Topological and Geometric Data Analysis

Últimos artículos

• The Shape of Adversarial Influence: Characterizing LLM Latent Spaces with Persistent Homology

  Aideen Fay, In\'es Garc\'ia-Redondo, Qiquan Wang, Haim Dubossarsky, Anthea Monod · 27 de abril de 2026

  Existing interpretability methods for Large Language Models (LLMs) predominantly capture linear directions or isolated features. This overlooks the high-dimensional, relational, and nonlinear geometry of model representations. We apply persistent homology (PH) to characterize how adversarial inputs …

• Spectral Kernel Dynamics for Planetary Surface Graphs: Distinction Dynamics and Topological Conservation

  Jnaneshwar Das · 24 de abril de 2026

  The spectral kernel field equation R[k] = T[k] lacks a conservation-law analog. We prove (i) the fixed-point flow is strictly volume-expanding (tr DF > 0), precluding automatic conservation, and (ii) the conservation deficit per mode equals the Hessian stability margin exactly: D_m = -Delta'. Closin…

• Fixation Sequences as Time Series: A Topological Approach to Dyslexia Detection

  Marius Huber, David R. Reich, Lena A. J\"ager · 24 de abril de 2026

  Persistent homology, a method from topological data analysis, extracts robust, multi-scale features from data. It produces stable representations of time series by applying varying thresholds to their values (a process known as a \textit{filtration}). We develop novel filtrations for time series and…

• Relocation of compact sets in $\mathbb{R}^n$ by diffeomorphisms and linear separability of datasets in $\mathbb{R}^n$

  Xiao-Song Yang, Xuan Zhou, Qi Zhou · 24 de abril de 2026

  Relocation of compact sets in an $n$-dimensional manifold by self-diffeomorphism is of its own interest as well as significant potential applications to data classification in data science. This paper presents a theory for relocating a finite number of compact sets in $\mathbb{R}^n$ to be relocated …

• Lorentz Framework for Semantic Segmentation

  Zahid Hasan, Masud Ahmed, Nirmalya Roy · 21 de abril de 2026

  Semantic segmentation in hyperbolic space enables compact modeling of hierarchical structure while providing inherent uncertainty quantification. Prior approaches predominantly rely on the Poincar\'e ball model, which suffers from numerical instability, optimization, and computational challenges. We…

• Contraction and Hourglass Persistence for Learning on Graphs, Simplices, and Cells

  Mattie Ji, Indradyumna Roy, Vikas Garg · 21 de abril de 2026

  Persistent homology (PH) encodes global information, such as cycles, and is thus increasingly integrated into graph neural networks (GNNs). PH methods in GNNs typically traverse an increasing sequence of subgraphs. In this work, we first expose limitations of this inclusion procedure. To remedy thes…

• Persistence-Augmented Neural Networks

  Elena Xinyi Wang, Arnur Nigmetov, Dmitriy Morozov · 21 de abril de 2026

  Topological Data Analysis (TDA) provides tools to describe the shape of data, but integrating topological features into deep learning pipelines remains challenging, especially when preserving local geometric structure rather than summarizing it globally. We propose a persistence-based data augmentat…

• Modern Structure-Aware Simplicial Spatiotemporal Neural Network

  Zhaobo Hu, Vincent Gauthier, Mehdi Naima · 20 de abril de 2026

  Spatiotemporal modeling has evolved beyond simple time series analysis to become fundamental in structural time series analysis. While current research extensively employs graph neural networks (GNNs) for spatial feature extraction with notable success, these networks are limited to capturing only p…

• Classification of Epileptic iEEG using Topological Machine Learning

  Sunia Tanweer, Narayan Puthanmadam Subramaniyam, Firas A. Khasawneh · 15 de abril de 2026

  Epileptic seizure detection from EEG signals remains challenging due to the high dimensionality and nonlinear, potentially stochastic, dynamics of neural activity. In this work, we investigate whether features derived from topological data analysis (TDA) can improve the classification of brain state…

• Learning Geometry and Topology via Multi-Chart Flows

  Hanlin Yu, S{\o}ren Hauberg, Marcelo Hartmann, Arto Klami, Georgios Arvanitidis · 14 de abril de 2026

  Real world data often lie on low-dimensional Riemannian manifolds embedded in high-dimensional spaces. This motivates learning degenerate normalizing flows that map between the ambient space and a low-dimensional latent space. However, if the manifold has a non-trivial topology, it can never be corr…

• NSFL: A Post-Training Neuro-Symbolic Fuzzy Logic Framework for Boolean Operators in Neural Embeddings

  Vladi Vexler, Ofer Idan, Gil Lederman, Dima Sivov · 14 de abril de 2026

  Standard dense retrievers lack a native calculus for multi-atom logical constraints. We introduce Neuro-Symbolic Fuzzy Logic (NSFL), a framework that adapts formal t-norms and t-conorms to neural embedding spaces without requiring retraining. NSFL operates as a first-order hybrid calculus: it anchor…

• Learning Discrete Diffusion of Graphs via Free-Energy Gradient Flows

  Dario Rancati, Jan Maas, Francesco Locatello · 14 de abril de 2026

  Diffusion-based models on continuous spaces have seen substantial recent progress through the mathematical framework of gradient flows, leveraging the Wasserstein-2 (${W}_2$) metric via the Jordan-Kinderlehrer-Otto (JKO) scheme. Despite the increasing popularity of diffusion models on discrete space…

• The Wasserstein transform

  Kun Jin, Facundo M\'emoli, Zane Smith, Zhengchao Wan · 14 de abril de 2026

  We introduce the Wasserstein Transform (WT), a general unsupervised framework for updating distance structures on given data sets with the purpose of enhancing features and denoising. Our framework represents each data point by a probability measure reflecting the neighborhood structure of the point…

• Diffusion Processes on Implicit Manifolds

  Victor Kawasaki-Borruat, Clara Grotehans, Pierre Vandergheynst, Adam Gosztolai · 9 de abril de 2026

  High-dimensional data are often modeled as lying near a low-dimensional manifold. We study how to construct diffusion processes on this data manifold in the implicit setting. That is, using only point cloud samples and without access to charts, projections, or other geometric primitives. Our main co…

• A Persistent Homology Design Space for 3D Point Cloud Deep Learning

  Prachi Kudeshia, Jiju Poovvancheri, Amr Ghoneim, Dong Chen · 7 de abril de 2026

  Persistent Homology (PH) offers stable, multi-scale descriptors of intrinsic shape structure by capturing connected components, loops, and voids that persist across scales, providing invariants that complement purely geometric representations of 3D data. Yet, despite strong theoretical guarantees an…

• A Spectral Framework for Multi-Scale Nonlinear Dimensionality Reduction

  Zeyang Huang, Angelos Chatzimparmpas, Thomas H\"ollt, Takanori Fujiwara · 6 de abril de 2026

  Dimensionality reduction (DR) is characterized by two longstanding trade-offs. First, there is a global-local preservation tension: methods such as t-SNE and UMAP prioritize local neighborhood preservation, yet may distort global manifold structure, while methods such as Laplacian Eigenmaps preserve…

• Tensor Computation of Euler Characteristic Functions and Transforms

  Jessi Cisewski-Kehe, Brittany Terese Fasy, Alexander McCleary, Eli Quist · 6 de abril de 2026

  The weighted Euler characteristic transform (WECT) and Euler characteristic function (ECF) have proven to be useful tools in a variety of applications. However, current methods for computing these functions are either not optimized for GPU computation or do not scale to higher-dimensional settings. …

• Denoising distances beyond the volumetric barrier

  Han Huang, Pakawut Jiradilok, Elchanan Mossel · 2 de abril de 2026

  We study the problem of reconstructing the latent geometry of a $d$-dimensional Riemannian manifold from a random geometric graph. While recent works have made significant progress in manifold recovery from random geometric graphs, and more generally from noisy distances, the precision of pairwise d…

• Diffusion Maps is not Dimensionality Reduction

  Julio Candanedo, Alejandro Pati\~no · 31 de marzo de 2026

  Diffusion maps (DMAP) are often used as a dimensionality-reduction tool, but more precisely they provide a spectral representation of the intrinsic geometry rather than a complete charting method. To illustrate this distinction, we study a Swiss roll with known isometric coordinates and compare DMAP…

• Graph Vector Field: A Unified Framework for Multimodal Health Risk Assessment from Heterogeneous Wearable and Environmental Data Streams

  Silvano Coletti, Francesca Fallucchi · 31 de marzo de 2026

  Digital health research has advanced dynamic graph-based disease models, topological learning on simplicial complexes, and multimodal mixture-of-experts architectures, but these strands remain largely disconnected. We propose Graph Vector Field (GVF), a framework that models health risk as a vector-…

• Persistence diagrams of random matrices via Morse theory: universality and a new spectral diagnostic

  Matthew Loftus · 31 de marzo de 2026

  We prove that the persistence diagram of the sublevel set filtration of the quadratic form f(x) = x^T M x restricted to the unit sphere S^{n-1} is analytically determined by the eigenvalues of the symmetric matrix M. By Morse theory, the diagram has exactly n-1 finite bars, with the k-th bar living …

• Binary Expansion Group Intersection Network

  Sicheng Zhou, Kai Zhang · 27 de marzo de 2026

  Conditional independence is central to modern statistics, but beyond special parametric families it rarely admits an exact covariance characterization. We introduce the binary expansion group intersection network (BEGIN), a distribution-free graphical representation for multivariate binary data and …

• TopoPilot: Reliable Conversational Workflow Automation for Topological Data Analysis and Visualization

  Nathaniel Gorski, Shusen Liu, Bei Wang · 27 de marzo de 2026

  Recent agentic systems demonstrate that large language models can generate scientific visualizations from natural language. However, reliability remains a major limitation: systems may execute invalid operations, introduce subtle but consequential errors, or fail to request missing information when …

• Quotient Geometry and Persistence-Stable Metrics for Swarm Configurations

  Mark M. Bailey · 20 de marzo de 2026

  Swarm and constellation reconfiguration can be viewed as motion of an unordered point configuration in an ambient space. Here, we provide persistence-stable, symmetry-invariant geometric representations for comparing and monitoring multi-agent configuration data. We introduce a quotient formation sp…

• $K-$means with learned metrics

  Pablo Groisman, Matthieu Jonckheere, Jordan Serres, Mariela Sued · 20 de marzo de 2026

  We study the Fr\'echet $k-$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 $k-$means are continuous with respect to the measured Gromov-Hausdorff topology. In this situation, we also prove a s…