transformation and lifting between topological domains. This enables, for example, to obtain richer data representations and more fine-grained analyses by mapping the topology and features of a graph to higher-order topological domains such as simplicial and cell complexes. The range of applicability of TopoBenchmarkX is demonstrated by benchmarking several TDL architectures for various tasks and datasets.

We introduce LeapfrogLayers, an invertible neural network architecture that can be trained to efficiently sample the topology of a 2D U(1) lattice gauge theory. We show an improvement in the integrated autocorrelation time of the topological charge when compared with traditional HMC, and look at how different quantities transform under our model. Our implementation is open source, and is publicly available on github at https://github.com/saforem2/l2hmc-qcd.

Topological data analysis (TDA) is an area of data science that focuses on using invariants from algebraic topology to provide multiscale shape descriptors for geometric data sets such as point clouds. One of the most important such descriptors is {\em persistent homology}, which encodes the change in shape as a filtration parameter changes; a typical parameter is the feature scale. For many data sets, it is useful to simultaneously vary multiple filtration parameters, for example feature scale and density. While the theoretical properties of single parameter persistent homology are well understood, less is known about the multiparameter case. In particular, a central question is the problem of representing multiparameter persistent homology by elements of a vector space for integration with standard machine learning algorithms. Existing approaches to this problem either ignore most of the multiparameter information to reduce to the one-parameter case or are heuristic and potentially unstable in the face of noise. In this article, we introduce a new general representation framework that leverages recent results on {\em decompositions} of multiparameter persistent homology. This framework is rich in information, fast to compute, and encompasses previous approaches. Moreover, we establish theoretical stability guarantees under this framework as well as efficient algorithms for practical computation, making this framework an applicable and versatile tool for analyzing geometric and point cloud data. We validate our stability results and algorithms with numerical experiments that demonstrate statistical convergence, prediction accuracy, and fast running times on several real data sets.

Contrastive Vision-Language Models (VLMs) have demonstrated strong zero-shot capabilities. However, their cross-modal alignment remains biased toward English due to limited multilingual multimodal data. Recent multilingual extensions have alleviated this gap but enforce instance-level alignment while neglecting the global geometry of the shared embedding space. We address this problem by introducing ToMCLIP (Topological Alignment for Multilingual CLIP), a topology-aware framework aligning embedding spaces with topology-preserving constraints. The proposed method applies persistent homology to define a topological alignment loss and approximates persistence diagram with theoretical error bounds using graph sparsification strategy. This work validates the proposed approach, showing enhanced structural coherence of multilingual representations, higher zero-shot accuracy on the CIFAR-100, and stronger multilingual retrieval performance on the xFlickr&CO. Beyond VLMs, the proposed approach provides a general method for incorporating topological alignment into representation learning. Code is available at https://github.com/junwon0/ToMCLIP.git.

The natural world is full of complex systems characterized by intricate relations between their components: from social interactions between individuals in a social network to electrostatic interactions between atoms in a protein. Topological Deep Learning (TDL) provides a comprehensive framework to process and extract knowledge from data associated with these systems, such as predicting the social community to which an individual belongs or predicting whether a protein can be a reasonable target for drug development. TDL has demonstrated theoretical and practical advantages that hold the promise of breaking ground in the applied sciences and beyond. However, the rapid growth of the TDL literature has also led to a lack of unification in notation and language across Topological Neural Network (TNN) architectures. This presents a real obstacle for building upon existing works and for deploying TNNs to new real-world problems. To address this issue, we provide an accessible introduction to TDL, and compare the recently published TNNs using a unified mathematical and graphical notation. Through an intuitive and critical review of the emerging field of TDL, we extract valuable insights into current challenges and exciting opportunities for future development.

Machine learning for point clouds has been attracting much attention, with many applications in various fields, such as shape recognition and material science. For enhancing the accuracy of such machine learning methods, it is often effective to incorporate global topological features, which are typically extracted by persistent homology. In the calculation of persistent homology for a point cloud, we choose a filtration for the point cloud, an increasing sequence of spaces. Since the performance of machine learning methods combined with persistent homology is highly affected by the choice of a filtration, we need to tune it depending on data and tasks. In this paper, we propose a framework that learns a filtration adaptively with the use of neural networks. In order to make the resulting persistent homology isometry-invariant, we develop a neural network architecture with such invariance. Additionally, we show a theoretical result on a finite-dimensional approximation of filtration functions, which justifies the proposed network architecture. Experimental results demonstrated the efficacy of our framework in several classification tasks.

We introduce a theoretical framework for differentiable surface evolution that allows discrete topology changes through the use of topological derivatives for variational optimization of image functionals. While prior methods for inverse rendering of geometry rely on silhouette gradients for topology changes, such signals are sparse. In contrast, our theory derives topological derivatives that relate the introduction of vanishing holes and phases to changes in image intensity. As a result, we enable differentiable shape perturbations in the form of hole or phase nucleation. We validate the proposed theory with optimization of closed curves in 2D and surfaces in 3D to lend insights into limitations of current methods and enable improved applications such as image vectorization, vector-graphics generation from text prompts, single-image reconstruction of shape ambigrams and multi-view 3D reconstruction.

Flow-matching methods for 3D shape assembly learn point-wise velocity fields that transport parts toward assembled configurations, yet they receive no explicit guidance about which cross-part interactions should drive the motion. We introduce TORA, a topology-first representation alignment framework that distills relational structure from a frozen pretrained 3D encoder into the flow-matching backbone during training. We first realize this via simple instantiation, token-wise cosine matching, which injects the learned geometric descriptors from the teacher representation. We then extend to employ a Centered Kernel Alignment (CKA) loss to match the similarity structure between student and teacher representations for enhanced topological alignment. Through systematic probing of diverse 3D encoders, we show that geometry- and contact-centric teacher properties, not semantic classification ability, govern alignment effectiveness, and that alignment is most beneficial at later transformer layers where spatial structure naturally emerges. TORA introduces zero inference overhead while yielding two consistent benefits: faster convergence (up to 6.9times) and improved accuracy in-distribution, along with greater robustness under domain shift. Experiments on five benchmarks spanning geometric, semantic, and inter-object assembly demonstrate state-of-the-art performance, with particularly pronounced gains in zero-shot transfer to unseen real-world and synthetic datasets. Project page: https://nahyuklee.github.io/tora.

We introduce the Mixed-Integer Quadratically Constrained Quadratic Programming framework for the quantum compilation problem and apply it in the context of topological quantum computing. In this setting, quantum gates are realized by sequences of elementary braids of quasiparticles with exotic fractional statistics in certain two-dimensional topological condensed matter systems, described by effective topological quantum field theories. We specifically focus on a non-semisimple version of topological field theory, which provides a foundation for an extended theory of Ising anyons and which has recently been shown by Iulianelli et al., Nature Communications {\bf 16}, 6408 (2025), to permit universal quantum computation. While the proofs of this pioneering result are existential in nature, the mixed integer programming provides an approach to explicitly construct quantum gates in topological systems. We demonstrate this by focusing specifically on the entangling controlled-NOT operation, and its local equivalence class, using braiding operations in the non-semisimple Ising system. This illustrates the utility of the Mixed-Integer Quadratically Constrained Quadratic Programming for topological quantum compilation.

3D cellular image segmentation methods are commonly divided into non-2D-based and 2D-based approaches, the latter reconstructing 3D shapes from the segmentation results of 2D layers. However, errors in 2D results often propagate, leading to oversegmentations in the final 3D results. To tackle this issue, we introduce an interpretable geometric framework that addresses the oversegmentations by correcting the 2D segmentation results based on geometric information from adjacent layers. Leveraging both geometric (layer-to-layer, 2D) and topological (3D shape) features, we use binary classification to determine whether neighboring cells should be stitched. We develop a pre-trained classifier on public plant cell datasets and validate its performance on animal cell datasets, confirming its effectiveness in correcting oversegmentations under the transfer learning setting. Furthermore, we demonstrate that our framework can be extended to correcting oversegmentation on non-2D-based methods. A clear pipeline is provided for end-users to build the pre-trained model to any labeled dataset.

In the literature, it has been shown that the evolution of the known explicit 3D surface to the target one can be learned from 2D images using the instantaneous flow field, where the known and target 3D surfaces may largely differ in topology. We are interested in capturing 4D shapes whose topology changes largely over time. We encounter that the straightforward extension of the existing 3D-based method to the desired 4D case performs poorly. In this work, we address the challenges in extending 3D neural evolution to 4D under large topological changes by proposing two novel modifications. More precisely, we introduce (i) a new architecture to discretize and encode the deformation and learn the SDF and (ii) a technique to impose the temporal consistency. (iii) Also, we propose a rendering scheme for color prediction based on Gaussian splatting. Furthermore, to facilitate learning directly from 2D images, we propose a learning framework that can disentangle the geometry and appearance from RGB images. This method of disentanglement, while also useful for the 4D evolution problem that we are concentrating on, is also novel and valid for static scenes. Our extensive experiments on various data provide awesome results and, most importantly, open a new approach toward reconstructing challenging scenes with significant topological changes and deformations. Our source code and the dataset are publicly available at https://github.com/insait-institute/N4DE.

Autonomous robots exploring unknown areas face a significant challenge -- navigating effectively without prior maps and with limited external feedback. This challenge intensifies in sparse reward environments, where traditional exploration techniques often fail. In this paper, we introduce TopoNav, a novel framework that empowers robots to overcome these constraints and achieve efficient, adaptable, and goal-oriented exploration. TopoNav's fundamental building blocks are active topological mapping, intrinsic reward mechanisms, and hierarchical objective prioritization. Throughout its exploration, TopoNav constructs a dynamic topological map that captures key locations and pathways. It utilizes intrinsic rewards to guide the robot towards designated sub-goals within this map, fostering structured exploration even in sparse reward settings. To ensure efficient navigation, TopoNav employs the Hierarchical Objective-Driven Active Topologies framework, enabling the robot to prioritize immediate tasks like obstacle avoidance while maintaining focus on the overall goal. We demonstrate TopoNav's effectiveness in simulated environments that replicate real-world conditions. Our results reveal significant improvements in exploration efficiency, navigational accuracy, and adaptability to unforeseen obstacles, showcasing its potential to revolutionize autonomous exploration in a wide range of applications, including search and rescue, environmental monitoring, and planetary exploration.

Protein representation learning (PRL) is crucial for understanding structure-function relationships, yet current sequence- and graph-based methods fail to capture the hierarchical organization inherent in protein structures. We introduce Topotein, a comprehensive framework that applies topological deep learning to PRL through the novel Protein Combinatorial Complex (PCC) and Topology-Complete Perceptron Network (TCPNet). Our PCC represents proteins at multiple hierarchical levels -- from residues to secondary structures to complete proteins -- while preserving geometric information at each level. TCPNet employs SE(3)-equivariant message passing across these hierarchical structures, enabling more effective capture of multi-scale structural patterns. Through extensive experiments on four PRL tasks, TCPNet consistently outperforms state-of-the-art geometric graph neural networks. Our approach demonstrates particular strength in tasks such as fold classification which require understanding of secondary structure arrangements, validating the importance of hierarchical topological features for protein analysis.

Existing alignment methods share a common topology of information flow, where reward information is collected from humans, modeled with preference learning, and used to tune language models. However, this shared topology has not been systematically characterized, nor have its alternatives been thoroughly explored, leaving the problems of low data efficiency and unreliable generalization unaddressed. As a solution, we introduce a theoretical framework for investigating reward generalization in reinforcement learning from human feedback (RLHF), focusing on the topology of information flow at both macro and micro levels. At the macro level, we portray the RLHF information flow as an autoencoding process over behavior distributions, formalizing the RLHF objective of distributional consistency between human preference and model behavior. At the micro level, we present induced Bayesian networks as a theory of reward generalization in RLHF, introducing fine-grained dataset topologies into generalization bounds. Combining analysis on both levels, we propose reward modeling from tree-structured preference information. It is shown to reduce reward uncertainty by up to Theta(log n/loglog n) times compared to baselines, where n is the dataset size. Validation on three NLP tasks shows that our tree-based reward model achieves an average win rate of 65% against baseline methods, thus improving reward generalization for free via topology design.

Knowledge graph embedding involves learning representations of entities -- the vertices of the graph -- and relations -- the edges of the graph -- such that the resulting representations encode the known factual information represented by the knowledge graph and can be used in the inference of new relations. We show that knowledge graph embedding is naturally expressed in the topological and categorical language of cellular sheaves: a knowledge graph embedding can be described as an approximate global section of an appropriate knowledge sheaf over the graph, with consistency constraints induced by the knowledge graph's schema. This approach provides a generalized framework for reasoning about knowledge graph embedding models and allows for the expression of a wide range of prior constraints on embeddings. Further, the resulting embeddings can be easily adapted for reasoning over composite relations without special training. We implement these ideas to highlight the benefits of the extensions inspired by this new perspective.

Although graph neural networks (GNNs) have achieved impressive achievements in graph classification, they often need abundant task-specific labels, which could be extensively costly to acquire. A credible solution is to explore additional labeled graphs to enhance unsupervised learning on the target domain. However, how to apply GNNs to domain adaptation remains unsolved owing to the insufficient exploration of graph topology and the significant domain discrepancy. In this paper, we propose Coupled Contrastive Graph Representation Learning (CoCo), which extracts the topological information from coupled learning branches and reduces the domain discrepancy with coupled contrastive learning. CoCo contains a graph convolutional network branch and a hierarchical graph kernel network branch, which explore graph topology in implicit and explicit manners. Besides, we incorporate coupled branches into a holistic multi-view contrastive learning framework, which not only incorporates graph representations learned from complementary views for enhanced understanding, but also encourages the similarity between cross-domain example pairs with the same semantics for domain alignment. Extensive experiments on popular datasets show that our CoCo outperforms these competing baselines in different settings generally.

Foundation models for single-cell RNA sequencing (scRNA-seq) have shown promising capabilities in capturing gene expression patterns. However, current approaches face critical limitations: they ignore biological prior knowledge encoded in gene regulatory relationships and fail to leverage multi-omics signals that could provide complementary regulatory insights. In this paper, we propose GRNFormer, a new framework that systematically integrates multi-scale Gene Regulatory Networks (GRNs) inferred from multi-omics data into RNA foundation model training. Our framework introduces two key innovations. First, we introduce a pipeline for constructing hierarchical GRNs that capture regulatory relationships at both cell-type-specific and cell-specific resolutions. Second, we design a structure-aware integration framework that addresses the information asymmetry in GRNs through two technical advances: (1) A graph topological adapter using multi-head cross-attention to weight regulatory relationships dynamically, and (2) a novel edge perturbation strategy that perturb GRNs with biologically-informed co-expression links to augment graph neural network training. Comprehensive experiments have been conducted on three representative downstream tasks across multiple model architectures to demonstrate the effectiveness of GRNFormer. It achieves consistent improvements over state-of-the-art (SoTA) baselines: 3.6% increase in drug response prediction correlation, 9.6% improvement in single-cell drug classification AUC, and 1.1% average gain in gene perturbation prediction accuracy.

The shift from monolithic LLMs to distributed multi-agent architectures demands new frameworks for verifying and securing autonomous coordination. Unlike traditional multi-agent systems focused on cooperative state alignment, modern LLM patterns: multi-agent debate, constitutional oversight, helper-critic loops-rely on adversarial critique for error correction and reasoning refinement. Since LLMs are dynamical systems whose latent states are imperfectly observable from verbalized outputs, securing these networks requires understanding both macroscopic topology and microscopic agent observability. This paper establishes a rigorous graph-theoretic framework for analyzing consensus in signed, directed interaction networks, bridging graph theory and LLM reasoning by formally mapping Transformer cross-entropy log-odds to the signed Laplacian. We characterize agreement stability through structural balance theory, showing how unbalanced critique cycles produce logical frustration and persistent reasoning oscillations, and prove that unobservable latent states from hidden system prompts act as topological Trojan horses that destabilize cooperative consensus. To resolve unobservable deadlocks, we restrict interaction topologies to chordal graphs and apply matrix decomposition with Gram-Schmidt orthogonalization, proving that rank-one spectral edge perturbations deterministically break expertise symmetry by shifting eigenvalues into the stable left-half plane. Core contributions include consensus theorems, polynomial-time Perfect Elimination Ordering verification algorithms, and large-scale empirical validation on clustered ensembles of LLaMA-3, Mistral, and Gemma agents.

Vision-Language Navigation in Continuous Environments (VLN-CE) requires an embodied agent to navigate towards target in continuous environments, following natural language instructions. While current graph-based methods offer an efficient, structured approach by abstracting the environment into a topological map and simplifying the action space to waypoint selection, they lag behind methods based on Large Vision-Language Models (LVLMs) in leveraging large-scale data and advanced training paradigms. In this paper, we try to bridge this gap by introducing ETP-R1, a framework that applies the paradigm of scaling up data and Reinforcement Fine-Tuning (RFT) to a graph-based VLN-CE model. To build a strong foundation, we first construct a high-quality, large-scale pretraining dataset using the Gemini API. This dataset consists of diverse, low-hallucination instructions for topological trajectories, providing rich supervision for our graph-based policy to map language to topological paths. This foundation is further strengthened by unifying data from both R2R and RxR tasks for joint pretraining. Building on this, we introduce a three-stage training paradigm, which culminates in the first application of closed-loop, online RFT to a graph-based VLN-CE model, powered by the Group Relative Policy Optimization (GRPO) algorithm. Extensive experiments demonstrate that our approach is highly effective, establishing new state-of-the-art performance across all major metrics on both the R2R-CE and RxR-CE benchmarks. Our code is available at https://github.com/Cepillar/ETP-R1.

Progress in computer-aided synthesis planning (CASP) is obscured by the lack of standardized evaluation infrastructure and the reliance on metrics that prioritize topological completion over chemical validity. We introduce RetroCast, a unified evaluation suite that standardizes heterogeneous model outputs into a common schema to enable statistically rigorous, apples-to-apples comparison. The framework includes a reproducible benchmarking pipeline with stratified sampling and bootstrapped confidence intervals, accompanied by SynthArena, an interactive platform for qualitative route inspection. We utilize this infrastructure to evaluate leading search-based and sequence-based algorithms on a new suite of standardized benchmarks. Our analysis reveals a divergence between "solvability" (stock-termination rate) and route quality; high solvability scores often mask chemical invalidity or fail to correlate with the reproduction of experimental ground truths. Furthermore, we identify a "complexity cliff" in which search-based methods, despite high solvability rates, exhibit a sharp performance decay in reconstructing long-range synthetic plans compared to sequence-based approaches. We release the full framework, benchmark definitions, and a standardized database of model predictions to support transparent and reproducible development in the field.

In fault-tolerant quantum computing with the surface code, non-Clifford gates are crucial for universal computation. However, implementing these gates using methods like magic state distillation and code switching requires significant resources. In this work, we propose a new protocol that combines magic state preparation and code switching to realize logical non-Clifford operations with the potential for fault tolerance. Our approach begins with a special logical state in the Z_4 surface code. By applying a sequence of transformations, the system goes through different topological codes, including the non-Abelian D_4 quantum double model. This process ultimately produces a magic state in a condensed Z_2 surface code, which enables the implementation of a logical T gate in the standard Z_2 surface code. In our analysis, we employ a framework where the topological codes are represented by their topological orders and all the transformations are considered as topological manipulations such as gauging symmetries and condensing anyons. This perspective is particularly useful for understanding code switching between topological codes.

Vector data is one of the two core data structures in geographic information science (GIS), essential for accurately storing and representing geospatial information. Shapefile, the most widely used vector data format, has become the industry standard supported by all major geographic information systems. However, processing this data typically requires specialized GIS knowledge and skills, creating a barrier for researchers from other fields and impeding interdisciplinary research in spatial data analysis. Moreover, while large language models (LLMs) have made significant advancements in natural language processing and task automation, they still face challenges in handling the complex spatial and topological relationships inherent in GIS vector data. To address these challenges, we propose ShapefileGPT, an innovative framework powered by LLMs, specifically designed to automate Shapefile tasks. ShapefileGPT utilizes a multi-agent architecture, in which the planner agent is responsible for task decomposition and supervision, while the worker agent executes the tasks. We developed a specialized function library for handling Shapefiles and provided comprehensive API documentation, enabling the worker agent to operate Shapefiles efficiently through function calling. For evaluation, we developed a benchmark dataset based on authoritative textbooks, encompassing tasks in categories such as geometric operations and spatial queries. ShapefileGPT achieved a task success rate of 95.24%, outperforming the GPT series models. In comparison to traditional LLMs, ShapefileGPT effectively handles complex vector data analysis tasks, overcoming the limitations of traditional LLMs in spatial analysis. This breakthrough opens new pathways for advancing automation and intelligence in the GIS field, with significant potential in interdisciplinary data analysis and application contexts.

Recent improvements in KG-to-text generation are due to additional auxiliary pre-training tasks designed to give the fine-tune task a boost in performance. These tasks require extensive computational resources while only suggesting marginal improvements. Here, we demonstrate that by fusing graph-aware elements into existing pre-trained language models, we are able to outperform state-of-the-art models and close the gap imposed by additional pre-training tasks. We do so by proposing a mask structure to capture neighborhood information and a novel type encoder that adds a bias to the graph-attention weights depending on the connection type. Experiments on two KG-to-text benchmark datasets show our models are competitive while involving fewer parameters and no additional pre-training tasks. By formulating the problem as a framework, we can interchange the various proposed components and begin interpreting KG-to-text generative models based on the topological and type information found in a graph.

Recent prevailing works on graph machine learning typically follow a similar methodology that involves designing advanced variants of graph neural networks (GNNs) to maintain the superior performance of GNNs on different graphs. In this paper, we aim to streamline the GNN design process and leverage the advantages of Large Language Models (LLMs) to improve the performance of GNNs on downstream tasks. We formulate a new paradigm, coined "LLMs-as-Consultants," which integrates LLMs with GNNs in an interactive manner. A framework named LOGIN (LLM Consulted GNN training) is instantiated, empowering the interactive utilization of LLMs within the GNN training process. First, we attentively craft concise prompts for spotted nodes, carrying comprehensive semantic and topological information, and serving as input to LLMs. Second, we refine GNNs by devising a complementary coping mechanism that utilizes the responses from LLMs, depending on their correctness. We empirically evaluate the effectiveness of LOGIN on node classification tasks across both homophilic and heterophilic graphs. The results illustrate that even basic GNN architectures, when employed within the proposed LLMs-as-Consultants paradigm, can achieve comparable performance to advanced GNNs with intricate designs. Our codes are available at https://github.com/QiaoYRan/LOGIN.

The enduring legacy of Euclidean geometry underpins classical machine learning, which, for decades, has been primarily developed for data lying in Euclidean space. Yet, modern machine learning increasingly encounters richly structured data that is inherently nonEuclidean. This data can exhibit intricate geometric, topological and algebraic structure: from the geometry of the curvature of space-time, to topologically complex interactions between neurons in the brain, to the algebraic transformations describing symmetries of physical systems. Extracting knowledge from such non-Euclidean data necessitates a broader mathematical perspective. Echoing the 19th-century revolutions that gave rise to non-Euclidean geometry, an emerging line of research is redefining modern machine learning with non-Euclidean structures. Its goal: generalizing classical methods to unconventional data types with geometry, topology, and algebra. In this review, we provide an accessible gateway to this fast-growing field and propose a graphical taxonomy that integrates recent advances into an intuitive unified framework. We subsequently extract insights into current challenges and highlight exciting opportunities for future development in this field.

Truck parking on freight corridors faces the major challenge of insufficient parking spaces. This is exacerbated by the Hour-of-Service (HOS) regulations, which often result in unauthorized parking practices, causing safety concerns. It has been shown that providing accurate parking usage prediction can be a cost-effective solution to reduce unsafe parking practices. In light of this, existing studies have developed various methods to predict the usage of a truck parking site and have demonstrated satisfactory accuracy. However, these studies focused on a single parking site, and few approaches have been proposed to predict the usage of multiple truck parking sites considering spatio-temporal dependencies, due to the lack of data. This paper aims to fill this gap and presents the Regional Temporal Graph Convolutional Network (RegT-GCN) to predict parking usage across the entire state to provide more comprehensive truck parking information. The framework leverages the topological structures of truck parking site locations and historical parking data to predict the occupancy rate considering spatio-temporal dependencies across a state. To achieve this, we introduce a Regional Decomposition approach, which effectively captures the geographical characteristics of the truck parking locations and their spatial correlations. Evaluation results demonstrate that the proposed model outperforms other baseline models, showing the effectiveness of our regional decomposition. The code is available at https://github.com/raynbowy23/RegT-GCN.

Despite the impressive advancements of Large Language Models (LLMs) in generating text, they are often limited by the knowledge contained in the input and prone to producing inaccurate or hallucinated content. To tackle these issues, Retrieval-augmented Generation (RAG) is employed as an effective strategy to enhance the available knowledge base and anchor the responses in reality by pulling additional texts from external databases. In real-world applications, texts are often linked through entities within a graph, such as citations in academic papers or comments in social networks. This paper exploits these topological relationships to guide the retrieval process in RAG. Specifically, we explore two kinds of topological connections: proximity-based, focusing on closely connected nodes, and role-based, which looks at nodes sharing similar subgraph structures. Our empirical research confirms their relevance to text relationships, leading us to develop a Topology-aware Retrieval-augmented Generation framework. This framework includes a retrieval module that selects texts based on their topological relationships and an aggregation module that integrates these texts into prompts to stimulate LLMs for text generation. We have curated established text-attributed networks and conducted comprehensive experiments to validate the effectiveness of this framework, demonstrating its potential to enhance RAG with topological awareness.

Training tool-use agents typically relies on outcome-based filtering: Supervised Fine-Tuning (SFT) on successful trajectories and Reinforcement Learning (RL) on pass-rate-selected tasks. However, this paradigm ignores interaction dynamics: successful trajectories may lack error recovery or exhibit redundancy, while pass rates fail to distinguish structurally informative tasks from trivial ones. We propose TopoCurate, an interaction-aware framework that projects multi-trial rollouts from the same task into a unified semantic quotient topology. By merging equivalent action-observation states, this projection transforms scattered linear trajectories into a structured manifold that explicitly captures how tool invocations and environmental responses drive the divergence between effective strategies and failure modes. Leveraging this representation, we introduce a dual-selection mechanism: for SFT, we prioritize trajectories demonstrating reflective recovery, semantic efficiency, and strategic diversity to mitigate covariate shift and mode collapse; for RL, we select tasks with high error branch ratios and strategic heterogeneity, maximizing gradient Signal-to-Noise Ratio to address vanishing signals in sparse-reward settings. Evaluations on BFCLv3 and Tau2 Bench show that TopoCurate achieves consistent gains of 4.2\% (SFT) and 6.9\% (RL) over state-of-the-art baselines. We will release the code and data soon for further investigations.

The 7-point checklist (7PCL) is a widely used diagnostic tool in dermoscopy for identifying malignant melanoma by assigning point values to seven specific attributes. However, the traditional 7PCL is limited to distinguishing between malignant melanoma and melanocytic Nevi, and falls short in scenarios where multiple skin diseases with appearances similar to melanoma coexist. To address this limitation, we propose a novel diagnostic framework that integrates a clinical knowledge-based topological graph (CKTG) with a gradient diagnostic strategy featuring a data-driven weighting system (GD-DDW). The CKTG captures both the internal and external relationships among the 7PCL attributes, while the GD-DDW emulates dermatologists' diagnostic processes, prioritizing visual observation before making predictions. Additionally, we introduce a multimodal feature extraction approach leveraging a dual-attention mechanism to enhance feature extraction through cross-modal interaction and unimodal collaboration. This method incorporates meta-information to uncover interactions between clinical data and image features, ensuring more accurate and robust predictions. Our approach, evaluated on the EDRA dataset, achieved an average AUC of 88.6%, demonstrating superior performance in melanoma detection and feature prediction. This integrated system provides data-driven benchmarks for clinicians, significantly enhancing the precision of melanoma diagnosis.

Diffusion models have made significant contributions to computer vision, sparking a growing interest in the community recently regarding the application of them to graph generation. Existing discrete graph diffusion models exhibit heightened computational complexity and diminished training efficiency. A preferable and natural way is to directly diffuse the graph within the latent space. However, due to the non-Euclidean structure of graphs is not isotropic in the latent space, the existing latent diffusion models effectively make it difficult to capture and preserve the topological information of graphs. To address the above challenges, we propose a novel geometrically latent diffusion framework HypDiff. Specifically, we first establish a geometrically latent space with interpretability measures based on hyperbolic geometry, to define anisotropic latent diffusion processes for graphs. Then, we propose a geometrically latent diffusion process that is constrained by both radial and angular geometric properties, thereby ensuring the preservation of the original topological properties in the generative graphs. Extensive experimental results demonstrate the superior effectiveness of HypDiff for graph generation with various topologies.

We tackle the problem of generating a complete vector map representation from aerial imagery in a single run: producing polygons for all land-cover classes with shared boundaries and without gaps or overlaps. Existing polygonization methods are typically class-specific; extending them to multiple classes via per-class runs commonly leads to topological inconsistencies, such as duplicated edges, gaps, and overlaps. We formalize this new task as All-Class Polygonal Vectorization (ACPV) and release the first public benchmark, Deventer-512, with standardized metrics jointly evaluating semantic fidelity, geometric accuracy, vertex efficiency, per-class topological fidelity and global topological consistency. To realize ACPV, we propose ACPV-Net, a unified framework introducing a novel Semantically Supervised Conditioning (SSC) mechanism coupling semantic perception with geometric primitive generation, along with a topological reconstruction that enforces shared-edge consistency by design. While enforcing such strict topological constraints, ACPV-Net surpasses all class-specific baselines in polygon quality across classes on Deventer-512. It also applies to single-class polygonal vectorization without any architectural modification, achieving the best-reported results on WHU-Building. Data, code, and models will be released at: https://github.com/HeinzJiao/ACPV-Net.

In this paper, we propose HiPoNet, an end-to-end differentiable neural network for regression, classification, and representation learning on high-dimensional point clouds. Our work is motivated by single-cell data which can have very high-dimensionality --exceeding the capabilities of existing methods for point clouds which are mostly tailored for 3D data. Moreover, modern single-cell and spatial experiments now yield entire cohorts of datasets (i.e., one data set for every patient), necessitating models that can process large, high-dimensional point-clouds at scale. Most current approaches build a single nearest-neighbor graph, discarding important geometric and topological information. In contrast, HiPoNet models the point-cloud as a set of higher-order simplicial complexes, with each particular complex being created using a reweighting of features. This method thus generates multiple constructs corresponding to different views of high-dimensional data, which in biology offers the possibility of disentangling distinct cellular processes. It then employs simplicial wavelet transforms to extract multiscale features, capturing both local and global topology from each view. We show that geometric and topological information is preserved in this framework both theoretically and empirically. We showcase the utility of HiPoNet on point-cloud level tasks, involving classification and regression of entire point-clouds in data cohorts. Experimentally, we find that HiPoNet outperforms other point-cloud and graph-based models on single-cell data. We also apply HiPoNet to spatial transcriptomics datasets using spatial coordinates as one of the views. Overall, HiPoNet offers a robust and scalable solution for high-dimensional data analysis.

The caliber and configuration of retinal blood vessels serve as important biomarkers for various diseases and medical conditions. A thorough analysis of the retinal vasculature requires the segmentation of the blood vessels and their classification into arteries and veins, typically performed on color fundus images obtained by retinography. However, manually performing these tasks is labor-intensive and prone to human error. While several automated methods have been proposed to address this task, the current state of art faces challenges due to manifest classification errors affecting the topological consistency of segmentation maps. In this work, we introduce RRWNet, a novel end-to-end deep learning framework that addresses this limitation. The framework consists of a fully convolutional neural network that recursively refines semantic segmentation maps, correcting manifest classification errors and thus improving topological consistency. In particular, RRWNet is composed of two specialized subnetworks: a Base subnetwork that generates base segmentation maps from the input images, and a Recursive Refinement subnetwork that iteratively and recursively improves these maps. Evaluation on three different public datasets demonstrates the state-of-the-art performance of the proposed method, yielding more topologically consistent segmentation maps with fewer manifest classification errors than existing approaches. In addition, the Recursive Refinement module within RRWNet proves effective in post-processing segmentation maps from other methods, further demonstrating its potential. The model code, weights, and predictions will be publicly available at https://github.com/j-morano/rrwnet.

Large Language Models (LLMs) increasingly rely on agentic capabilities-iterative retrieval, tool use, and decision-making-to overcome the limits of static, parametric knowledge. Yet existing agentic frameworks treat external information as unstructured text and fail to leverage the topological dependencies inherent in real-world data. To bridge this gap, we introduce Agentic Graph Learning (AGL), a paradigm that reframes graph learning as an interleaved process of topology-aware navigation and LLM-based inference. Specifically, we propose AgentGL, the first reinforcement learning (RL)-driven framework for AGL. AgentGL equips an LLM agent with graph-native tools for multi-scale exploration, regulates tool usage via search-constrained thinking to balance accuracy and efficiency, and employs a graph-conditioned curriculum RL strategy to stabilize long-horizon policy learning without step-wise supervision. Across diverse Text-Attributed Graph (TAG) benchmarks and multiple LLM backbones, AgentGL substantially outperforms strong GraphLLMs and GraphRAG baselines, achieving absolute improvements of up to 17.5% in node classification and 28.4% in link prediction. These results demonstrate that AGL is a promising frontier for enabling LLMs to autonomously navigate and reason over complex relational environments. The code is publicly available at https://github.com/sunyuanfu/AgentGL.

Verifying the reliability of complex, multi-step reasoning in Large Language Models (LLMs) remains a fundamental challenge, as existing methods often lack both faithfulness and precision. To address this issue, we propose the Graph of Verification (GoV) framework. GoV offers three key contributions: First, it explicitly models the underlying deductive process as a directed acyclic graph (DAG), whether this structure is implicit or explicitly constructed. Second, it enforces a topological order over the DAG to guide stepwise verification. Third, GoV introduces the notion of customizable node blocks, which flexibly define the verification granularity, from atomic propositions to full paragraphs, while ensuring that all requisite premises derived from the graph are provided as contextual input for each verification unit. We evaluate GoV on the Number Triangle Summation task and the ProcessBench benchmark with varying levels of reasoning complexity. Experimental results show that GoV substantially improves verification accuracy, faithfulness, and error localization when compared to conventional end-to-end verification approaches. Our code and data are available at https://github.com/Frevor/Graph-of-Verification.

Accurate and robust simultaneous localization and mapping (SLAM) is crucial for autonomous mobile systems, typically achieved by leveraging the geometric features of the environment. Incorporating semantics provides a richer scene representation that not only enhances localization accuracy in SLAM but also enables advanced cognitive functionalities for downstream navigation and planning tasks. Existing point-wise semantic LiDAR SLAM methods often suffer from poor efficiency and generalization, making them less robust in diverse real-world scenarios. In this paper, we propose a semantic graph-enhanced SLAM framework, named SG-SLAM, which effectively leverages the geometric, semantic, and topological characteristics inherent in environmental structures. The semantic graph serves as a fundamental component that facilitates critical functionalities of SLAM, including robust relocalization during odometry failures, accurate loop closing, and semantic graph map construction. Our method employs a dual-threaded architecture, with one thread dedicated to online odometry and relocalization, while the other handles loop closure, pose graph optimization, and map update. This design enables our method to operate in real time and generate globally consistent semantic graph maps and point cloud maps. We extensively evaluate our method across the KITTI, MulRAN, and Apollo datasets, and the results demonstrate its superiority compared to state-of-the-art methods. Our method has been released at https://github.com/nubot-nudt/SG-SLAM.

Large Language Model-based Multi-Agent Systems (LLM-MAS) are increasingly applied to complex collaborative scenarios. However, their collaborative mechanisms may cause minor inaccuracies to gradually solidify into system-level false consensus through iteration. Such risks are difficult to trace since errors can propagate and amplify through message dependencies. Existing protections often rely on single-agent validation or require modifications to the collaboration architecture, which can weaken effective information flow and may not align with natural collaboration processes in real tasks. To address this, we propose a propagation dynamics model tailored for LLM-MAS that abstracts collaboration as a directed dependency graph and provides an early-stage risk criterion to characterize amplification risk. Through experiments on six mainstream frameworks, we identify three vulnerability classes: cascade amplification, topological sensitivity, and consensus inertia. We further instantiate an attack where injecting just a single atomic error seed leads to widespread failure. In response, we introduce a genealogy-graph-based governance layer, implemented as a message-layer plugin, that suppresses both endogenous and exogenous error amplification without altering the collaboration architecture. Experiments show that this approach raises the defense success rate from a baseline of 0.32 to over 0.89 and significantly mitigates the cascading spread of minor errors.

Circuit link prediction, which identifies missing component connections from incomplete netlists, is crucial in analog circuit design automation. However, existing methods face three main challenges: 1) Insufficient use of topological patterns in circuit graphs reduces prediction accuracy; 2) Data scarcity due to the complexity of annotations hinders model generalization; 3) Limited adaptability to various netlist formats restricts model flexibility. We propose Graph Neural Networks Based Analog Circuit Link Prediction (GNN-ACLP), a graph neural networks (GNNs) based method featuring three innovations to tackle these challenges. First, we introduce the SEAL (learning from Subgraphs, Embeddings, and Attributes for Link prediction) framework and achieve port-level accuracy in circuit link prediction. Second, we propose Netlist Babel Fish, a netlist format conversion tool that leverages retrieval-augmented generation (RAG) with a large language model (LLM) to enhance the compatibility of netlist formats. Finally, we build a comprehensive dataset, SpiceNetlist, comprising 775 annotated circuits of 7 different types across 10 component classes. Experiments demonstrate accuracy improvements of 16.08% on SpiceNetlist, 11.38% on Image2Net, and 16.01% on Masala-CHAI compared to the baseline in intra-dataset evaluation, while maintaining accuracy from 92.05% to 99.07% in cross-dataset evaluation, demonstrating robust feature transfer capabilities. However, its linear computational complexity makes processing large-scale netlists challenging and requires future addressing.

Hierarchical graph pooling(HGP) are designed to consider the fact that conventional graph neural networks(GNN) are inherently flat and are also not multiscale. However, most HGP methods suffer not only from lack of considering global topology of the graph and focusing on the feature learning aspect, but also they do not align local and global features since graphs should inherently be analyzed in a multiscale way. LGRPool is proposed in the present paper as a HGP in the framework of expectation maximization in machine learning that aligns local and global aspects of message passing with each other using a regularizer to force the global topological information to be inline with the local message passing at different scales through the representations at different layers of HGP. Experimental results on some graph classification benchmarks show that it slightly outperforms some baselines.

Living systems face both environmental complexity and limited access to free-energy resources. Survival under these conditions requires a control system that can activate, or deploy, available perception and action resources in a context specific way. We show here that when systems are described as executing active inference driven by the free-energy principle (and hence can be considered Bayesian prediction-error minimizers), their control flow systems can always be represented as tensor networks (TNs). We show how TNs as control systems can be implmented within the general framework of quantum topological neural networks, and discuss the implications of these results for modeling biological systems at multiple scales.

Global perception is essential for embodied agents in 360° spaces, yet current affordance grounding remains largely object-centric and restricted to perspective views. To bridge this gap, we introduce a novel task: Holistic Affordance Grounding in 360° Indoor Environments. This task faces unique challenges, including severe geometric distortions from Equirectangular Projection (ERP), semantic dispersion, and cross-scale alignment difficulties. We propose PanoAffordanceNet, an end-to-end framework featuring a Distortion-Aware Spectral Modulator (DASM) for latitude-dependent calibration and an Omni-Spherical Densification Head (OSDH) to restore topological continuity from sparse activations. By integrating multi-level constraints comprising pixel-wise, distributional, and region-text contrastive objectives, our framework effectively suppresses semantic drift under low supervision. Furthermore, we construct 360-AGD, the first high-quality panoramic affordance grounding dataset. Extensive experiments demonstrate that PanoAffordanceNet significantly outperforms existing methods, establishing a solid baseline for scene-level perception in embodied intelligence. The source code and benchmark dataset will be made publicly available at https://github.com/GL-ZHU925/PanoAffordanceNet.

Graph representation learning has emerged as a powerful technique for addressing real-world problems. Various downstream graph learning tasks have benefited from its recent developments, such as node classification, similarity search, and graph classification. However, prior arts on graph representation learning focus on domain specific problems and train a dedicated model for each graph dataset, which is usually non-transferable to out-of-domain data. Inspired by the recent advances in pre-training from natural language processing and computer vision, we design Graph Contrastive Coding (GCC) -- a self-supervised graph neural network pre-training framework -- to capture the universal network topological properties across multiple networks. We design GCC's pre-training task as subgraph instance discrimination in and across networks and leverage contrastive learning to empower graph neural networks to learn the intrinsic and transferable structural representations. We conduct extensive experiments on three graph learning tasks and ten graph datasets. The results show that GCC pre-trained on a collection of diverse datasets can achieve competitive or better performance to its task-specific and trained-from-scratch counterparts. This suggests that the pre-training and fine-tuning paradigm presents great potential for graph representation learning.

We introduce a new generative model that combines latent diffusion with persistent homology to create 3D shapes with high diversity, with a special emphasis on their topological characteristics. Our method involves representing 3D shapes as implicit fields, then employing persistent homology to extract topological features, including Betti numbers and persistence diagrams. The shape generation process consists of two steps. Initially, we employ a transformer-based autoencoding module to embed the implicit representation of each 3D shape into a set of latent vectors. Subsequently, we navigate through the learned latent space via a diffusion model. By strategically incorporating topological features into the diffusion process, our generative module is able to produce a richer variety of 3D shapes with different topological structures. Furthermore, our framework is flexible, supporting generation tasks constrained by a variety of inputs, including sparse and partial point clouds, as well as sketches. By modifying the persistence diagrams, we can alter the topology of the shapes generated from these input modalities.

Artificial intelligence in construction increasingly depends on structured representations such as Building Information Models and knowledge graphs, yet early-stage building designs are predominantly created as flexible boundary-representation (B-rep) models that lack explicit spatial, semantic, and performance structure. This paper presents a robust, fully automated framework that transforms unstructured B-rep geometry into knowledge-graph-based Building Information Models and further into executable Building Energy Models. The framework enables artificial intelligence to explicitly interpret building elements, spatial topology, and their associated thermal and performance attributes. It integrates automated geometry cleansing, multiple auto space-generation strategies, graph-based extraction of space and element topology, ontology-aligned knowledge modeling, and reversible transformation between ontology-based BIM and EnergyPlus energy models. Validation on parametric, sketch-based, and real-world building datasets demonstrates high robustness, consistent topological reconstruction, and reliable performance-model generation. By bridging design models, BIM, and BEM, the framework provides an AI-oriented infrastructure that extends BIM- and graph-based intelligence pipelines to flexible early-stage design geometry, enabling performance-driven design exploration and optimization by learning-based methods.

Graph foundation models represent a transformative paradigm for learning transferable representations across diverse graph domains. Recent methods leverage large language models to unify graph and text modalities into a shared representation space using contrastive learning. However, systematic evaluations reveal significant performance degradation at structural boundaries where distinct topological patterns converge, with accuracy losses exceeding 20 percentage points. This issue arises from a key limitation: current methods assume all graph structures can be encoded within a single Euclidean space. In reality, tree structures require hyperbolic geometry to preserve hierarchical branching, while cyclic patterns depend on spherical geometry for closure properties. At structural boundaries, nodes experience conflicting geometric constraints that uniform encoding spaces cannot resolve. This raises a crucial challenge: Can alignment frameworks be designed to respect the intrinsic geometric diversity of graph structures? We introduce GraphShaper, a geometry-aware framework that enhances graph encoding through multi-geometric specialization. Our approach employs expert networks tailored to different geometric spaces, dynamically computing fusion weights to adaptively integrate geometric properties based on local structural characteristics. This adaptive fusion preserves structural integrity before alignment with text embeddings. Extensive experiments demonstrate that GraphShaper achieves 9.47\% accuracy improvements on citation networks and 7.63\% on social networks in zero-shot settings.

In this work, we present two results: The first result is the formalization of Tutte's theorem in Lean, a key theorem concerning matchings in graph theory. As this formalization is ready to be integrated in Lean's mathlib, it provides a valuable step in the path towards formalizing research-level mathematics in this area. The second result is a framework for doing educational formalization projects. This framework provides a structure to learn to formalize mathematics with minimal teacher input. This framework applies to both traditional academic settings and independent community-driven environments. We demonstrate the framework's use by connecting it to the process of formalizing Tutte's theorem.

Complex prediction models such as deep learning are the output from fitting machine learning, neural networks, or AI models to a set of training data. These are now standard tools in science. A key challenge with the current generation of models is that they are highly parameterized, which makes describing and interpreting the prediction strategies difficult. We use topological data analysis to transform these complex prediction models into pictures representing a topological view. The result is a map of the predictions that enables inspection. The methods scale up to large datasets across different domains and enable us to detect labeling errors in training data, understand generalization in image classification, and inspect predictions of likely pathogenic mutations in the BRCA1 gene.

Adversarially perturbed images of text can cause sophisticated OCR systems to produce misleading or incorrect transcriptions from seemingly invisible changes to humans. Some of these perturbations even survive physical capture, posing security risks to high-stakes applications such as document processing, license plate recognition, and automated compliance systems. Existing defenses, such as adversarial training, input preprocessing, or post-recognition correction, are often model-specific, computationally expensive, and affect performance on unperturbed inputs while remaining vulnerable to unseen or adaptive attacks. To address these challenges, TopoReformer is introduced, a model-agnostic reformation pipeline that mitigates adversarial perturbations while preserving the structural integrity of text images. Topology studies properties of shapes and spaces that remain unchanged under continuous deformations, focusing on global structures such as connectivity, holes, and loops rather than exact distance. Leveraging these topological features, TopoReformer employs a topological autoencoder to enforce manifold-level consistency in latent space and improve robustness without explicit gradient regularization. The proposed method is benchmarked on EMNIST, MNIST, against standard adversarial attacks (FGSM, PGD, Carlini-Wagner), adaptive attacks (EOT, BDPA), and an OCR-specific watermark attack (FAWA).

Persistent homology, a technique from computational topology, has recently shown strong empirical performance in the context of graph classification. Being able to capture long range graph properties via higher-order topological features, such as cycles of arbitrary length, in combination with multi-scale topological descriptors, has improved predictive performance for data sets with prominent topological structures, such as molecules. At the same time, the theoretical properties of persistent homology have not been formally assessed in this context. This paper intends to bridge the gap between computational topology and graph machine learning by providing a brief introduction to persistent homology in the context of graphs, as well as a theoretical discussion and empirical analysis of its expressivity for graph learning tasks.

With the development of large language models (LLMs), particularly with the introduction of the long reasoning chain technique, the reasoning ability of LLMs in complex problem-solving has been significantly enhanced. While acknowledging the power of long reasoning chains, we cannot help but wonder: Why do different reasoning chains perform differently in reasoning? What components of the reasoning chains play a key role? Existing studies mainly focus on evaluating reasoning chains from a functional perspective, with little attention paid to their structural mechanisms. To address this gap, this work is the first to analyze and evaluate the quality of the reasoning chain from a structural perspective. We apply persistent homology from Topological Data Analysis (TDA) to map reasoning steps into semantic space, extract topological features, and analyze structural changes. These changes reveal semantic coherence, logical redundancy, and identify logical breaks and gaps. By calculating homology groups, we assess connectivity and redundancy at various scales, using barcode and persistence diagrams to quantify stability and consistency. Our results show that the topological structural complexity of reasoning chains correlates positively with accuracy. More complex chains identify correct answers sooner, while successful reasoning exhibits simpler topologies, reducing redundancy and cycles, enhancing efficiency and interpretability. This work provides a new perspective on reasoning chain quality assessment and offers guidance for future optimization.

Data quality is crucial for the successful training, generalization and performance of artificial intelligence models. Furthermore, it is known that the leading approaches in artificial intelligence are notoriously data-hungry. In this paper, we propose the use of small training datasets towards faster training. Specifically, we provide a novel topological method based on morphisms between persistence modules to measure the training data quality with respect to the complete dataset. This way, we can provide an explanation of why the chosen training dataset will lead to poor performance.

In a novel application of the tools of topological data analysis (TDA) to nonperturbative quantum gravity, we introduce a new class of observables that allows us to assess whether quantum spacetime really resembles a ``quantum foam" near the Planck scale. The key idea is to investigate the Betti numbers of coarse-grained path integral histories, regularized in terms of dynamical triangulations, as a function of the coarse-graining scale. In two dimensions our analysis exhibits the well-known fractal structure of Euclidean quantum gravity.

We consider three monads on Top, the category of topological spaces, which formalize topological aspects of probability and possibility in categorical terms. The first one is the Hoare hyperspace monad H, which assigns to every space its space of closed subsets equipped with the lower Vietoris topology. The second is the monad V of continuous valuations, also known as the extended probabilistic powerdomain. We construct both monads in a unified way in terms of double dualization. This reveals a close analogy between them, and allows us to prove that the operation of taking the support of a continuous valuation is a morphism of monads from V to H. In particular, this implies that every H-algebra (topological complete semilattice) is also a V-algebra. Third, we show that V can be restricted to a submonad of tau-smooth probability measures on Top. By composing these two morphisms of monads, we obtain that taking the support of a tau-smooth probability measure is also a morphism of monads.

Generation of graphs is a major challenge for real-world tasks that require understanding the complex nature of their non-Euclidean structures. Although diffusion models have achieved notable success in graph generation recently, they are ill-suited for modeling the topological properties of graphs since learning to denoise the noisy samples does not explicitly learn the graph structures to be generated. To tackle this limitation, we propose a generative framework that models the topology of graphs by explicitly learning the final graph structures of the diffusion process. Specifically, we design the generative process as a mixture of endpoint-conditioned diffusion processes which is driven toward the predicted graph that results in rapid convergence. We further introduce a simple parameterization of the mixture process and develop an objective for learning the final graph structure, which enables maximum likelihood training. Through extensive experimental validation on general graph and 2D/3D molecule generation tasks, we show that our method outperforms previous generative models, generating graphs with correct topology with both continuous (e.g. 3D coordinates) and discrete (e.g. atom types) features. Our code is available at https://github.com/harryjo97/GruM.

We propose a novel approach for preserving topological structures of the input space in latent representations of autoencoders. Using persistent homology, a technique from topological data analysis, we calculate topological signatures of both the input and latent space to derive a topological loss term. Under weak theoretical assumptions, we construct this loss in a differentiable manner, such that the encoding learns to retain multi-scale connectivity information. We show that our approach is theoretically well-founded and that it exhibits favourable latent representations on a synthetic manifold as well as on real-world image data sets, while preserving low reconstruction errors.

In this paper, we introduce fundamental notions of homotopy theory, including homotopy excision and the Freudenthal suspension theorem. We then explore framed cobordism and its connection to stable homotopy groups of spheres through the Pontryagin-Thom construction. Using this framework, we compute the stable stems in dimensions 0, 1, and 2. This work is primarily expository, revisiting proofs from Pont1 with slight modifications incorporating modern notation. Furthermore, in the final section, we discuss 2-dimensional framed manifolds with Arf invariant one and examine why the result of Pont2 regarding π_2^S is incorrect.

Structural topology optimization (TO) is central to engineering design but remains computationally intensive due to complex physics and hard constraints. Existing deep-learning methods are limited to fixed square grids, a few hand-coded boundary conditions, and post-hoc optimization, preventing general deployment. We introduce Optimize Any Topology (OAT), a foundation-model framework that directly predicts minimum-compliance layouts for arbitrary aspect ratios, resolutions, volume fractions, loads, and fixtures. OAT combines a resolution- and shape-agnostic autoencoder with an implicit neural-field decoder and a conditional latent-diffusion model trained on OpenTO, a new corpus of 2.2 million optimized structures covering 2 million unique boundary-condition configurations. On four public benchmarks and two challenging unseen tests, OAT lowers mean compliance up to 90% relative to the best prior models and delivers sub-1 second inference on a single GPU across resolutions from 64 x 64 to 256 x 256 and aspect ratios as high as 10:1. These results establish OAT as a general, fast, and resolution-free framework for physics-aware topology optimization and provide a large-scale dataset to spur further research in generative modeling for inverse design. Code & data can be found at https://github.com/ahnobari/OptimizeAnyTopology.

There is a growing body of work that leverages features extracted via topological data analysis to train machine learning models. While this field, sometimes known as topological machine learning (TML), has seen some notable successes, an understanding of how the process of learning from topological features differs from the process of learning from raw data is still limited. In this work, we begin to address one component of this larger issue by asking whether a model trained with topological features learns internal representations of data that are fundamentally different than those learned by a model trained with the original raw data. To quantify ``different'', we exploit two popular metrics that can be used to measure the similarity of the hidden representations of data within neural networks, neural stitching and centered kernel alignment. From these we draw a range of conclusions about how training with topological features does and does not change the representations that a model learns. Perhaps unsurprisingly, we find that structurally, the hidden representations of models trained and evaluated on topological features differ substantially compared to those trained and evaluated on the corresponding raw data. On the other hand, our experiments show that in some cases, these representations can be reconciled (at least to the degree required to solve the corresponding task) using a simple affine transformation. We conjecture that this means that neural networks trained on raw data may extract some limited topological features in the process of making predictions.

We give a conceptual treatment of the notion of joints, marginals, and independence in the setting of categorical probability. This is achieved by endowing the usual probability monads (like the Giry monad) with a monoidal and an opmonoidal structure, mutually compatible (i.e. a bimonoidal structure). If the underlying monoidal category is cartesian monoidal, a bimonoidal structure is given uniquely by a commutative strength. However, if the underlying monoidal category is not cartesian monoidal, a strength is not enough to guarantee all the desired properties of joints and marginals. A bimonoidal structure is then the correct requirement for the more general case. We explain the theory and the operational interpretation, with the help of the graphical calculus for monoidal categories. We give a definition of stochastic independence based on the bimonoidal structure, compatible with the intuition and with other approaches in the literature for cartesian monoidal categories. We then show as an example that the Kantorovich monad on the category of complete metric spaces is a bimonoidal monad for a non-cartesian monoidal structure.

Topological materials present unconventional electronic properties that make them attractive for both basic science and next-generation technological applications. The majority of currently known topological materials have been discovered using methods that involve symmetry-based analysis of the quantum wavefunction. Here we use machine learning to develop a simple-to-use heuristic chemical rule that diagnoses with a high accuracy whether a material is topological using only its chemical formula. This heuristic rule is based on a notion that we term topogivity, a machine-learned numerical value for each element that loosely captures its tendency to form topological materials. We next implement a high-throughput procedure for discovering topological materials based on the heuristic topogivity-rule prediction followed by ab initio validation. This way, we discover new topological materials that are not diagnosable using symmetry indicators, including several that may be promising for experimental observation.

Image-based 3D reconstruction has increasingly stunning results over the past few years with the latest improvements in computer vision and graphics. Geometry and topology are two fundamental concepts when dealing with 3D mesh structures. But the latest often remains a side issue in the 3D mesh-based reconstruction literature. Indeed, performing per-vertex elementary displacements over a 3D sphere mesh only impacts its geometry and leaves the topological structure unchanged and fixed. Whereas few attempts propose to update the geometry and the topology, all need to lean on costly 3D ground-truth to determine the faces/edges to prune. We present in this work a method that aims to refine the topology of any 3D mesh through a face-pruning strategy that extensively relies upon 2D alpha masks and camera pose information. Our solution leverages a differentiable renderer that renders each face as a 2D soft map. Its pixel intensity reflects the probability of being covered during the rendering process by such a face. Based on the 2D soft-masks available, our method is thus able to quickly highlight all the incorrectly rendered faces for a given viewpoint. Because our module is agnostic to the network that produces the 3D mesh, it can be easily plugged into any self-supervised image-based (either synthetic or natural) 3D reconstruction pipeline to get complex meshes with a non-spherical topology.

Modern representation learning increasingly relies on unsupervised and self-supervised methods trained on large-scale unlabeled data. While these approaches achieve impressive generalization across tasks and domains, evaluating embedding quality without labels remains an open challenge. In this work, we propose Persistence, a topology-aware metric based on persistent homology that quantifies the geometric structure and topological richness of embedding spaces in a fully unsupervised manner. Unlike metrics that assume linear separability or rely on covariance structure, Persistence captures global and multi-scale organization. Empirical results across diverse domains show that Persistence consistently achieves top-tier correlations with downstream performance, outperforming existing unsupervised metrics and enabling reliable model and hyperparameter selection.

A suitable feature representation that can both preserve the data intrinsic information and reduce data complexity and dimensionality is key to the performance of machine learning models. Deeply rooted in algebraic topology, persistent homology (PH) provides a delicate balance between data simplification and intrinsic structure characterization, and has been applied to various areas successfully. However, the combination of PH and machine learning has been hindered greatly by three challenges, namely topological representation of data, PH-based distance measurements or metrics, and PH-based feature representation. With the development of topological data analysis, progresses have been made on all these three problems, but widely scattered in different literatures. In this paper, we provide a systematical review of PH and PH-based supervised and unsupervised models from a computational perspective. Our emphasizes are the recent development of mathematical models and tools, including PH softwares and PH-based functions, feature representations, kernels, and similarity models. Essentially, this paper can work as a roadmap for the practical application of PH-based machine learning tools. Further, we consider different topological feature representations in different machine learning models, and investigate their impacts on the protein secondary structure classification.

Topological magnons give rise to possibilities for engineering novel spintronics devices with critical applications in quantum information and computation, due to its symmetry-protected robustness and low dissipation. However, to make reliable and systematic predictions about material realization of topological magnons has been a major challenge, due to the lack of neutron scattering data for most materials. In this work, we significantly advance the symmetry-based approach for identifying topological magnons through developing a fully automated algorithm, utilizing the theory of symmetry indicators, that enables a highly efficient and large-scale search for candidate materials hosting field-induced topological magnons. This progress not only streamlines the discovery process but also expands the scope of materials exploration beyond previous manual or traditional methods, offering a powerful tool for uncovering novel topological phases in magnetic systems. Performing a large-scale search over all 1649 magnetic materials in Bilbao Crystallographic Server with a commensurate magnetic order, we discover 387 candidate materials for topological magnons, significantly expanding the pool of topological magnon materials. We further discuss examples and experimental accessibility of the candidate materials, shedding light on future experimental realizations of topological magnons in magnetic materials.

Topological materials provide a platform that utilizes the geometric characteristics of structured materials to control the flow of waves, enabling unidirectional and protected transmission that is immune to defects or impurities. The topologically designed photonic materials can carry quantum states and electromagnetic energy, benefiting nanolasers or quantum photonic systems. This article reviews recent advances in the topological applications of photonic materials for radiative heat transfer, especially in the near field. When the separation distance between media is considerably smaller than the thermal wavelength, the heat transfer exhibits super-Planckian behavior that surpasses Planck's blackbody predictions. Near-field thermal radiation in subwavelength systems supporting surface modes has various applications, including nanoscale thermal management and energy conversion. Photonic materials and structures that support topological surface states show immense potential for enhancing or suppressing near-field thermal radiation. We present various topological effects, such as periodic and quasi-periodic nanoparticle arrays, Dirac and Weyl semimetal-based materials, structures with broken global symmetries, and other topological insulators, on near-field heat transfer. Also, the possibility of realizing near-field thermal radiation in such topological materials for alternative thermal management and heat flux guiding in nano-scale systems is discussed based on the existing technology.

The structure of topology underpins much of the research on performance and robustness, yet available topology data are typically scarce, necessitating the generation of synthetic graphs with desired properties for testing or release. Prior diffusion-based approaches either embed conditions into the diffusion model, requiring retraining for each attribute and hindering real-time applicability, or use classifier-based guidance post-training, which does not account for topology scale and practical constraints. In this paper, we show from a discrete perspective that gradients from a pre-trained graph-level classifier can be incorporated into the discrete reverse diffusion posterior to steer generation toward specified structural properties. Based on this insight, we propose Classifier-guided Conditional Topology Generation with Persistent Homology (CoPHo), which builds a persistent homology filtration over intermediate graphs and interprets features as guidance signals that steer generation toward the desired properties at each denoising step. Experiments on four generic/network datasets demonstrate that CoPHo outperforms existing methods at matching target metrics, and we further validate its transferability on the QM9 molecular dataset.

The inference of topological principles is a key problem in structured reconstruction. We observe that wrongly predicted topological relationships are often incurred by the lack of holistic geometry clues in low-level features. Inspired by the fact that massive signals can be compactly described with frequency analysis, we experimentally explore the efficiency and tendency of learning structure geometry in the frequency domain. Accordingly, we propose a frequency-domain feature learning strategy (F-Learn) to fuse scattered geometric fragments holistically for topology-intact structure reasoning. Benefiting from the parsimonious design, the F-Learn strategy can be easily deployed into a deep reconstructor with a lightweight model modification. Experiments demonstrate that the F-Learn strategy can effectively introduce structure awareness into geometric primitive detection and topology inference, bringing significant performance improvement to final structured reconstruction. Code and pre-trained models are available at https://github.com/Geo-Tell/F-Learn.

The discoveries of intrinsically magnetic topological materials, including semimetals with a large anomalous Hall effect and axion insulators, have directed fundamental research in solid-state materials. Topological quantum chemistry has enabled the understanding of and the search for paramagnetic topological materials. Using magnetic topological indices obtained from magnetic topological quantum chemistry (MTQC), here we perform a high-throughput search for magnetic topological materials based on first-principles calculations. We use as our starting point the Magnetic Materials Database on the Bilbao Crystallographic Server, which contains more than 549 magnetic compounds with magnetic structures deduced from neutron-scattering experiments, and identify 130 enforced semimetals (for which the band crossings are implied by symmetry eigenvalues), and topological insulators. For each compound, we perform complete electronic structure calculations, which include complete topological phase diagrams using different values of the Hubbard potential. Using a custom code to find the magnetic co-representations of all bands in all magnetic space groups, we generate data to be fed into the algorithm of MTQC to determine the topology of each magnetic material. Several of these materials display previously unknown topological phases, including symmetry-indicated magnetic semimetals, three-dimensional anomalous Hall insulators and higher-order magnetic semimetals. We analyse topological trends in the materials under varying interactions: 60 per cent of the 130 topological materials have topologies sensitive to interactions, and the others have stable topologies under varying interactions. We provide a materials database for future experimental studies and open-source code for diagnosing topologies of magnetic materials.

Using a set of LambdaCDM simulations of cosmic structure formation, we study the evolving connectivity and changing topological structure of the cosmic web using state-of-the-art tools of multiscale topological data analysis (TDA). We follow the development of the cosmic web topology in terms of the evolution of Betti number curves and feature persistence diagrams of the three (topological) classes of structural features: matter concentrations, filaments and tunnels, and voids. The Betti curves specify the prominence of features as a function of density level, and their evolution with cosmic epoch reflects the changing network connections between these structural features. The persistence diagrams quantify the longevity and stability of topological features. In this study we establish, for the first time, the link between persistence diagrams, the features they show, and the gravitationally driven cosmic structure formation process. By following the diagrams' development over cosmic time, the link between the multiscale topology of the cosmic web and the hierarchical buildup of cosmic structure is established. The sharp apexes in the diagrams are intimately related to key transitions in the structure formation process. The apex in the matter concentration diagrams coincides with the density level at which, typically, they detach from the Hubble expansion and begin to collapse. At that level many individual islands merge to form the network of the cosmic web and a large number of filaments and tunnels emerge to establish its connecting bridges. The location trends of the apex possess a self-similar character that can be related to the cosmic web's hierarchical buildup. We find that persistence diagrams provide a significantly higher and more profound level of information on the structure formation process than more global summary statistics like Euler characteristic or Betti numbers.

Transaction data from digital payment systems can be used to study economic processes at such a detail that was not possible previously. Here, we analyse the data from Sarafu token network, a community inclusion currency in Kenya. During the COVID-19 emergency, the Sarafu was disbursed as part of a humanitarian aid project. In this work, the transactions are analysed using network science. A topological categorisation is defined to identify cyclic and acyclic components. Furthermore, temporal aspects of circulation taking place within these components are considered. The significant presence of different types of strongly connected components as compared to randomized null models shows the importance of cycles in this economic network. Especially, indicating their key role in currency recirculation. In some acyclic components, the most significant triad suggests the presence of a group of users collecting currency from accounts active only once, hinting at a misuse of the system. In some other acyclic components, small isolated groups of users were active only once, suggesting the presence of users only interested in trying out the system. The methods used in this paper can answer specific questions related to user activities, currency design, and assessment of monetary interventions. Our methodology provides a general quantitative tool for analysing the behaviour of users in a currency network.

Topological flat bands (FBs) offer an ideal platform for realizing exotic topological phases, such as fractional Chern insulators, yet their realization with both exact flatness and stable topology in local lattice models has been long hindered by fundamental no-go theorems. Here, we overcome this barrier by demonstrating the existence of critical topological FBs (CTFBs) in finite-range hopping models. They saturate the no-go theorems via a unique structure of Bloch wavefunctions: While continuous over the whole Brillouin zone, the wavefunctions are non-analytic at isolated band touching points, thereby relaxing the inherent restrictions on the coexistence of exact flatness and stable topology. We establish a general principle to construct CTFBs, as well as their parent Hamiltonians, that carry desired topological invariants in given space groups. Explicit examples exhibiting Chern numbers, strong Z_2 index, and crystalline-symmetry-protected invariants in two and three dimensions are provided. Furthermore, an automated algorithm identifies more than 50,000 robust, symmetry-indicated CTFBs. Filling such CTFBs yields short-range entangled topological states that exhibit power-law correlations. Crucially, all filled CTFB states admit exact tensor-network representations with finite bond dimensions, providing a tractable starting point for exploring strongly correlated topological matter.

Persistent homology (PH) provides topological descriptors for geometric data, such as weighted graphs, which are interpretable, stable to perturbations, and invariant under, e.g., relabeling. Most applications of PH focus on the one-parameter case -- where the descriptors summarize the changes in topology of data as it is filtered by a single quantity of interest -- and there is now a wide array of methods enabling the use of one-parameter PH descriptors in data science, which rely on the stable vectorization of these descriptors as elements of a Hilbert space. Although the multiparameter PH (MPH) of data that is filtered by several quantities of interest encodes much richer information than its one-parameter counterpart, the scarceness of stability results for MPH descriptors has so far limited the available options for the stable vectorization of MPH. In this paper, we aim to bring together the best of both worlds by showing how the interpretation of signed barcodes -- a recent family of MPH descriptors -- as signed measures leads to natural extensions of vectorization strategies from one parameter to multiple parameters. The resulting feature vectors are easy to define and to compute, and provably stable. While, as a proof of concept, we focus on simple choices of signed barcodes and vectorizations, we already see notable performance improvements when comparing our feature vectors to state-of-the-art topology-based methods on various types of data.

Knot diagrams are among the most common visual tools in topology. Computer programs now make it possible to draw, manipulate and render them digitally, which proves to be useful in knot theory teaching and research. Still, an openly available tool to manipulate knot diagrams in a real-time, interactive way is yet to be developed. We introduce a method of operating on the geometry of the knot diagram itself without any underlying three-dimensional structure that can underpin such an application. This allows us to directly interact with vector graphics knot diagrams while at the same time computing knot invariants in ways proposed by previous work. An implementation of this method is provided.

We present EuLearn, the first surface datasets equitably representing a diversity of topological types. We designed our embedded surfaces of uniformly varying genera relying on random knots, thus allowing our surfaces to knot with themselves. EuLearn contributes new topological datasets of meshes, point clouds, and scalar fields in 3D. We aim to facilitate the training of machine learning systems that can discern topological features. We experimented with specific emblematic 3D neural network architectures, finding that their vanilla implementations perform poorly on genus classification. To enhance performance, we developed a novel, non-Euclidean, statistical sampling method adapted to graph and manifold data. We also introduce adjacency-informed adaptations of PointNet and Transformer architectures that rely on our non-Euclidean sampling strategy. Our results demonstrate that incorporating topological information into deep learning workflows significantly improves performance on these otherwise challenging EuLearn datasets.

We introduce a Heegaard-Floer homology functor from the category of oriented links in closed 3-manifolds and oriented surface cobordisms in 4-manifolds connecting them to the category of F[v]-modules and F[v]-homomorphisms between them, where F is the field with two elements. In comparison with previously defined TQFTs for decorated links and link cobordisms, the construction of this paper has the advantage of being independent from the decoration. Some of the basic properties of this functor are also explored.

We introduce the category of information structures, whose objects are suitable diagrams of measurable sets that encode the possible outputs of a given family of observables and their mutual relationships of refinement; they serve as mathematical models of contextuality in classical and quantum settings. Each information structure can be regarded as a ringed site with trivial topology; the structure ring is generated by the observables themselves and its multiplication corresponds to joint measurement. We extend Baudot and Bennequin's definition of information cohomology to this setting, as a derived functor in the category of modules over the structure ring, and show explicitly that the bar construction gives a projective resolution in that category, recovering in this way the cochain complexes previously considered in the literature. Finally, we study the particular case of a one-parameter family of coefficients made of functions of probability distributions. The only 1-cocycles are Shannon entropy or Tsallis alpha-entropy, depending on the value of the parameter.

Scientific inquiry requires systems-level reasoning that integrates heterogeneous experimental data, cross-domain knowledge, and mechanistic evidence into coherent explanations. While Large Language Models (LLMs) offer inferential capabilities, they often depend on retrieval-augmented contexts that lack structural depth. Traditional Knowledge Graphs (KGs) attempt to bridge this gap, yet their pairwise constraints fail to capture the irreducible higher-order interactions that govern emergent physical behavior. To address this, we introduce a methodology for constructing hypergraph-based knowledge representations that faithfully encode multi-entity relationships. Applied to a corpus of ~1,100 manuscripts on biocomposite scaffolds, our framework constructs a global hypergraph of 161,172 nodes and 320,201 hyperedges, revealing a scale-free topology (power law exponent ~1.23) organized around highly connected conceptual hubs. This representation prevents the combinatorial explosion typical of pairwise expansions and explicitly preserves the co-occurrence context of scientific formulations. We further demonstrate that equipping agentic systems with hypergraph traversal tools, specifically using node-intersection constraints, enables them to bridge semantically distant concepts. By exploiting these higher-order pathways, the system successfully generates grounded mechanistic hypotheses for novel composite materials, such as linking cerium oxide to PCL scaffolds via chitosan intermediates. This work establishes a "teacherless" agentic reasoning system where hypergraph topology acts as a verifiable guardrail, accelerating scientific discovery by uncovering relationships obscured by traditional graph methods.

We present TopoMortar, a brick wall dataset that is the first dataset specifically designed to evaluate topology-focused image segmentation methods, such as topology loss functions. TopoMortar enables to investigate in two ways whether methods incorporate prior topological knowledge. First, by eliminating challenges seen in real-world data, such as small training set, noisy labels, and out-of-distribution test-set images, that, as we show, impact the effectiveness of topology losses. Second, by allowing to assess in the same dataset topology accuracy across dataset challenges, isolating dataset-related effects from the effect of incorporating prior topological knowledge. In these two experiments, it is deliberately difficult to improve topology accuracy without actually using topology information, thus, permitting to attribute an improvement in topology accuracy to the incorporation of prior topological knowledge. To this end, TopoMortar includes three types of labels (accurate, noisy, pseudo-labels), two fixed training sets (large and small), and in-distribution and out-of-distribution test-set images. We compared eight loss functions on TopoMortar, and we found that clDice achieved the most topologically accurate segmentations, Skeleton Recall loss performed best particularly with noisy labels, and the relative advantageousness of the other loss functions depended on the experimental setting. Additionally, we show that simple methods, such as data augmentation and self-distillation, can elevate Cross entropy Dice loss to surpass most topology loss functions, and that those simple methods can enhance topology loss functions as well. clDice and Skeleton Recall loss, both skeletonization-based loss functions, were also the fastest to train, making this type of loss function a promising research direction. TopoMortar and our code can be found at https://github.com/jmlipman/TopoMortar

We explore the use of topological data analysis (TDA) combined with machine learning for discriminating standard model backgrounds from the invisible decay of the Z^prime boson associated with monophoton emission at a 3 TeV muon collider. Reconstructed events are mapped into a six-dimensional kinematic space and aggregated into bags of events, from which persistent homology is used to extract Betti number distributions. Within the Multiple Instance Learning paradigm, classifiers trained on these topological descriptors demonstrate significantly improved classification accuracy compared to the conventional ML approaches based on event-wise kinematic inputs. We also draw exclusion contours at 95\% CL in the (m_{Z^prime}, m_chi) parameter space, highlighting the potential of topological features to extend the discovery reach of future collider experiments.

The Polymorphic Combinatorial Framework (PCF) leverages Large Language Models (LLMs) and mathematical frameworks to guide the meta-prompt enabled design of solution spaces and adaptive AI agents for complex, dynamic environments. Unlike static agent architectures, PCF enables real-time parameter reconfiguration through mathematically-grounded combinatorial spaces, allowing agents to adapt their core behavioral traits dynamically. Grounded in combinatorial logic, topos theory, and rough fuzzy set theory, PCF defines a multidimensional SPARK parameter space (Skills, Personalities, Approaches, Resources, Knowledge) to capture agent behaviors. This paper demonstrates how LLMs can parameterize complex spaces and estimate likely parameter values/variabilities. Using PCF, we parameterized mock caf\'e domains (five levels of complexity), estimated variables/variabilities, and conducted over 1.25 million Monte Carlo simulations. The results revealed trends in agent adaptability and performance across the five complexity tiers, with diminishing returns at higher complexity levels highlighting thresholds for scalable designs. PCF enables the generation of optimized agent configurations for specific scenarios while maintaining logical consistency. This framework supports scalable, dynamic, explainable, and ethical AI applications in domains like customer service, healthcare, robotics, and collaborative systems, paving the way for adaptable and cooperative next-generation polymorphic agents.

Random walks are widely used for mining networks due to the computational efficiency of computing them. For instance, graph representation learning learns a d-dimensional embedding space, so that the nodes that tend to co-occur on random walks (a proxy of being in the same network neighborhood) are close in the embedding space. Specific local network topology (i.e., structure) influences the co-occurrence of nodes on random walks, so random walks of limited length capture only partial topological information, hence diminishing the performance of downstream methods. We explicitly capture all topological neighborhood information and improve performance by introducing orbit adjacencies that quantify the adjacencies of two nodes as co-occurring on a given pair of graphlet orbits, which are symmetric positions on graphlets (small, connected, non-isomorphic, induced subgraphs of a large network). Importantly, we mathematically prove that random walks on up to k nodes capture only a subset of all the possible orbit adjacencies for up to k-node graphlets. Furthermore, we enable orbit adjacency-based analysis of networks by developing an efficient GRaphlet-orbit ADjacency COunter (GRADCO), which exhaustively computes all 28 orbit adjacency matrices for up to four-node graphlets. Note that four-node graphlets suffice, because real networks are usually small-world. In large networks on around 20,000 nodes, GRADCOcomputesthe28matricesinminutes. Onsixrealnetworksfromvarious domains, we compare the performance of node-label predictors obtained by using the network embeddings based on our orbit adjacencies to those based on random walks. We find that orbit adjacencies, which include those unseen by random walks, outperform random walk-based adjacencies, demonstrating the importance of the inclusion of the topological neighborhood information that is unseen by random walks.

Vision-language models (VLMs) have made significant strides in reasoning, yet they often struggle with complex multimodal tasks and tend to generate overly verbose outputs. A key limitation is their reliance on chain-of-thought (CoT) reasoning, despite many tasks benefiting from alternative topologies like trees or graphs. To address this, we introduce STELAR-Vision, a training framework for topology-aware reasoning. At its core is TopoAug, a synthetic data pipeline that enriches training with diverse topological structures. Using supervised fine-tuning and reinforcement learning, we post-train Qwen2VL models with both accuracy and efficiency in mind. Additionally, we propose Frugal Learning, which reduces output length with minimal accuracy loss. On MATH-V and VLM-S2H, STELAR-Vision improves accuracy by 9.7% over its base model and surpasses the larger Qwen2VL-72B-Instruct by 7.3%. On five out-of-distribution benchmarks, it outperforms Phi-4-Multimodal-Instruct by up to 28.4% and LLaMA-3.2-11B-Vision-Instruct by up to 13.2%, demonstrating strong generalization. Compared to Chain-Only training, our approach achieves 4.3% higher overall accuracy on in-distribution datasets and consistently outperforms across all OOD benchmarks. We have released datasets, and code will be available.

In this paper, we propose REASON, a novel framework that enables the automatic discovery of both intra-level (i.e., within-network) and inter-level (i.e., across-network) causal relationships for root cause localization. REASON consists of Topological Causal Discovery and Individual Causal Discovery. The Topological Causal Discovery component aims to model the fault propagation in order to trace back to the root causes. To achieve this, we propose novel hierarchical graph neural networks to construct interdependent causal networks by modeling both intra-level and inter-level non-linear causal relations. Based on the learned interdependent causal networks, we then leverage random walks with restarts to model the network propagation of a system fault. The Individual Causal Discovery component focuses on capturing abrupt change patterns of a single system entity. This component examines the temporal patterns of each entity's metric data (i.e., time series), and estimates its likelihood of being a root cause based on the Extreme Value theory. Combining the topological and individual causal scores, the top K system entities are identified as root causes. Extensive experiments on three real-world datasets with case studies demonstrate the effectiveness and superiority of the proposed framework.

With their non-Abelian topological charges, real multi-bandgap systems challenge the conventional topological phase classifications. As the minimal sector of multi-bandgap systems, real triple degeneracies (RTPs), which serve as real 'Weyl points', lay the foundation for the research on real topological phases. However, experimental demonstration of physical systems with global band configurations consisting of multiple RTPs in crystals has not been reported. Here we present experimental evidence of RTPs in photonic meta-crystals, characterizing them using the Euler number, and establishing their connection with both Abelian and non-Abelian charges. By considering RTPs as the basic elements, we further propose the concept of a topological compound, akin to a chemical compound, where we find that certain phases are not topologically allowed. The topological classification of RTPs in crystals demonstrated in our work plays a similar role as the 'no-go' theorem in Weyl systems.

Many clustering schemes are defined by optimizing an objective function defined on the partitions of the underlying set of a finite metric space. In this paper, we construct a framework for studying what happens when we instead impose various structural conditions on the clustering schemes, under the general heading of functoriality. Functoriality refers to the idea that one should be able to compare the results of clustering algorithms as one varies the data set, for example by adding points or by applying functions to it. We show that within this framework, one can prove a theorems analogous to one of J. Kleinberg, in which for example one obtains an existence and uniqueness theorem instead of a non-existence result. We obtain a full classification of all clustering schemes satisfying a condition we refer to as excisiveness. The classification can be changed by varying the notion of maps of finite metric spaces. The conditions occur naturally when one considers clustering as the statistical version of the geometric notion of connected components. By varying the degree of functoriality that one requires from the schemes it is possible to construct richer families of clustering schemes that exhibit sensitivity to density.

In this article, we introduce a new parameterized family of topological invariants, taking the form of candidate decompositions, for multi-parameter persistence modules. We prove that our candidate decompositions are controllable approximations: when restricting to modules that can be decomposed into interval summands, we establish theoretical results about the approximation error between our candidate decompositions and the true underlying module in terms of the standard interleaving and bottleneck distances. Moreover, even when the underlying module does not admit such a decomposition, our candidate decompositions are nonetheless stable invariants; small perturbations in the underlying module lead to small perturbations in the candidate decomposition. Then, we introduce MMA (Multipersistence Module Approximation): an algorithm for computing stable instances of such invariants, which is based on fibered barcodes and exact matchings, two constructions that stem from the theory of single-parameter persistence. By design, MMA can handle an arbitrary number of filtrations, and has bounded complexity and running time. Finally, we present empirical evidence validating the generalization capabilities and running time speed-ups of MMA on several data sets.

In this paper we study holomorphic properties of infinite dimensional spin factors. Among the infinite dimensional Banach spaces with homogeneous open unit balls, we show that the spin factors are natural outlier spaces in which to ask the question (as was proved in the early 1970s for Hilbert spaces): Do biholomorphic automorphisms g of the open unit ball B have fixed points in overline B? In this paper, for infinite dimensional spin factors, we provide reasonable conditions on g that allow us to explicitly construct fixed points of g lying on partial B. En route, we also prove that every spin factor has the density property. In another direction, we focus on (compact) holomorphic maps f:Brightarrow B, having no fixed point in B and examine the sequence of iterates (f^n). As (f^n) does not generally converge, we instead trace the target set T(f) of f, that is, the images of all accumulation points of (f^n)_n, for any topology finer than the topology of pointwise convergence on B. We prove for a spin factor that T(f) lies on the boundary of a single bidisc unique to f.

One of the main goals and challenges of materials discovery is to find the best candidates for each interest property or application. Machine learning rises in this context to efficiently optimize this search, exploring the immense materials space, consisting of simultaneously the atomic, compositional, and structural spaces. Topological insulators, presenting symmetry-protected metallic edge states, are a promising class of materials for different applications. However, further, development is limited by the scarcity of viable candidates. Here we present and discuss machine learning-accelerated strategies for searching the materials space for two-dimensional topological materials. We show the importance of detailed investigations of each machine learning component, leading to different results. Using recently created databases containing thousands of ab initio calculations of 2D materials, we train machine learning models capable of determining the electronic topology of materials, with an accuracy of over 90%. We can then generate and screen thousands of novel materials, efficiently predicting their topological character without the need for a priori structural knowledge. We discover 56 non-trivial materials, of which 17 novel insulating candidates for further investigation, for which we corroborate their topological properties with density functional theory calculations. This strategy is 10times more efficient than the trial-and-error approach while few orders of magnitude faster and is a proof of concept for guiding improved materials discovery search strategies.

In this paper we demonstrate how sub-Riemannian geometry can be used for manifold learning and surface reconstruction by combining local linear approximations of a point cloud to obtain lower dimensional bundles. Local approximations obtained by local PCAs are collected into a rank k tangent subbundle on R^d, k<d, which we call a principal subbundle. This determines a sub-Riemannian metric on R^d. We show that sub-Riemannian geodesics with respect to this metric can successfully be applied to a number of important problems, such as: explicit construction of an approximating submanifold M, construction of a representation of the point-cloud in R^k, and computation of distances between observations, taking the learned geometry into account. The reconstruction is guaranteed to equal the true submanifold in the limit case where tangent spaces are estimated exactly. Via simulations, we show that the framework is robust when applied to noisy data. Furthermore, the framework generalizes to observations on an a priori known Riemannian manifold.

Graph neural networks achieve high accuracy in link prediction by jointly leveraging graph topology and node attributes. Topology, however, is represented indirectly; state-of-the-art methods based on subgraph classification label nodes with distance to the target link, so that, although topological information is present, it is tempered by pooling. This makes it challenging to leverage features like loops and motifs associated with network formation mechanisms. We propose a link prediction algorithm based on a new pooling scheme called WalkPool. WalkPool combines the expressivity of topological heuristics with the feature-learning ability of neural networks. It summarizes a putative link by random walk probabilities of adjacent paths. Instead of extracting transition probabilities from the original graph, it computes the transition matrix of a "predictive" latent graph by applying attention to learned features; this may be interpreted as feature-sensitive topology fingerprinting. WalkPool can leverage unsupervised node features or be combined with GNNs and trained end-to-end. It outperforms state-of-the-art methods on all common link prediction benchmarks, both homophilic and heterophilic, with and without node attributes. Applying WalkPool to a set of unsupervised GNNs significantly improves prediction accuracy, suggesting that it may be used as a general-purpose graph pooling scheme.

We introduce Diagram of Thought (DoT), a framework that models iterative reasoning in large language models (LLMs) as the construction of a directed acyclic graph (DAG) within a single model. Unlike traditional approaches that represent reasoning as linear chains or trees, DoT organizes propositions, critiques, refinements, and verifications into a cohesive DAG structure, allowing the model to explore complex reasoning pathways while maintaining logical consistency. Each node in the diagram corresponds to a proposition that has been proposed, critiqued, refined, or verified, enabling the LLM to iteratively improve its reasoning through natural language feedback. By leveraging auto-regressive next-token prediction with role-specific tokens, DoT facilitates seamless transitions between proposing ideas and critically evaluating them, providing richer feedback than binary signals. Furthermore, we formalize the DoT framework using Topos Theory, providing a mathematical foundation that ensures logical consistency and soundness in the reasoning process. This approach enhances both the training and inference processes within a single LLM, eliminating the need for multiple models or external control mechanisms. DoT offers a conceptual framework for designing next-generation reasoning-specialized models, emphasizing training efficiency, robust reasoning capabilities, and theoretical grounding. The code is available at https://github.com/diagram-of-thought/diagram-of-thought.

Time series reasoning treats time as a first-class axis and incorporates intermediate evidence directly into the answer. This survey defines the problem and organizes the literature by reasoning topology with three families: direct reasoning in one step, linear chain reasoning with explicit intermediates, and branch-structured reasoning that explores, revises, and aggregates. The topology is crossed with the main objectives of the field, including traditional time series analysis, explanation and understanding, causal inference and decision making, and time series generation, while a compact tag set spans these axes and captures decomposition and verification, ensembling, tool use, knowledge access, multimodality, agent loops, and LLM alignment regimes. Methods and systems are reviewed across domains, showing what each topology enables and where it breaks down in faithfulness or robustness, along with curated datasets, benchmarks, and resources that support study and deployment (https://github.com/blacksnail789521/Time-Series-Reasoning-Survey). Evaluation practices that keep evidence visible and temporally aligned are highlighted, and guidance is distilled on matching topology to uncertainty, grounding with observable artifacts, planning for shift and streaming, and treating cost and latency as design budgets. We emphasize that reasoning structures must balance capacity for grounding and self-correction against computational cost and reproducibility, while future progress will likely depend on benchmarks that tie reasoning quality to utility and on closed-loop testbeds that trade off cost and risk under shift-aware, streaming, and long-horizon settings. Taken together, these directions mark a shift from narrow accuracy toward reliability at scale, enabling systems that not only analyze but also understand, explain, and act on dynamic worlds with traceable evidence and credible outcomes.

We define and develop a homotopy invariant notion for the topological complexity of a map f:X to Y, denoted TC(f), that interacts with TC(X) and TC(Y) in the same way cat(f) interacts with cat(X) and cat(Y). Furthermore, TC(f) and cat(f) satisfy the same inequalities as TC(X) and cat(X). We compare it to other invariants defined in the papers [15,16,17,18,20]. We apply TC(f) to studying group homomorphisms f:Hto G.

The field of face recognition (FR) has undergone significant advancements with the rise of deep learning. Recently, the success of unsupervised learning and graph neural networks has demonstrated the effectiveness of data structure information. Considering that the FR task can leverage large-scale training data, which intrinsically contains significant structure information, we aim to investigate how to encode such critical structure information into the latent space. As revealed from our observations, directly aligning the structure information between the input and latent spaces inevitably suffers from an overfitting problem, leading to a structure collapse phenomenon in the latent space. To address this problem, we propose TopoFR, a novel FR model that leverages a topological structure alignment strategy called PTSA and a hard sample mining strategy named SDE. Concretely, PTSA uses persistent homology to align the topological structures of the input and latent spaces, effectively preserving the structure information and improving the generalization performance of FR model. To mitigate the impact of hard samples on the latent space structure, SDE accurately identifies hard samples by automatically computing structure damage score (SDS) for each sample, and directs the model to prioritize optimizing these samples. Experimental results on popular face benchmarks demonstrate the superiority of our TopoFR over the state-of-the-art methods. Code and models are available at: https://github.com/modelscope/facechain/tree/main/face_module/TopoFR.

We explore local vs. global evolution of knowledge systems through the framework of socio-epistemic networks (SEN), applying two complementary methods to a corpus of scientific texts. The framework comprises three interconnected layers-social, semiotic (material), and semantic-proposing a multilayered approach to understanding structural developments of knowledge. To analyse diachronic changes on the semantic layer, we first use information-theoretic measures based on relative entropy to detect semantic shifts, assess their significance, and identify key driving features. Second, variations in document embedding densities reveal changes in semantic neighbourhoods, tracking how concentration of similar documents increase, remain stable, or disperse. This enables us to trace document trajectories based on content (topics) or metadata (authorship, institution). Case studies of Joseph Silk and Hans-J\"urgen Treder illustrate how individual scholar's work aligns with broader disciplinary shifts in general relativity and gravitation research, demonstrating the applications, limitations, and further potential of this approach.

Modeling medical vessel-like anatomy is challenging due to its intricate topology and sensitivity to dataset shifts. Consequently, task-specific models often suffer from topological inconsistencies, including artificial disconnections and spurious merges. Motivated by the promise of multimodal large language models (MLLMs) for zero-shot generalization, we propose TubeMLLM, a unified foundation model that couples structured understanding with controllable generation for medical vessel-like anatomy. By integrating topological priors through explicit natural language prompting and aligning them with visual representations in a shared-attention architecture, TubeMLLM significantly enhances topology-aware perception. Furthermore, we construct TubeMData, a pionner multimodal benchmark comprising comprehensive topology-centric tasks, and introduce an adaptive loss weighting strategy to emphasize topology-critical regions during training. Extensive experiments on fifteen diverse datasets demonstrate our superiority. Quantitatively, TubeMLLM achieves state-of-the-art out-of-distribution performance, substantially reducing global topological discrepancies on color fundus photography (decreasing the β_{0} number error from 37.42 to 8.58 compared to baselines). Notably, TubeMLLM exhibits exceptional zero-shot cross-modality transferring ability on unseen X-ray angiography, achieving a Dice score of 67.50% while significantly reducing the β_{0} error to 1.21. TubeMLLM also maintains robustness against degradations such as blur, noise, and low resolution. Furthermore, in topology-aware understanding tasks, the model achieves 97.38% accuracy in evaluating mask topological quality, significantly outperforming standard vision-language baselines.

This paper presents a framework that leverages pre-trained foundation models for robotic manipulation without domain-specific training. The framework integrates off-the-shelf models, combining multimodal perception from foundation models with a general-purpose reasoning model capable of robust task sequencing. Scene graphs, dynamically maintained within the framework, provide spatial awareness and enable consistent reasoning about the environment. The framework is evaluated through a series of tabletop robotic manipulation experiments, and the results highlight its potential for building robotic manipulation systems directly on top of off-the-shelf foundation models.

Due to the advantages of hypergraphs in modeling high-order relationships in complex systems, they have been applied to higher-order clustering, hypergraph neural networks and computer vision. These applications rely heavily on access to high-quality, large-scale real-world hypergraph data. Yet, compared to traditional pairwise graphs, real hypergraph datasets remain scarce in both scale and diversity. This shortage significantly limits the development and evaluation of advanced hypergraph learning algorithms. Therefore, how to quickly generate large-scale hypergraphs that conform to the characteristics of real networks is a crucial task that has not received sufficient attention. Motivated by recent advances in large language models (LLMs), particularly their capabilities in semantic reasoning, structured generation, and simulating human behavior, we investigate whether LLMs can facilitate hypergraph generation from a fundamentally new perspective. We introduce HyperLLM, a novel LLM-driven hypergraph generator that simulates the formation and evolution of hypergraphs through a multi-agent collaboration. The framework integrates prompts and structural feedback mechanisms to ensure that the generated hypergraphs reflect key real-world patterns. Extensive experiments across diverse datasets demonstrate that HyperLLM achieves superior fidelity to structural and temporal hypergraph patterns, while requiring minimal statistical priors. Our findings suggest that LLM-based frameworks offer a promising new direction for hypergraph modeling.

The Shadowing Principle of Manin has proved a valuable tool for addressing questions of quantitative topology raised by Gromov in the late 1900s. The principle informally provides a way for bounded algebraic maps between differential graded algebras to be translated into nearby genuine maps between their geometric realizations. We extend this principle to finite towers of principal K(G,n) fibrations, and in particular apply this construction to nilpotent spaces. As a specific application of the extended principle, we provide upper bounds on the asymptotic behavior of volumes of nullhomotopies of Lipschitz maps into nilpotent spaces. We further refine these bounds in the case when c = 1 to nearly meet those of the simply connected setting. We similarly refine these bounds in the event the target space is coformal, and demonstrate that the bounds in this setting are nearly sharp.

Real-valued functions on geometric data -- such as node attributes on a graph -- can be optimized using descriptors from persistent homology, allowing the user to incorporate topological terms in the loss function. When optimizing a single real-valued function (the one-parameter setting), there is a canonical choice of descriptor for persistent homology: the barcode. The operation mapping a real-valued function to its barcode is differentiable almost everywhere, and the convergence of gradient descent for losses using barcodes is relatively well understood. When optimizing a vector-valued function (the multiparameter setting), there is no unique choice of descriptor for multiparameter persistent homology, and many distinct descriptors have been proposed. This calls for the development of a general framework for differentiability and optimization that applies to a wide range of multiparameter homological descriptors. In this article, we develop such a framework and show that it encompasses well-known descriptors of different flavors, such as signed barcodes and the multiparameter persistence landscape. We complement the theory with numerical experiments supporting the idea that optimizing multiparameter homological descriptors can lead to improved performances compared to optimizing one-parameter descriptors, even when using the simplest and most efficiently computable multiparameter descriptors.

Characteristic classes, which are abstract topological invariants associated with vector bundles, have become an important notion in modern physics with surprising real-world consequences. As a representative example, the incredible properties of topological insulators, which are insulators in their bulk but conductors on their surface, can be completely characterized by a specific characteristic class associated with their electronic band structure, the first Chern class. Given their importance to next generation computing and the computational challenge of calculating them using first-principles approaches, there is a need to develop machine learning approaches to predict the characteristic classes associated with a material system. To aid in this program we introduce the {Haldane bundle dataset}, which consists of synthetically generated complex line bundles on the 2-torus. We envision this dataset, which is not as challenging as noisy and sparsely measured real-world datasets but (as we show) still difficult for off-the-shelf architectures, to be a testing ground for architectures that incorporate the rich topological and geometric priors underlying characteristic classes.

Topology reasoning aims to comprehensively understand road scenes and present drivable routes in autonomous driving. It requires detecting road centerlines (lane) and traffic elements, further reasoning their topology relationship, i.e., lane-lane topology, and lane-traffic topology. In this work, we first present that the topology score relies heavily on detection performance on lane and traffic elements. Therefore, we introduce a powerful 3D lane detector and an improved 2D traffic element detector to extend the upper limit of topology performance. Further, we propose TopoMLP, a simple yet high-performance pipeline for driving topology reasoning. Based on the impressive detection performance, we develop two simple MLP-based heads for topology generation. TopoMLP achieves state-of-the-art performance on OpenLane-V2 benchmark, i.e., 41.2% OLS with ResNet-50 backbone. It is also the 1st solution for 1st OpenLane Topology in Autonomous Driving Challenge. We hope such simple and strong pipeline can provide some new insights to the community. Code is at https://github.com/wudongming97/TopoMLP.

A challenge towards developing NLP systems for the world's languages is understanding how they generalize to typological differences relevant for real-world applications. To this end, we propose M2C, a morphologically-aware framework for behavioral testing of NLP models. We use M2C to generate tests that probe models' behavior in light of specific linguistic features in 12 typologically diverse languages. We evaluate state-of-the-art language models on the generated tests. While models excel at most tests in English, we highlight generalization failures to specific typological characteristics such as temporal expressions in Swahili and compounding possessives in Finish. Our findings motivate the development of models that address these blind spots.

We study the implications of the modeling choice to use a graph, instead of a hypergraph, to represent real-world interconnected systems whose constituent relationships are of higher order by nature. Such a modeling choice typically involves an underlying projection process that maps the original hypergraph onto a graph, and is common in graph-based analysis. While hypergraph projection can potentially lead to loss of higher-order relations, there exists very limited studies on the consequences of doing so, as well as its remediation. This work fills this gap by doing two things: (1) we develop analysis based on graph and set theory, showing two ubiquitous patterns of hyperedges that are root to structural information loss in all hypergraph projections; we also quantify the combinatorial impossibility of recovering the lost higher-order structures if no extra help is provided; (2) we still seek to recover the lost higher-order structures in hypergraph projection, and in light of (1)'s findings we propose to relax the problem into a learning-based setting. Under this setting, we develop a learning-based hypergraph reconstruction method based on an important statistic of hyperedge distributions that we find. Our reconstruction method is evaluated on 8 real-world datasets under different settings, and exhibits consistently good performance. We also demonstrate benefits of the reconstructed hypergraphs via use cases of protein rankings and link predictions.

A central question for neuroscience is how to characterize brain representations of perceptual and cognitive content. An ideal characterization should distinguish different functional regions with robustness to noise and idiosyncrasies of individual brains that do not correspond to computational differences. Previous studies have characterized brain representations by their representational geometry, which is defined by the representational dissimilarity matrix (RDM), a summary statistic that abstracts from the roles of individual neurons (or responses channels) and characterizes the discriminability of stimuli. Here we explore a further step of abstraction: from the geometry to the topology of brain representations. We propose topological representational similarity analysis (tRSA), an extension of representational similarity analysis (RSA) that uses a family of geo-topological summary statistics that generalizes the RDM to characterize the topology while de-emphasizing the geometry. We evaluate this new family of statistics in terms of the sensitivity and specificity for model selection using both simulations and functional MRI (fMRI) data. In the simulations, the ground truth is a data-generating layer representation in a neural network model and the models are the same and other layers in different model instances (trained from different random seeds). In fMRI, the ground truth is a visual area and the models are the same and other areas measured in different subjects. Results show that topology-sensitive characterizations of population codes are robust to noise and interindividual variability and maintain excellent sensitivity to the unique representational signatures of different neural network layers and brain regions.

Leaf-lesion segmentation is topology-sensitive: small merges, splits, or false holes can be biologically meaningful descriptors of biochemical pathways, yet they are weakly penalized by standard pixel-wise losses in Euclidean latents. I explore HyperTopo-Adapters, a lightweight, parameter-efficient head trained on top of a frozen vision encoder, which embeds features on a product manifold -- hyperbolic + Euclidean + spherical (H + E + S) -- to encourage hierarchical separation (H), local linear detail (E), and global closure (S). A topology prior complements Dice/BCE in two forms: (i) persistent-homology (PH) distance for evaluation and selection, and (ii) a differentiable surrogate that combines a soft Euler-characteristic match with total variation regularization for stable training. I introduce warm-ups for both the hyperbolic contrastive term and the topology prior, per-sample evaluation of structure-aware metrics (Boundary-F1, Betti errors, PD distance), and a min-PD within top-K Dice rule for checkpoint selection. On a Kaggle leaf-lesion dataset (N=2,940), early results show consistent gains in boundary and topology metrics (reducing Delta beta_1 hole error by 9%) while Dice/IoU remain competitive. The study is diagnostic by design: I report controlled ablations (curvature learning, latent dimensions, contrastive temperature, surrogate settings), and ongoing tests varying encoder strength (ResNet-50, DeepLabV3, DINOv2/v3), input resolution, PH weight, and partial unfreezing of late blocks. The contribution is an open, reproducible train/eval suite (available at https://github.com/ChimdiWalter/HyperTopo-Adapters) that isolates geometric/topological priors and surfaces failure modes to guide stronger, topology-preserving architectures.

In this paper we introduce a novel family of attributed graphs for the purpose of shape discrimination. Our graphs typically arise from variations on the Mapper graph construction, which is an approximation of the Reeb graph for point cloud data. Our attributions enrich these constructions with (persistent) homology in ways that are provably stable, thereby recording extra topological information that is typically lost in these graph constructions. We provide experiments which illustrate the use of these invariants for shape representation and classification. In particular, we obtain competitive shape classification results when using our topologically attributed graphs as inputs to a simple graph neural network classifier.

Foundation models applied to bio-molecular space hold promise to accelerate drug discovery. Molecular representation is key to building such models. Previous works have typically focused on a single representation or view of the molecules. Here, we develop a multi-view foundation model approach, that integrates molecular views of graph, image and text. Single-view foundation models are each pre-trained on a dataset of up to 200M molecules and then aggregated into combined representations. Our multi-view model is validated on a diverse set of 18 tasks, encompassing ligand-protein binding, molecular solubility, metabolism and toxicity. We show that the multi-view models perform robustly and are able to balance the strengths and weaknesses of specific views. We then apply this model to screen compounds against a large (>100 targets) set of G Protein-Coupled receptors (GPCRs). From this library of targets, we identify 33 that are related to Alzheimer's disease. On this subset, we employ our model to identify strong binders, which are validated through structure-based modeling and identification of key binding motifs.

Recent large vision-language models (LVLMs) have demonstrated strong potential for device control. However, existing research has primarily focused on point-and-click (PnC) interaction, while remote-control (RC) interaction commonly encountered in everyday TV usage remains largely underexplored. To fill this gap, we introduce TVWorld, an offline graph-based abstraction of real-world TV navigation that enables reproducible and deployment-free evaluation. On this basis, we derive two complementary benchmarks that comprehensively assess TV-use capabilities: TVWorld-N for topology-aware navigation and TVWorld-G for focus-aware grounding. These benchmarks expose a key limitation of existing agents: insufficient topology awareness for focus-based, long-horizon TV navigation. Motivated by this finding, we propose a Topology-Aware Training framework that injects topology awareness into LVLMs. Using this framework, we develop TVTheseus, a foundation model specialized for TV navigation. TVTheseus achieves a success rate of 68.3% on TVWorld-N, surpassing strong closed-source baselines such as Gemini 3 Flash and establishing state-of-the-art (SOTA) performance. Additional analyses further provide valuable insights into the development of effective TV-use agents.

Hyperbolic space has proven to be well-suited for capturing hierarchical relations in data, such as trees and directed acyclic graphs. Prior work introduced the concept of entailment cones, which uses partial orders defined by nested cones in the Poincar\'e ball to model hierarchies. Here, we introduce the ``shadow cones" framework, a physics-inspired entailment cone construction. Specifically, we model partial orders as subset relations between shadows formed by a light source and opaque objects in hyperbolic space. The shadow cones framework generalizes entailment cones to a broad class of formulations and hyperbolic space models beyond the Poincar\'e ball. This results in clear advantages over existing constructions: for example, shadow cones possess better optimization properties over constructions limited to the Poincar\'e ball. Our experiments on datasets of various sizes and hierarchical structures show that shadow cones consistently and significantly outperform existing entailment cone constructions. These results indicate that shadow cones are an effective way to model partial orders in hyperbolic space, offering physically intuitive and novel insights about the nature of such structures.

A standard Variational Autoencoder, with a Euclidean latent space, is structurally incapable of capturing topological properties of certain datasets. To remove topological obstructions, we introduce Diffusion Variational Autoencoders with arbitrary manifolds as a latent space. A Diffusion Variational Autoencoder uses transition kernels of Brownian motion on the manifold. In particular, it uses properties of the Brownian motion to implement the reparametrization trick and fast approximations to the KL divergence. We show that the Diffusion Variational Autoencoder is capable of capturing topological properties of synthetic datasets. Additionally, we train MNIST on spheres, tori, projective spaces, SO(3), and a torus embedded in R3. Although a natural dataset like MNIST does not have latent variables with a clear-cut topological structure, training it on a manifold can still highlight topological and geometrical properties.

Recently, studies exemplified by Hyper-Connections (HC) have extended the ubiquitous residual connection paradigm established over the past decade by expanding the residual stream width and diversifying connectivity patterns. While yielding substantial performance gains, this diversification fundamentally compromises the identity mapping property intrinsic to the residual connection, which causes severe training instability and restricted scalability, and additionally incurs notable memory access overhead. To address these challenges, we propose Manifold-Constrained Hyper-Connections (mHC), a general framework that projects the residual connection space of HC onto a specific manifold to restore the identity mapping property, while incorporating rigorous infrastructure optimization to ensure efficiency. Empirical experiments demonstrate that mHC is effective for training at scale, offering tangible performance improvements and superior scalability. We anticipate that mHC, as a flexible and practical extension of HC, will contribute to a deeper understanding of topological architecture design and suggest promising directions for the evolution of foundational models.

We develop a discipline-agnostic emergence calculus that treats theories as fixed points of idempotent operators acting on descriptions. We show that, once processes are composable but access to the underlying system is mediated by a bounded observational interface, a canonical toolkit of six closure-changing primitives (P1--P6) is unavoidable. The framework unifies order-theoretic closure operators with dynamics-induced endomaps E_{τ,f} built from a Markov kernel, a coarse-graining lens, and a time scale τ. We introduce a computable total-variation idempotence defect for E_{τ,f}; small retention error implies approximate idempotence and yields stable "objects" packaged at the chosen τ within a fixed lens. For directionality, we define an arrow-of-time functional as the path-space KL divergence between forward and time-reversed trajectories and prove it is monotone under coarse-graining (data processing); we also formalize a protocol-trap audit showing that protocol holonomy alone cannot sustain asymmetry without a genuine affinity in the lifted dynamics. Finally, we prove a finite forcing-style counting lemma: relative to a partition-based theory, definable predicate extensions are exponentially rare, giving a clean anti-saturation mechanism for strict ladder climbing.

A key challenge in artificial intelligence is the creation of systems capable of autonomously advancing scientific understanding by exploring novel domains, identifying complex patterns, and uncovering previously unseen connections in vast scientific data. In this work, we present SciAgents, an approach that leverages three core concepts: (1) the use of large-scale ontological knowledge graphs to organize and interconnect diverse scientific concepts, (2) a suite of large language models (LLMs) and data retrieval tools, and (3) multi-agent systems with in-situ learning capabilities. Applied to biologically inspired materials, SciAgents reveals hidden interdisciplinary relationships that were previously considered unrelated, achieving a scale, precision, and exploratory power that surpasses traditional human-driven research methods. The framework autonomously generates and refines research hypotheses, elucidating underlying mechanisms, design principles, and unexpected material properties. By integrating these capabilities in a modular fashion, the intelligent system yields material discoveries, critique and improve existing hypotheses, retrieve up-to-date data about existing research, and highlights their strengths and limitations. Our case studies demonstrate scalable capabilities to combine generative AI, ontological representations, and multi-agent modeling, harnessing a `swarm of intelligence' similar to biological systems. This provides new avenues for materials discovery and accelerates the development of advanced materials by unlocking Nature's design principles.