Topological and Geometric Data Analysis

Latest papers

• Topology-Preserving Deep Joint Source-Channel Coding for Semantic Communication

  Omar Erak, Omar Alhussein, Fang Fang, Sami Muhaidat · 19 March 2026

  Many wireless vision applications, such as autonomous driving, require preservation of global structural information rather than only per-pixel fidelity. However, existing Deep joint source-channel coding (DeepJSCC) schemes mainly optimize pixel-wise losses and provide no explicit protection of conn…

• Cohomological Obstructions to Global Counterfactuals: A Sheaf-Theoretic Foundation for Generative Causal Models

  Rui Wu, Hong Xie, Yongjun Li · 19 March 2026

  Current continuous generative models (e.g., Diffusion Models, Flow Matching) implicitly assume that locally consistent causal mechanisms naturally yield globally coherent counterfactuals. In this paper, we prove that this assumption fails fundamentally when the causal graph exhibits non-trivial homo…

• The Causal Uncertainty Principle: Manifold Tearing and the Topological Limits of Counterfactual Interventions

  Rui Wu, Hong Xie, Yongjun Li · 19 March 2026

  Judea Pearl's do-calculus provides a foundation for causal inference, but its translation to continuous generative models remains fraught with geometric challenges. We establish the fundamental limits of such interventions. We define the Counterfactual Event Horizon and prove the Manifold Tearing Th…

• Manifold-Matching Autoencoders

  Laurent Cheret, Vincent L\'etourneau, Isar Nejadgholi, Chris Drummond, Hussein Al Osman, Maia Fraser · 18 March 2026

  We study a simple unsupervised regularization scheme for autoencoders called Manifold-Matching (MMAE): we align the pairwise distances in the latent space to those of the input data space by minimizing mean squared error. Because alignment occurs on pairwise distances rather than coordinates, it can…

• Learning Topology-Driven Multi-Subspace Fusion for Grassmannian Deep Network

  Xuan Yu, Tianyang Xu · 18 March 2026

  Grassmannian manifold offers a powerful carrier for geometric representation learning by modelling high-dimensional data as low-dimensional subspaces. However, existing approaches predominantly rely on static single-subspace representations, neglecting the dynamic interplay between multiple subspace…

• $K-$means with leraned metrics

  Pablo Groisman, Matthieu Jonckheere, Jordan Serres, Mariela Sued · 17 March 2026

  We study the Fr\'echet {\it k-}means of a metric measure space when both the measure and the distance are unknown and have to be estimated. We prove a general result that states that the {\it k-}means are continuous with respect to the measured Gromov-Hausdorff topology. In this situation, we also p…

• Persistence Spheres: a Bi-continuous Linear Representation of Measures for Partial Optimal Transport

  Matteo Pegoraro · 17 March 2026

  We improve and extend persistence spheres, introduced in~\cite{pegoraro2025persistence}. Persistence spheres map an integrable measure $\mu$ on the upper half-plane, including persistence diagrams (PDs) as counting measures, to a function $S(\mu)\in C(\mathbb{S}^2)$, and the map is stable with respe…

• TopoCL: Topological Contrastive Learning for Medical Imaging

  Guangyu Meng, Pengfei Gu, Peixian Liang, John P. Lalor, Erin Wolf Chambers, Danny Z. Chen · 17 March 2026

  Contrastive learning (CL) has become a powerful approach for learning representations from unlabeled images. However, existing CL methods focus predominantly on visual appearance features while neglecting topological characteristics (e.g., connectivity patterns, boundary configurations, cavity forma…

• Topology-Preserving Data Augmentation for Ring-Type Polygon Annotations

  Sudip Laudari, Sang Hun Baek · 17 March 2026

  Geometric data augmentation is widely used in segmentation pipelines and typically assumes that polygon annotations represent simply connected regions. However, in structured domains such as architectural floorplan analysis, ring-type regions are often encoded as a single cyclic polygon chain connec…

• Interpretable Classification of Time Series Using Euler Characteristic Surfaces

  Salam Rabindrajit Luwang, Sushovan Majhi, Vishal Mandal, Atish J. Mitra, Md. Nurujjaman, Buddha Nath Sharma · 17 March 2026

  Persistent homology (PH) -- the conventional method in topological data analysis -- is computationally expensive, requires further vectorization of its signatures before machine learning (ML) can be applied, and captures information along only the spatial axis. For time series data, we propose Euler…

• Brain Tumor Classification from 3D MRI Using Persistent Homology and Betti Features: A Topological Data Analysis Approach on BraTS2020

  Faisal Ahmed · 17 March 2026

  Accurate and interpretable brain tumor classification from medical imaging remains a challenging problem due to the high dimensionality and complex structural patterns present in magnetic resonance imaging (MRI). In this study, we propose a topology-driven framework for brain tumor classification ba…

• Beyond Means: Topological Causal Effects under Persistent-Homology Ignorability

  Amir Saki, Usef Faghihi · 17 March 2026

  Average treatment effects (ATE) and conditional average treatment effects (CATE) are foundational causal estimands, but they target changes in expected outcomes and can miss treatment-induced changes in the shape of outcome distributions. A canonical failure mode occurs when control outcomes are uni…

• Local Urysohn Width: A Topological Complexity Measure for Classification

  Xin Li · 17 March 2026

  We introduce \emph{local Urysohn width}, a complexity measure for classification problems on metric spaces. Unlike VC dimension, fat-shattering dimension, and Rademacher complexity, which characterize the richness of hypothesis \emph{classes}, Urysohn width characterizes the topological-geometric co…

• SuperLocalMemory V3: Information-Geometric Foundations for Zero-LLM Enterprise Agent Memory

  Varun Pratap Bhardwaj · 17 March 2026

  Persistent memory is a central capability for AI agents, yet the mathematical foundations of memory retrieval, lifecycle management, and consistency remain unexplored. Current systems employ cosine similarity for retrieval, heuristic decay for salience, and provide no formal contradiction detection.…

• A Geometrically-Grounded Drive for MDL-Based Optimization in Deep Learning

  Ming Lei, Shufan Wu, Christophe Baehr · 16 March 2026

  This paper introduces a novel optimization framework that fundamentally integrates the Minimum Description Length (MDL) principle into the training dynamics of deep neural networks. Moving beyond its conventional role as a model selection criterion, we reformulate MDL as an active, adaptive driving …

• The Density of Cross-Persistence Diagrams and Its Applications

  Alexander Mironenko, Evgeny. Burnaev, Serguei Barannikov · 13 March 2026

  Topological Data Analysis (TDA) provides powerful tools to explore the shape and structure of data through topological features such as clusters, loops, and voids. Persistence diagrams are a cornerstone of TDA, capturing the evolution of these features across scales. While effective for analyzing in…

• Busemann Functions in the Wasserstein Space: Existence, Closed-Forms, and Applications to Slicing

  Cl\'ement Bonet, Elsa Cazelles, Lucas Drumetz, Nicolas Courty · 13 March 2026

  The Busemann function has recently found much interest in a variety of geometric machine learning problems, as it naturally defines projections onto geodesic rays of Riemannian manifolds and generalizes the notion of hyperplanes. As several sources of data can be conveniently modeled as probability …

• Beyond Scalars: Evaluating and Understanding LLM Reasoning via Geometric Progress and Stability

  Xinyan Jiang, Ninghao Liu, Di Wang, Lijie Hu · 12 March 2026

  Evaluating LLM reliability via scalar probabilities often fails to capture the structural dynamics of reasoning. We introduce TRACED, a framework that assesses reasoning quality through theoretically grounded geometric kinematics. By decomposing reasoning traces into Progress (displacement) and Stab…

• Topological descriptors of foot clearance gait dynamics improve differential diagnosis of Parkinsonism

  Jhonathan Barrios, Wolfram Erlhagen, Miguel F. Gago, Estela Bicho, Flora Ferreira · 9 March 2026

  Differential diagnosis among parkinsonian syndromes remains a clinical challenge due to overlapping motor symptoms and subtle gait abnormalities. Accurate differentiation is crucial for treatment planning and prognosis. While gait analysis is a well established approach for assessing motor impairmen…

• A Novel Patch-Based TDA Approach for Computed Tomography Imaging

  Dashti A. Ali, Aras T. Asaad, Jacob J. Peoples, Mohammad Hamghalam, Natalie Gangai, Richard K. G. Do, Alice C. Wei, Amber L. Simpson · 9 March 2026

  The development of machine learning (ML) models based on computed tomography (CT) imaging has been a major focus due to the promise that imaging holds for diagnosis, staging, and prognostication. These models often rely on the extraction of hand-crafted features, incorporating robust feature enginee…

• Topology Structure Optimization of Reservoirs Using GLMY Homology

  Yu Chen, Shengwei Wang, Hongwei Lin · 6 March 2026

  Reservoir is an efficient network for time series processing. It is well known that network structure is one of the determinants of its performance. However, the topology structure of reservoirs, as well as their performance, is hard to analyzed, due to the lack of suitable mathematical tools. In th…

• Discovering mathematical concepts through a multi-agent system

  Daattavya Aggarwal, Oisin Kim, Carl Henrik Ek, Challenger Mishra · 6 March 2026

  Mathematical concepts emerge through an interplay of processes, including experimentation, efforts at proof, and counterexamples. In this paper, we present a new multi-agent model for computational mathematical discovery based on this observation. Our system, conceived with research in mind, poses i…

• Axiomatic On-Manifold Shapley via Optimal Generative Flows

  Cenwei Zhang, Lin Zhu, Manxi Lin, Lei You · 6 March 2026

  Shapley-based attribution is critical for post-hoc XAI but suffers from off-manifold artifacts due to heuristic baselines. While generative methods attempt to address this, they often introduce geometric inefficiency and discretization drift. We propose a formal theory of on-manifold Aumann-Shapley …

• The Theory behind UMAP?

  David Wegmann · 5 March 2026

  In 2018, McInnes et al. introduced a dimensionality reduction algorithm called UMAP, which enjoys wide popularity among data scientists. Their work introduces a finite variant of a functor called the metric realization, based on an unpublished draft by Spivak. This draft contains many errors, most o…

• GeoTop: Advancing Image Classification with Geometric-Topological Analysis

  Mariem Abaach, Ian Morilla · 5 March 2026

  A fundamental challenge in diagnostic imaging is the phenomenon of topological equivalence, where benign and malignant structures share global topology but differ in critical geometric detail, leading to diagnostic errors in both conventional and deep learning models. We introduce GeoTop, a mathemat…