Topological and Geometric Data Analysis

Neueste Paper

• CutisAI: Deep Learning Framework for Automated Dermatology and Cancer Screening

  Rohit Kaushik, Eva Kaushik · 7. Januar 2026

  The rapid growth of dermatological imaging and mobile diagnostic tools calls for systems that not only demonstrate empirical performance but also provide strong theoretical guarantees. Deep learning models have shown high predictive accuracy; however, they are often criticized for lacking well, cali…

• Hierarchical topological clustering

  Ana Carpio, Gema Duro · 6. Januar 2026

  Topological methods have the potential of exploring data clouds without making assumptions on their the structure. Here we propose a hierarchical topological clustering algorithm that can be implemented with any distance choice. The persistence of outliers and clusters of arbitrary shape is inferred…

• Efficient Cover Construction for Ball Mapper via Accelerated Range Queries

  Jay-Anne Bulauan, John Rick Manzanares · 6. Januar 2026

  Ball Mapper is an widely used tool in topological data analysis for summarizing the structure of high-dimensional data through metric-based coverings and graph representations. A central computational bottleneck in Ball Mapper is the construction of the underlying cover, which requires repeated rang…

• Warp-Cortex: An Asynchronous, Memory-Efficient Architecture for Million-Agent Cognitive Scaling on Consumer Hardware

  Jorge L. Ruiz Williams · 6. Januar 2026

  Current multi-agent Large Language Model (LLM) frameworks suffer from linear memory scaling, rendering "System 2" parallel reasoning impractical on consumer hardware. We present Warp Cortex, an asynchronous architecture that theoretically enables million-agent cognitive scaling by decoupling agent l…

• Streaming Sliced Optimal Transport

  Khai Nguyen · 5. Januar 2026

  Sliced optimal transport (SOT), or sliced Wasserstein (SW) distance, is widely recognized for its statistical and computational scalability. In this work, we further enhance computational scalability by proposing the first method for estimating SW from sample streams, called \emph{streaming sliced W…

• On the geometry and topology of representations: the manifolds of modular addition

  Gabriela Moisescu-Pareja, Gavin McCracken, Harley Wiltzer, Vincent L\'etourneau, Colin Daniels, Doina Precup, Jonathan Love · 1. Januar 2026

  The Clock and Pizza interpretations, associated with architectures differing in either uniform or learnable attention, were introduced to argue that different architectural designs can yield distinct circuits for modular addition. In this work, we show that this is not the case, and that both unifor…

• Topological Spatial Graph Coarsening

  Anna Calissano, Etienne Lasalle · 1. Januar 2026

  Spatial graphs are particular graphs for which the nodes are localized in space (e.g., public transport network, molecules, branching biological structures). In this work, we consider the problem of spatial graph reduction, that aims to find a smaller spatial graph (i.e., with less nodes) with the s…

• Frequent subgraph-based persistent homology for graph classification

  Xinyang Chen, Ama\"el Broustet, Guoting Chen · 1. Januar 2026

  Persistent homology (PH) has recently emerged as a powerful tool for extracting topological features. Integrating PH into machine learning and deep learning models enhances topology awareness and interpretability. However, most PH methods on graphs rely on a limited set of filtrations, such as degre…

• Visual Language Hypothesis

  Xiu Li · 30. Dezember 2025

  We study visual representation learning from a structural and topological perspective. We begin from a single hypothesis: that visual understanding presupposes a semantic language for vision, in which many perceptual observations correspond to a small number of discrete semantic states. Together wit…

• Persistent Homology via Finite Topological Spaces

  Sel\c{c}uk Kayacan · 30. Dezember 2025

  We propose a functorial framework for persistent homology based on finite topological spaces and their associated posets. Starting from a finite metric space, we associate a filtration of finite topologies whose structure maps are continuous identity maps. By passing functorially to posets and to si…

• Top-K Exterior Power Persistent Homology: Algorithm, Structure, and Stability

  Yoshihiro Maruyama · 24. Dezember 2025

  Exterior powers play important roles in persistent homology in computational geometry. In the present paper we study the problem of extracting the $K$ longest intervals of the exterior-power layers of a tame persistence module. We prove a structural decomposition theorem that organizes the exterior-…

• Algorithm for Interpretable Graph Features via Motivic Persistent Cohomology

  Yoshihiro Maruyama · 24. Dezember 2025

  We present the Chromatic Persistence Algorithm (CPA), an event-driven method for computing persistent cohomological features of weighted graphs via graphic arrangements, a classical object in computational geometry. We establish rigorous complexity results: CPA is exponential in the worst case, fixe…

• CoPHo: Classifier-guided Conditional Topology Generation with Persistent Homology

  Gongli Xi, Ye Tian, Mengyu Yang, Zhenyu Zhao, Yuchao Zhang, Xiangyang Gong, Xirong Que, Wendong Wang · 24. Dezember 2025

  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…

• Comparing Dynamical Models Through Diffeomorphic Vector Field Alignment

  Ruiqi Chen (Division of Biology and Biomedical Sciences, Washington University in St. Louis), Giacomo Vedovati (Department of Electrical and Systems Engineering, Washington University in St. Louis), Todd Braver (Department of Psychological and Brain Sciences, Washington University in St. Louis), ShiNung Ching (Department of Electrical and Systems Engineering, Washington University in St. Louis) · 23. Dezember 2025

  Dynamical systems models such as recurrent neural networks (RNNs) are increasingly popular in theoretical neuroscience for hypothesis-generation and data analysis. Evaluating the dynamics in such models is key to understanding their learned generative mechanisms. However, such evaluation is impeded …

• Incremental Generation is Necessary and Sufficient for Universality in Flow-Based Modelling

  Hossein Rouhvarzi, Anastasis Kratsios · 22. Dezember 2025

  Incremental flow-based denoising models have reshaped generative modelling, but their empirical advantage still lacks a rigorous approximation-theoretic foundation. We show that incremental generation is necessary and sufficient for universal flow-based generation on the largest natural class of sel…

• Persistent Multiscale Density-based Clustering

  Dani\"el Bot, Leland McInnes, Jan Aerts · 19. Dezember 2025

  Clustering is a cornerstone of modern data analysis. Detecting clusters in exploratory data analyses (EDA) requires algorithms that make few assumptions about the data. Density-based clustering algorithms are particularly well-suited for EDA because they describe high-density regions, assuming only …

• Topological Metric for Unsupervised Embedding Quality Evaluation

  Aleksei Shestov, Anton Klenitskiy, Daria Denisova, Amurkhan Dzagkoev, Daniil Petrovich, Andrey Savchenko, Maksim Makarenko · 18. Dezember 2025

  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…

• Hierarchical Persistence Velocity for Network Anomaly Detection: Theory and Applications to Cryptocurrency Markets

  Omid Khormali · 17. Dezember 2025

  We introduce the Overlap-Weighted Hierarchical Normalized Persistence Velocity (OW-HNPV), a novel topological data analysis method for detecting anomalies in time-varying networks. Unlike existing methods that measure cumulative topological presence, we introduce the first velocity-based perspective…

• TUN: Detecting Significant Points in Persistence Diagrams with Deep Learning

  Yu Chen, Hongwei Lin · 17. Dezember 2025

  Persistence diagrams (PDs) provide a powerful tool for understanding the topology of the underlying shape of a point cloud. However, identifying which points in PDs encode genuine signals remains challenging. This challenge directly hinders the practical adoption of topological data analysis in many…

• Topologically-Stabilized Graph Neural Networks: Empirical Robustness Across Domains

  Jelena Losic · 17. Dezember 2025

  Graph Neural Networks (GNNs) have become the standard for graph representation learning but remain vulnerable to structural perturbations. We propose a novel framework that integrates persistent homology features with stability regularization to enhance robustness. Building on the stability theorems…

• Interval Fisher's Discriminant Analysis and Visualisation

  Diogo Pinheiro, M. Ros\'ario Oliveira, Igor Kravchenko, Lina Oliveira · 16. Dezember 2025

  In Data Science, entities are typically represented by single valued measurements. Symbolic Data Analysis extends this framework to more complex structures, such as intervals and histograms, that express internal variability. We propose an extension of multiclass Fisher's Discriminant Analysis to in…

• The Geometry of Intelligence: Deterministic Functional Topology as a Foundation for Real-World Perception

  Eduardo Di Santi · 16. Dezember 2025

  Real-world physical processes do not generate arbitrary variability: their signals concentrate on compact and low-variability subsets of functional space. This geometric structure enables rapid generalization from a few examples in both biological and artificial systems. This work develops a deter…

• A Novel Patch-Based TDA Approach for Computed Tomography

  Dashti A. Ali, Aras T. Asaad, Jacob J. Peoples, Mohammad Hamghalam, Alex Robins, Mane Piliposyan, Richard K. G. Do, Natalie Gangai, Yun S. Chun, Ahmad Bashir Barekzai, Jayasree Chakraborty, Hala Khasawneh, Camila Vilela, Natally Horvat, Jo\~ao Miranda, Alice C. Wei, Amber L. Simpson · 16. Dezember 2025

  The development of machine learning (ML) models based on computed tomography (CT) imaging modality has been a major focus of recent research in the medical imaging domain. Incorporating robust feature engineering approach can highly improve the performance of these models. Topological data analysis …

• Continuous Edit Distance, Geodesics and Barycenters of Time-varying Persistence Diagrams

  Sebastien Tchitchek, Mohamed Kissi, Julien Tierny · 16. Dezember 2025

  We introduce the Continuous Edit Distance (CED), a geodesic and elastic distance for time-varying persistence diagrams (TVPDs). The CED extends edit-distance ideas to TVPDs by combining local substitution costs with penalized deletions/insertions, controlled by two parameters: \(\alpha\) (trade-off …

• A General Algorithm for Detecting Higher-Order Interactions via Random Sequential Additions

  Ahmad Shamail, Claire McWhite · 15. Dezember 2025

  Many systems exhibit complex interactions between their components: some features or actions amplify each other's effects, others provide redundant information, and some contribute independently. We present a simple geometric method for discovering interactions and redundancies: when elements are ad…