Topological and Geometric Data Analysis

Últimos artículos

• ToMAToMP: Robust and Multi-Parameter Topological Clustering

  Ludo Andrianirina, Mathieu Carri\`ere · 15 de mayo de 2026

  Topological clustering, and its main algorithm ToMATo, is a clustering method from Topological Data Analysis (TDA) which has been applied successfully in several applications during the last few years. This is due to its high versatility, as clusters are detected from the persistent components in th…

• TopoPrimer: The Missing Topological Context in Forecasting Models

  Zara Zetlin, Kayhan Moharreri, Maria Safi · 15 de mayo de 2026

  We introduce TopoPrimer, a framework that makes the global topological structure of the series population an explicit input to any forecasting model. TopoPrimer improves accuracy across diverse domains, stabilizes forecasts under seasonal demand spikes, and closes the cold-start gap. Precomputed…

• Distance-Matrix Wasserstein Statistics for Scalable Gromov--Wasserstein Learning

  Ao Xu, Tieru Wu · 15 de mayo de 2026

  Gromov--Wasserstein (GW) distances compare graphs, shapes, and point clouds through internal distances, without requiring a common coordinate system. This invariance is powerful, but discrete GW is a nonconvex quadratic optimal transport problem and is difficult to estimate at scale. We propose \emp…

• Beyond Explained Variance: A Cautionary Tale of PCA

  Gionni Marchetti · 14 de mayo de 2026

  We address shortcomings of principal component analysis (PCA) for visualizing high-dimensional data lying on a nonlinear low-dimensional manifold via two-dimensional scatterplots, focusing on a fossil teeth dataset from the early mammalian insectivore Kuehneotherium. While the PCA scatterplot report…

• From Baselines to Transport Geodesics: Axiomatic Attribution via Optimal Generative Flows

  Cenwei Zhang, Lin Zhu, Manxi Lin, Lei You · 14 de mayo de 2026

  Feature attributions often hide a critical modeling choice: they explain a prediction along a counterfactual path from a reference state to an input. Different baselines, interpolations, and generative trajectories define different paths and can therefor produce different explanations. We study this…

• Towards Scalable Persistence-Based Topological Optimization

  Abderrahim Bendahi, Alexandre Duplessis, Arnaud Fickinger · 13 de mayo de 2026

  Persistence-based topological optimization deforms a point cloud $X \subset \mathbb{R}^d$ by minimizing objectives of the form $L(X) = \ell(\mathrm{Dgm}(X))$, where $\mathrm{Dgm}(X)$ is a persistence diagram. In practice, optimization is limited by two coupled issues: persistent homology is typicall…

• Chordless cycle filtrations for dimensionality detection in complex networks via topological data analysis

  Aina Ferr\`a Marc\'us, Robert Jankowski, Meritxell Vila Mi\~nana, Carles Casacuberta, M. \'Angeles Serrano · 11 de mayo de 2026

  Many complex networks, ranging from social to biological systems, exhibit structural patterns consistent with an underlying hyperbolic geometry. Revealing the dimensionality of this latent space can disentangle the structural complexity of communities, impact efficient network navigation, and fundam…

• Structural Learning Theory: A Metric-Topology Factorization Approach

  Xin Li · 8 de mayo de 2026

  Learning in structured, multi-context, or non-stationary environments involves two orthogonal difficulties. The first is \emph{metric}: once the correct context is known, how hard is prediction within it? This is the domain of Statistical Learning Theory (SLT). The second is \emph{structural}: how m…

• Topological Signatures of Grokking

  Yifan Tang, Qiquan Wang, In\'es Garc\'ia-Redondo, Anthea Monod · 8 de mayo de 2026

  We study the grokking phenomenon through the lens of topology. Using persistent homology on point clouds derived from the embedding matrices of a range of models trained on modular arithmetic with varying primes, we identify a clear and consistent topological signature of grokking: a sharp increase …

• Consistent Geometric Deep Learning via Hilbert Bundles and Cellular Sheaves

  Kartik Tandon, Julian Gould, Tanishq Bhatia, Francesca Dominici, Alejandro Ribeiro, Claudio Battiloro · 8 de mayo de 2026

  Modern deep learning architectures increasingly contend with sophisticated signals that are natively infinite-dimensional, such as time series, probability distributions, or operators, and are defined over irregular domains. Yet, a unified learning theory for these settings has been lacking. To star…

• Empirical Evidence for Simply Connected Decision Regions in Image Classifiers

  Arjhun Swaminathan, Mete Akg\"un · 8 de mayo de 2026

  Understanding the topology of decision regions is central to explaining the inner workings of deep neural networks. Prior empirical work has provided evidence that these regions are path connected. We study a stronger topological question: whether closed loops inside a decision region can be contrac…

• Local-Order Auxiliary Losses Can Improve Autoencoder Reconstruction

  Harvey Dam, Martin Burtscher, Tripti Agarwal, Ganesh Gopalakrishnan · 8 de mayo de 2026

  Mean-squared error is the default objective for training autoencoders, yet compressed reconstructions often depend not only on pointwise accuracy but also on preserving local spatial order. We study whether structural auxiliary losses can improve, rather than trade off against, MSE in finite-capacit…

• A Mean Curvature Approach to Boundary Detection: Geometric Insights for Unsupervised Learning

  Alexandre L. M. Levada · 7 de mayo de 2026

  Accurate boundary detection in high-dimensional data remains a central challenge in unsupervised learning, particularly in the presence of non-linear structures and heterogeneous densities. In this work, we introduce Mean Curvature Boundary Points (MCBP), a novel geometric framework grounded in Geom…

• Local Intrinsic Dimension Unveils Hallucinations in Diffusion Models

  Bartlomiej Sobieski, Matthew Tivnan, Dawid P{\l}udowski, Micha{\l} Jan W{\l}odarczyk, Pengfei Jin, Przemyslaw Biecek, Quanzheng Li · 7 de mayo de 2026

  Diffusion models are prone to generating structural hallucinations - samples that match the statistical properties of the training data yet defy underlying structural rules, resulting in anomalies like hands with more than five fingers. Recent research studied this failure mode from several viewpoin…

• Global and Local Topology-Aware Attention with Persistent Homology and Euler Biases for Time-Series Forecasting

  Usef Faghihi, Amir Saki · 6 de mayo de 2026

  Scientific time series often encode predictive geometric structure, including connectivity, cycles, shell-like geometry, directional changes, and nonlinear neighborhoods, that standard dot-product attention does not explicitly represent. We introduce a topology-aware attention framework that adds su…

• A Closed-Form Adaptive-Landmark Kernel for Certified Point-Cloud and Graph Classification

  Sushovan Majhi, Atish Mitra, \v{Z}iga Virk, Pramita Bagchi · 6 de mayo de 2026

  We introduce PALACE (Persistence Adaptive-Landmark Analytic Classification Engine), the data-adaptive companion to PLACE, paying a small cross-validation tier on three knobs (budget, radii, bandwidth; $\leq 5$ choices each). A cover-theoretic core (Lebesgue-number criterion on the landmark cover) yi…

• Predicting Euler Characteristics and Constructing Topological Structure Using Machine Learning Techniques

  Gyunghun Yu (Department of Physics, Kyung Hee University, Seoul, South Korea), Seong Min Park (Department of Physics, Kyung Hee University, Seoul, South Korea), Han Gyu Yoon (Department of Physics, Kyung Hee University, Seoul, South Korea), Tae Jung Moon (Department of Physics, Kyung Hee University, Seoul, South Korea), Jun Woo Choi (Center for Spintronics, Korea Institute of Science and Technology, Seoul, South Korea), Hee Young Kwon (Center for Spintronics, Korea Institute of Science and Technology, Seoul, South Korea), Changyeon Won (Department of Physics, Kyung Hee University, Seoul, South Korea) · 6 de mayo de 2026

  This study proposes a novel approach to extract topological properties, specifically the Euler characteristic, from input images using neural networks without relying on large pre-existing datasets but with a single geometric image. Inspired by solid-state physics, where topological properties of ma…

• A Closed-Form Persistence-Landmark Pipeline for Certified Point-Cloud and Graph Classification

  Sushovan Majhi, Atish Mitra, \v{Z}iga Virk, Pramita Bagchi · 5 de mayo de 2026

  We introduce PLACE (Persistence-Landmark Analytic Classification Engine), a closed-form pipeline for classifying point clouds and graphs through their persistent-homology signatures. Three quantitative guarantees -- a margin-based excess-risk rate, a closed-form descriptor-selection rule, and a per-…

• Weisfeiler Lehman Test on Combinatorial Complexes: Generalized Expressive Power of Topological Neural Networks

  Jiawen Chen, Qi Shao, Duxin Chen, Wenwu Yu · 4 de mayo de 2026

  Combinatorial complexes have unified set-based (e.g., graphs, hypergraphs) and part-whole (e.g., simplicial, cellular complexes) structures into a common topological framework. Existing topological neural networks and Weisfeiler-Lehman variants remain fragmented, lacking a unified theoretical founda…

• Monitoring Neural Training with Topology: A Footprint-Predictable Collapse Index

  Alexander Kalinowski · 1 de mayo de 2026

  Representational collapse, where embeddings become anisotropic and lose multi-scale structure, can erode downstream performance long before performance metrics react. We propose an online, topology-aware monitor for evolving neural representations that couples Modular Morse Homology Maintenance (MMH…

• How Hard Is Continuous Clustering? Lower Bounds from the Existential Theory of the Reals

  Angshul Majumdar · 1 de mayo de 2026

  This paper studies the computational difficulty of clustering problems that are defined directly on a continuous probability density. Rather than working with finite samples, we assume the density is given as a polynomial and ask whether it contains certain cluster structures. Four natural questions…

• Topology-Aware Representation Alignment for Semi-Supervised Vision-Language Learning

  Junwon You, Mihyun Jang, Sangwoo Mo, Jae-Hun Jung · 30 de abril de 2026

  Vision-language models have shown strong performance, but they often generalize poorly to specialized domains. While semi-supervised vision-language learning mitigates this limitation by leveraging a small set of labeled image-text pairs together with abundant unlabeled images, existing methods rema…

• DiRe-RAPIDS: Topology-faithful dimensionality reduction at scale

  Alexander Kolpakov, Igor Rivin · 29 de abril de 2026

  Dimensionality reduction methods such as UMAP and t-SNE are central tools for visualising high-dimensional data, but their local-neighborhood objectives can preserve sampling noise while distorting global topology. We show that standard local metrics reward this noise memorisation: top-performing em…

• The Shape of Attraction in UMAP: Exploring the Embedding Forces in Dimensionality Reduction

  Mohammad Tariqul Islam, Jason W. Fleischer · 28 de abril de 2026

  Uniform manifold approximation and projection (UMAP) is among the most popular neighbor embedding methods. The method samples pairs of point indices according to similarities in the high-dimensional space, and applies attractive and repulsive forces to their coordinates in the low-dimensional embedd…

• Geodesics in the Deep Linear Network

  Alan Chen · 28 de abril de 2026

  We derive a general system of ODEs and associated explicit solutions in a special case for geodesics between full rank matrices in the deep linear network geometry. In the process, we find horizontal straight lines in the invariant balanced manifold that remain geodesics under Riemannian submersion.…