Topological and Geometric Data Analysis

Últimos artículos

• Constructing VAE Latent Spaces with Prescribed Topology

  Jilles S. van Hulst, Jakub M. Tomczak, W. P. M. H. Heemels, Duarte J. Antunes · 8 de junio de 2026

  Variational autoencoders (VAEs) learn low-dimensional latent representations of high-dimensional data. When the data lies on a manifold with non-Euclidean topology, the standard Gaussian prior introduces a topological mismatch that degrades reconstruction quality and prevents faithful representation…

• On Out-of-sample Embedding in UMAP

  Mohammad Tariqul Islam, Jason W. Fleischer · 4 de junio de 2026

  Neighbor embedding algorithms reveal correlations in high-dimensional data by constructing an equivalent graph representation in a lower-dimensional space. An increasingly popular algorithm is Uniform Manifold Learning and Projection (UMAP), which uses algebraic topology to map distances between the…

• Incremental Sheaf Cohomology on Cellular Complexes: O(1)-in-n Lazy Edit Processing under Bounded Local Geometry

  Jason L. Volk · 4 de junio de 2026

  We present an algorithmic framework for incremental maintenance of first sheaf cohomology $H^1(X; \mathcal{F})$ on dynamically evolving 1-dimensional cellular complexes equipped with finite-dimensional cellular sheaves. The classical computation of $H^1$ via factorization of the coboundary matrix re…

• Theoretical Aspects of Lie Groupoid and Lie Algebroid Equivariant Convolutional Neural Networks

  Michael Astwood · 3 de junio de 2026

  We introduce Lie groupoid equivariant neural networks as a specialization of recently proposed topological category-equivariant neural networks to the differentiable setting. Lie groupoid equivariant neural networks are composed from Lie groupoid lifting convolutions and Lie groupoid convolution lay…

• Learning Coherent Representations: A Topological Approach to Interpretability

  Sigurd Gaukstad, Melvin Vaupel, Valdemar Karg{\aa}rd Olsen, Erik Hermansen, Benjamin Dunn · 3 de junio de 2026

  Deep neural networks learn representations where individual features often lack interpretable meaning; a single neuron may activate for scattered, unrelated inputs. We introduce coherence, a geometric property inspired by neural coding in the brain, where neurons like grid cells and head direction c…

• Graph Mamba Survival Analysis Based on Topology-Aware ordering

  Yuanfang Chen, Peiqiang Yan, Yuntao Shou, Qian Zhao, Xiangyong Cao · 3 de junio de 2026

  In computational pathology, Whole Slide Images (WSIs) survival analysis is crucial for patient prognosis assessment, but it faces multiple technical challenges. Although the Transformer captures long-range dependencies through its self-attention mechanism, its $O(N^2)$ time complexity causes a sever…

• Mathematical Morphology in Machine Learning

  Erick Oliveira Rodrigues, Aura Conci · 1 de junio de 2026

  This work introduces mathematical morphology-an established visual computing theory-into machine learning to exploit shape and density aspects often overlooked by standard techniques. We propose a fast clustering algorithm based on morphological reconstruction that accurately preserves cluster shape…

• Discovering a Zeta Map Algorithm on Dyck Paths via Mechanistic Interpretability

  Xiaoyu Huang, Blake Jackson, Kyu-Hwan Lee · 1 de junio de 2026

  Machine learning is increasingly used in mathematical discovery, but in mathematics the desired output is often not a prediction itself, but an explicit construction that can be checked independently. We study this setting through the zeta map on Dyck paths, a classical bijection in the combinatoric…

• A Quotient Homology Theory of Representation in Neural Networks

  Kosio Beshkov · 29 de mayo de 2026

  Previous research has proven that the set of maps implemented by neural networks with a ReLU activation function is identical to the set of piecewise linear continuous maps. Furthermore, such networks induce a hyperplane arrangement splitting the input domain of the network into convex polyhedra $G_…

• TopoGeoScore: A Self-Supervised Source-Only Geometric Framework for OOD Checkpoint Selection

  Farid Hazratian, Ali Zia, Hien Duy Nguyen · 29 de mayo de 2026

  Out-of-distribution (OOD) robustness is difficult to diagnose when target-domain labels are unavailable. We consider a more restrictive source-only variant of unsupervised accuracy estimation: selecting robust checkpoints using only source-domain representations, with no target samples or target lab…

• The Shape of Reasoning: Topological Analysis of Reasoning Traces in Large Language Models

  Xue Wen Tan, Nathaniel Tan, Galen Lee, Stanley Kok · 28 de mayo de 2026

  Evaluating the quality of reasoning traces from large language models remains understudied, labor-intensive, and unreliable: current practice relies on expert rubrics, manual annotation, and slow pairwise judgments. Automated efforts are dominated by graph-based proxies that quantify structural conn…

• Topology-Driven Transferability Estimation of Medical Foundation Models for Segmentation

  Jiaqi Tang, Shaoyang Zhang, Xiaoqi Wang, Jiaying Zhou, Yang Liu, Qingchao Chen · 26 de mayo de 2026

  The advent of large-scale self-supervised learning (SSL) has produced a vast zoo of medical foundation models. However, selecting optimal medical foundation models for specific segmentation tasks remains a computational bottleneck. Existing Transferability Estimation (TE) metrics, primarily designed…

• Detecting Metastable Basins in High Dimensions via Marginal Trajectory Distribution Discrimination

  Taj Jones-McCormick · 26 de mayo de 2026

  We study the problem of identifying dynamically distinct basins of attraction in high dimensional time-homogeneous Markov processes using only trajectory sampling. This problem is fundamental in the analysis of metastable dynamical systems, where the process rapidly mixes within basins while transit…

• TopoAlign: Topology-Aware Visual Representation Alignment

  Xinyuan Yan, Rita Sevastjanova, Mennatallah El-Assady, Bei Wang · 26 de mayo de 2026

  Neural networks encode inputs as high-dimensional vectors, known as representations, that capture how models process data by encoding task-relevant structure and semantics. Representation alignment refers to the degree to which different models, layers, or training conditions produce similar represe…

• Train-Free Segmentation in MRI with Cubical Persistent Homology

  Anton Fran\c{c}ois, Rapha\"el Tinarrage · 26 de mayo de 2026

  We investigate a framework for train-free MRI segmentation based on Topological Data Analysis. The pipeline proceeds in three steps, first identifying the whole object to segment via automatic thresholding, then detecting a distinctive subset whose topology is known in advance, and finally deducing …

• Any-Dimensional Invariant Universality

  Shengtai Yao, Eitan Levin, Mateo D\'iaz · 25 de mayo de 2026

  Several machine learning models are defined for inputs of any size, such as graphs with different numbers of nodes and point clouds containing varying numbers of points. The universality properties of such any-dimensional models remain poorly understood, as universality is traditionally studied for …

• Commutator-Induced Uncertainty in VAEs

  Tahereh Dehdarirad, Michael Felsberg, Gabriel Eilertsen, Ziliang Xiong · 25 de mayo de 2026

  Variational autoencoders (VAEs) often struggle to represent non-commutative structure in learned latent spaces. Symmetry-aware VAEs commonly address this issue by enforcing commutativity through algebraic regularization, which is appropriate for commutative transformation groups but can suppress mea…

• Axiomatizing Neural Networks via Pursuit of Subspaces

  Mehmet Yamac, Mert Duman, Ugur Akpinar, Felix Rojas Casadiego, Serkan Kiranyaz, Marcel van Gerven, Moncef Gabbouj · 21 de mayo de 2026

  While deep neural networks have achieved remarkable success across a wide range of domains, their underlying mechanisms remain poorly understood, and they are often regarded as black boxes. This gap between empirical performance and theoretical understanding poses a challenge analogous to the pre-ax…

• Euclidean Embedding of Data Using Local Distances

  Dimitris Arabadjis · 20 de mayo de 2026

  We study the problem of recovering a globally consistent Euclidean embedding of data, given only a local distance graph and propose a method that optimally represents these distances. The method operates solely on a neighborhood graph weighted by pairwise distances, without requiring any prior vecto…

• AdaGraph: A Graph-Native Clustering Algorithm That Overcomes the Curse of Dimensionality and Enables Scientific Discovery

  Ahmed Elmahdi · 19 de mayo de 2026

  We present AdaGraph, a graph-native clustering algorithm born from the Structure-Centric Machine Learning (SC-ML) paradigm -- a new field of unsupervised learning that replaces geometry-centric (distance-based) computation with structure-centric (topology-based) computation, fundamentally dissolving…

• Understanding In-Context Learning on Structured Manifolds: Bridging Attention to Kernel Methods

  Zhaiming Shen, Alexander Hsu, Rongjie Lai, Wenjing Liao · 19 de mayo de 2026

  While in-context learning (ICL) has achieved remarkable success in natural language and vision domains, its theoretical understanding-particularly in the context of structured geometric data-remains unexplored. This paper initiates a theoretical study of ICL for regression of H\"older functions on m…

• PFlow-T: A Persistence-Driven Forward Process for Topology-Controlled Generation

  Snigdha Chandan Khilar · 19 de mayo de 2026

  Current topology aware diffusion models face an architectural mismatch by using Gaussian noise for corruption while recovering structural features through conditional side channels To fix this we introduce PFlow T a generative model that bases its forward process entirely on persistent homology In P…

• Topo-GS: Continuous Volumetric Embedding of High-Dimensional Data via Topological Gaussian Splatting

  Jo\~ao Paulo Gois, Luis Gustavo Nonato · 19 de mayo de 2026

  Dimensionality reduction algorithms map high-dimensional data into visualizable 2D or 3D spaces, but traditionally rely on a discrete point-cloud paradigm. This discrete abstraction is susceptible to visual occlusion and artificial discontinuities, often failing to represent the continuous density o…

• Topological Data Analysis combined with Machine Learning for Predicting Permeability of Porous Media

  Ebru Dagdelen, Catherin Neena Lalu, Aakash Karlekar, Manav Arora, Matthew Illingworth, Jonathan Jaquette, Linda Cummings, Lou Kondic · 19 de mayo de 2026

  Flow in porous media is difficult to address using standard analytical or numerical methods due to its complexity. However, since synthetic representations of porous media are easy to produce and data from physical experiments are becoming more widely available, the problem is well-suited to studies…

• Topology-Preserving Polygon Augmentation for Segmentation in Structured Visual Domains

  Sudip Laudari, Sang Hun Baek · 18 de mayo de 2026

  Geometric data augmentation is widely used in segmentation workflows, but polygon annotations are often assumed to remain valid after transformation. This assumption can fail in structured domains such as architectural floorplan analysis, where a region may contain an interior void encoded as part o…