Topological and Geometric Data Analysis

Neueste Paper

• Learning Topology-Aware Representations via Test-Time Adaptation for Anomaly Segmentation

  Ali Zia, Usman Ali, Abdul Rehman, Umer Ramzan, Kang Han, Muhammad Faheem, Shahnawaz Qureshi, Wei Xiang · 29. Juni 2026

  Test-time adaptation (TTA) has emerged as a promising paradigm for mitigating distribution shifts in deep models. However, existing TTA approaches for anomaly segmentation remain limited by their reliance on pixel-level heuristics, such as confidence thresholding or entropy minimisation, which fail …

• TopoCast: A Topological Fidelity Framework for Evaluating Transformer-Based Time Series Forecasting

  Sandeepa Weerasekara, Sandareka Wickramanayake · 25. Juni 2026

  Deep learning-based models have achieved state-of-the-art performance in Time Series Forecasting (TSF), yet their evaluation remains dominated by pointwise error metrics such as Mean Squared Error (MSE), which quantify numerical accuracy but overlook structural properties of the forecast signal, inc…

• Exact and Approximate Range Queries for Efficient Ball Mapper Construction

  Jay-Anne Bulauan, John Rick Manzanares · 23. Juni 2026

  Ball Mapper is a tool in topological data analysis that summarizes a finite metric dataset by covering it with metric balls and encoding their overlaps as a graph. Its construction requires repeated fixed-radius range queries, which can become computationally expensive for large or high-dimensional …

• Measuring What Persists: Conditioning Mechanisms and a Geometric Framework for AI Agent Identity

  Andrew Tanner · 23. Juni 2026

  AI agents in long-context applications drift from their specified identity. Current methods detect this only after qualitative degradation is visible. We present a geometric framework for measuring identity structure using $\sqrt{\mathrm{JSD}}$ metric spaces and magnitude homology from enriched cate…

• Topological Data Analysis for High-Dimensional Dynamic Process Monitoring

  Angan Mukherjee, Tyler A. Soderstrom, Michael J. Kurtz, Victor M. Zavala · 19. Juni 2026

  Real-time process monitoring requires methods that extract actionable information from high-dimensional time-series data. In this work, we present a new approach for process monitoring that combines tools of topological data analysis (TDA) and machine learning. In the proposed approach, we represent…

• Tracking Representation Dynamics in Large Language Models with Persistent Homology

  Naman Malhotra, Jay Ambadkar, Abhinav Gupta, Kushal Kasivel, Abbas Schwarz, Kamillo Ferry, Anthea Monod · 19. Juni 2026

  Large language models are commonly aligned through supervised fine-tuning, yet little is known about how their internal representations evolve during this process. We study alignment dynamics using persistent homology by tracking the topology of activation spaces throughout fine-tuning. Across four …

• Provable quantum speedups for computing persistence in topological data analysis

  Casper Gyurik, Alexander Schmidhuber, Robbie King, Vedran Dunjko, Ryu Hayakawa · 18. Juni 2026

  Topological data analysis (TDA) aims to extract noise-robust features from a data set by examining the number and persistence of holes in its topology. We provide an efficient quantum algorithm for a computational problem closely related to a core task in TDA -- determining whether a given hole pers…

• Unreduced Persistence Diagrams for Topological Machine Learning

  Nicole Abreu, Parker B. Edwards, Francis Motta · 18. Juni 2026

  Supervised machine learning pipelines trained on features derived from persistent homology have been experimentally observed to ignore much of the information contained in a persistence diagram. Computing persistence diagrams is often the most computationally demanding step in such a pipeline, howev…

• Fisher Width: A Geometric Measure of Complexity on Statistical Manifolds

  Vu Khac Ky · 18. Juni 2026

  Gaussian width is a central geometric complexity measure in high-dimensional probability, compressed sensing, convex optimization, and learning theory. It quantifies the average extent of a set along random directions, thereby capturing the effective dimension of constraint sets, hypothesis classes,…

• Non-negative Matrix Factorisation with Topological Regularisation

  Matias de Jong van Lier, Shizuo Kaji, Keunsu Kim · 17. Juni 2026

  We investigate the learning of interpretable bases in non-negative matrix factorisation (NMF) by regularising the topology of the learned basis functions. Our approach is motivated by the observation that many data modalities can be viewed as non-negative functions on a structured domain, where the …

• A homotopy-type-theoretic generalization of neurosymbolic inference

  Fernando Zhapa-Camacho, Robert Hoehndorf · 17. Juni 2026

  A wide range of neurosymbolic (NeSy) systems compute one functional: a belief-weighted sum of a logical quantity over a space of $\sigma$-structures, of which weighted model counting, fuzzy logic, and probabilistic logic are special cases. This account is built on sets, and a set deliberately forget…

• The Data Manifold under the Microscope

  Marios Koulakis, Constantin Seibold · 16. Juni 2026

  A significant gap exists between theory and practice in deep learning. Generalization and approximation error bounds are often derived for simplified models or are too loose to be informative. Many rely on the manifold hypothesis and on geometric regularity such as intrinsic dimension, curvature, an…

• Analytic Torsion and Spectral Gap Capture Persistent-Laplacian Performance

  Jernej Grlj, Aaron D. Lauda · 16. Juni 2026

  While persistent Laplacians (PL) offer a richer geometric representation of data than persistent homology, utilizing their full eigenspectrum for learning tasks is often hampered by high dimensionality and the ``varying length'' problem across different filtration scales. We propose a compact spectr…

• PHINN: Persistent Homology Inspired Neural Network for Rare-Event Time Series Generation

  Emre Yusuf, Ren Takahashi, Jayabrata Bhaduri · 16. Juni 2026

  Rare events in time series are critical to model but hard to learn due to data scarcity. Current generative models struggle with extreme values. We observe that rare events leave distinct topological fingerprints - transitions in Betti numbers from point-cloud embeddings - that are more stable and d…

• Learning Topological Representations for Molecular Dynamics

  Dominik Geng, Florian Graf, Martin Uray, Roland Kwitt · 16. Juni 2026

  Molecular dynamics (MD) simulations generate trajectories in a high-dimensional configuration space whose analysis critically depends on molecular descriptors, typically handcrafted observables or learned kinetic embeddings. Designing descriptors that are both expressive and broadly applicable, howe…

• Topological Flow Matching

  Kacper Wyrwal, \.Ismail \.Ilkan Ceylan, Alexander Tong · 16. Juni 2026

  Flow matching is a powerful generative modeling framework, valued for its simplicity and strong empirical performance. However, its standard formulation treats signals on structured spaces, such as fMRI data on brain graphs, as points in Euclidean space, overlooking the rich topological features of …

• Discovery under Hypothesis Redundancy: A Geometric Theory of Discovery Bottlenecks

  Li Xia, Baoxun Wang · 15. Juni 2026

  Scientific discovery saturates when new hypotheses cease to provide independent information, even if the nominal hypothesis space remains large. We study hybrid discovery systems that combine structured local search with LLM-generated non-local proposals and pose the Search Compression Hypothesis: n…

• From Persistence to Survival: Hypothesis Testing, Effect Sizes and Vectorisation for Topological Features

  Juliette Murris, Bernadette Stolz, Karsten Borgwardt · 11. Juni 2026

  Persistence diagrams are common representations in topological data analysis, but they do not naturally live in a vector space, and the statistical tools developed for comparing them have largely evolved separately from those used for downstream prediction. We introduce STRAND (Survival Topological …

• Moonshine: An Autonomous Mathematical Research Agent Centered on Conjecture Generation

  Xiaoyang Chen, Xiang Jiang · 10. Juni 2026

  Moonshine is an autonomous agent whose central objective is to generate mathematical conjectures. Its core capability is to extract structure from classical problems, distill new concepts, and formulate conjectures of mathematical significance. Rather than treating the solution of a single propositi…

• $k$-Nearest Neighbors in Gromov--Wasserstein Space

  Kaitlyn Hohmeier, Nicolas Fraiman, Caroline Moosmueller · 10. Juni 2026

  The Gromov--Wasserstein (GW) distance provides a framework for comparing metric measure spaces, regardless of their underlying structure or geometry. For network-based data, it enables direct comparisons of graphs with different numbers of nodes, without requiring an embedding or other abstraction. …

• MAD: Manifold Attracted Diffusion

  Dennis Elbr\"achter, Giovanni S. Alberti, Matteo Santacesaria · 10. Juni 2026

  Score-based diffusion models are a highly effective method for generating samples from a distribution of images. We consider scenarios where the training data comes from a noisy version of the target distribution, and present an efficiently implementable modification of the inference procedure to ge…

• Topological Neural Operators

  Lennart Bastian, Samuel Leventhal, Mustafa Hajij, Tolga Birdal · 9. Juni 2026

  We introduce Topological Neural Operators (TNOs), a principled framework for operator learning on cell complexes that lifts neural operators (NOs) from functions on points and/or edges to topological domains. TNOs represent data as features defined on cells of varying dimension and model their inter…

• The Topological Dual of a Dataset: A Logic-to-Topology Encoding for AlphaGeometry-Style Data

  Anthony Bordg · 9. Juni 2026

  AlphaGeometry represents a milestone in neuro-symbolic reasoning, yet its architecture faces a log-linear scaling bottleneck within its symbolic deduction engine that limits its efficiency as problem complexity increases. Recent technical reports suggest that current domain-specific languages may be…

• Barycentric Projections of Optimal Transport Plans on Riemannian Manifolds

  Kisung You · 9. Juni 2026

  Optimal transport couplings are probabilistic objects, while many learning pipelines require deterministic maps. In Euclidean space, barycentric projection converts a coupling into a map by taking conditional expectations, but on a Riemannian manifold curvature and cut loci make this operation nontr…

• MS-COOT: Comparing Morse-Smale Complexes with Co-Optimal Transport

  Guangyu Meng, Mingzhe Li, Erin Wolf Chambers · 9. Juni 2026

  Understanding and comparing structures in scalar fields is a central challenge in scientific visualization, with applications ranging from feature analysis to temporal and structural comparison. The Morse-Smale (MS) complex provides a natural representation by decomposing a scalar field into regions…