Topological and Geometric Data Analysis

Últimos artículos

• Stochastic Dynamics on Persistence Diagram Space via Reinforcement Learning

  Farzana Nasrin · 7 de agosto de 2026

  Persistence diagrams (PDs) provide stable and interpretable summaries of multiscale topological structure. While substantial progress has been made in the statistical analysis of PDs, existing literature often treats diagrams as static objects and provide limited frameworks for probabilistic modelin…

• Topological Simplification in Predictive Coding Networks

  Adam Shaw, Jiayu Li, Michael Sperling, Michael Kim, Alvin Jin · 5 de agosto de 2026

  We study the topology of learned representations in predictive coding networks (PCNs), a neuro-inspired bidirectional architecture, using a quantitative layer-wise persistent homology analysis. We train well-performing PCNs on a synthetic classification dataset ($\geq 99.9\%$ test accuracy) and on M…

• How fine a change can moments see? A scale law for detecting distribution shift, with a kernel calibration rule

  Adel Kaleche · 4 de agosto de 2026

  Detecting that a stream of high-dimensional embeddings has changed is usually framed as a choice of statistic. We give a scale law that constrains any moment-based choice and test it against topological alternatives. The law: certifying a feature of spatial scale eps carrying mass fraction f require…

• TAGTorch: A PyTorch Library for Geometry, Topology, and Symmetry-Aware Machine Learning

  Brendan Kennedy, Tegan Emerson, Gregory Roek, Emilie Purvine, Henry Kvinge · 3 de agosto de 2026

  Over the last decade, neural networks have been applied to an increasingly diverse range of applications, including data with rich geometric, topological, or symmetry-related structure. As a result, researchers have increasingly drawn inspiration from topology, algebra, and geometry. Despite this ri…

• Voronoi Histograms for Adaptive Vectorization of Expected Persistence Diagrams

  Kaifeng Zhang, Kai Ming Ting · 30 de julio de 2026

  Persistence Diagram (PD) is known to capture point cloud topology effectively, but its computation has high time complexity. Expected Persistence Diagram (EPD) has been developed to reduce the time cost by studying the topology of multiple subsets of a point cloud and it serves as a distribution of …

• Topological Data Analysis and Graph-Theoretic Approaches for Tennis Match Prediction

  Jake Schwaderer, Alexander Bastien, Omid Khormali, Alejandro Navarrete, Mia Pesavento, Angelika Elderbrook · 28 de julio de 2026

  We present two approaches for predicting tennis match outcomes using topological data analysis and graph theory on ATP singles matches from 2000-2025. The first method applies lower-star filtration to player competitive networks, extracting topological features through persistent homology using four…

• Queryable Self-Organizing Maps: A Database Abstraction for Topology-Driven Data Exploration

  Denis Mayr Lima Martins, Gottfried Vossen · 28 de julio de 2026

  Self-Organizing Maps (SOMs) have long been used as exploratory tools for high-dimensional data: they organize objects into a two-dimensional topology that reveals clusters, gradients, sparse regions, dense regions, and boundaries. Yet, in modern data systems, SOMs are typically trained and visualize…

• Do Sheaf Neural Networks Use Holonomy? A Measure--Intervene--Control Study

  Ankit Grover, R\'emi Bourgerie · 23 de julio de 2026

  Geometric architectures are often justified by internal mechanisms such as rotations, yet task performance alone cannot show whether those mechanisms drive predictions. Using sheaf neural networks (SNNs) as a testbed, we introduce the first basis-independent measurement of trained triangle-loop prod…

• Topology-Driven Clustering: Enhancing Performance with Betti Number Filtration

  Arghya Pratihar, Kushal Bose, Swagatam Das · 22 de julio de 2026

  Clustering aims at partitioning data points into groups of similar objects without knowing about the class labels. However, clustering datasets with complex geometric structures, such as nonconvex shapes, multiple scales, or intertwined manifolds, remains challenging for traditional algorithms that …

• Topological Signatures of Context-Level Reliability in TabPFN

  James Hu, Mahdi Ghelichi · 21 de julio de 2026

  TabPFN is a transformer-based foundation model for tabular prediction that performs inference without task-specific training by conditioning on a support set and query inputs. Despite its strong empirical performance, its internal behavior on structurally difficult tabular geometries remains poorly …

• Learning in Infinitesimal Non-Compositional Sketches

  Sridhar Mahadevan · 17 de julio de 2026

  This paper develops a categorical framework -- Learning in Infinitesimal Non-Compositional Sketches (LINCS) -- as the repair of non-compositionality: failures of diagrams to factor through quotient sketches lifted to the tangent category setting. Machine learning problems are specified as sketches: …

• Quantum Topological Data Encoding

  Adam Weso{\l}owski, Dimitrios Thanos, Daniel Leykam, Lirand\"e Pira · 16 de julio de 2026

  Many datasets encountered across a wide range of domains possess rich geometric and topological structure that is difficult to capture using conventional vector-based representations. Quantum machine learning offers the possibility of processing high-dimensional data in Hilbert spaces, but its pract…

• Depth-Efficient Quantum Topological Data Analysis for Regime-Specific Detection of Financial Stress

  Arul Rhik Mazumder, Shreyan Ronit Mazumder · 14 de julio de 2026

  We present, to our knowledge, the first adaptation of Pauli Correlation Encoding (PCE) to quantum topological data analysis, reformulating Betti number estimation as a depth-efficient variational optimization over a compressed qubit register. From a Takens embedding and Vietoris--Rips filtration of …

• Computation, Condensation, and the Incompleteness Between Them: A Coupled Foundation of Intelligence

  Xin Li · 10 de julio de 2026

  The theory of computation was built to answer Turing's question: what is effectively calculable by an unbounded, immortal, disembodied agent following rules? Intelligence answers a different question (nature's): what can a \emph{finite}, mortal, energy-limited agent do quickly enough to survive in a…

• Tangent classes of matroids and wonderful compactifications

  Ronnie Cheng, Shurui Liu, Guoxiong Gao · 8 de julio de 2026

  For every loopless matroid $M$ and every Feichtner--Yuzvinsky building set $\mathcal{G}$ containing the top flat, we construct an integral tangent class $T_{M,\mathcal{G}}^{\mathbb{Z}}\in K_{\mathbb{Z}}(M,\mathcal{G})$; in the realizable case it specializes to the class of the tangent bundle of the …

• On the convergence of graph Laplacians with a symmetric divergence

  Liane Xu · 8 de julio de 2026

  When analyzing a manifold learning algorithm for data lying on a smooth, compact, connected Riemannian submanifold $(\mathcal{M}, g)$ of $\mathbb{R}^d$, a key estimate for the geodesic distance $d_g$ is that there exists $K > 0$ such that $0 \leq d_g(p, q)^2 - \|p-q\|^2 \leq K d_g(p, q)^4$ for all $…

• Level-Crossing Density as a Mesh-Free High-Frequency Auxiliary Loss for Implicit Neural Representations

  Gunner Levi Howe · 8 de julio de 2026

  The Minkowski functionals of a field's excursion sets -- area, boundary measure, and Euler characteristic -- describe its level-set morphology; the Euler characteristic is the cheapest handle on topology. We derive smooth Monte-Carlo estimators for all three of a continuous neural field, evaluated a…

• Information-Geometric Superposed Vowel Evaluation: Part 1. Moraic Syllabary (Japanese)

  Yusei Tamura, Shigekazu Ishihara, Ken Ito · 7 de julio de 2026

  This paper explains the principles and provides examples of a new method for distinguishing between FAKE human speech synthesized by generative AI and natural speech. Since synthetic speech is generated based on information from a limited set of training spectra, the variety of vowels - which are ke…

• Complexity of Normalized Persistence Problems for Topological Data Analysis and Local Hamiltonians

  Dominic Lowe, M. S. Kim, Roberto Bondesan, Ryu Hayakawa · 7 de julio de 2026

  Topological data analysis (TDA) is a machine learning technique that uses topology to extract patterns from data and has shown the potential to exhibit quantum advantage. A key concept in TDA is persistent homology, which measures the robustness of topological information at different lengthscales. …

• Is the Geometry Doing the Work? An Operating-Point Audit of Hierarchy in Hyperbolic Vision-Language Models

  Jaeyoung Kim, Eunseok Kim, Dongsuk Jang · 7 de julio de 2026

  Whether a hyperbolic representation model uses its geometry cannot be read off its curvature parameter: what matters is the dimensionless operating point $\sqrt{c}\rho$ and whether the radial and cone machinery is active there. We develop a battery of necessary-condition diagnostics and audit three …

• A simplex-based measure of symmetry

  Egor Bakaev, Amir Yehudayoff · 7 de julio de 2026

  For compact convex sets $L,K \subset \mathbb{R}^n$, denote by $\lambda_K(L)$ the smallest size of a homothet of $K$ that contains $L$. We define a measure of symmetry based on the $n$-simplex $\Delta = \Delta^n \subset \mathbb{R}^n$ as the ratio \[ \rho_\Delta(L):=\frac{\lambda_{-\Delta}(L)}{\lambda…

• Topological data analysis using persistent discrete homology

  Chris Kapulkin, Nathan Kershaw · 7 de julio de 2026

  We propose persistent discrete homology as a tool for topological data analysis and discuss its advantages over the existing methods. In particular, we provide empirical evidence that persistent discrete homology is more noise-resistant than persistent homology of the Vietoris-Rips complex for data …

• Topological Void Analysis A Mathematical Framework for Systematic Technical Innovation Discovery in Knowledge Spaces

  Kris Pan · 2 de julio de 2026

  Identifying where to innovate in a dense technical domain - such as operating systems or hardware/software co-design - is fundamentally a search problem in a high-dimensional knowledge space. Existing approaches rely on keyword search, citation proximity, or human intuition, none of which formalise …

• Low-dimensional topology of deep neural networks

  Junyu Ren, Lek-Heng Lim · 1 de julio de 2026

  We study layered models, including feedforward networks, ResNets, and transformers, by limiting each layer to a width of $d = 3$, i.e., $\mathbb{R}^3$ as representation space. This allows us to track how a neural network changes low-dimensional topological invariants through its layers. Just about a…

• Sample Complexity of Scientific Discovery: PAC Learnability of Compositional Function Trees

  \c{S}uayp Talha Kocabay, Talha R\"uzgar Akku\c{s}, Kerem Yal\c{c}{\i}n · 30 de junio de 2026

  Scientific discovery via symbolic regression is often viewed as statistically and computationally intractable because the hypothesis space of expressions grows combinatorially with depth. This paper revisits the statistical side through the lens of PAC learning, focusing on compositional function tr…