Topological and Geometric Data Analysis

Últimos artículos

• Loss Barcode: A Topological Measure of Escapability in Loss Landscapes

  Serguei Barannikov, Daria Voronkova, Alexander Mironenko, Ilya Trofimov, Alexander Korotin, Grigorii Sotnikov, Evgeny Burnaev · 4 de marzo de 2026

  Neural network training is commonly based on SGD. However, the understanding of SGD's ability to converge to good local minima, given the non-convex nature of loss functions and the intricate geometric characteristics of loss landscapes, remains limited. In this paper, we apply topological data anal…

• Coalgebras for categorical deep learning: Representability and universal approximation

  Dragan Ma\v{s}ulovi\'c · 4 de marzo de 2026

  Categorical deep learning (CDL) has recently emerged as a framework that leverages category theory to unify diverse neural architectures. While geometric deep learning (GDL) is grounded in the specific context of invariants of group actions, CDL aims to provide domain-independent abstractions for re…

• Learning Global Hypothesis Space for Enhancing Synergistic Reasoning Chain

  Jiaquan Zhang, Chaoning Zhang, Shuxu Chen, Xudong Wang, Zhenzhen Huang, Pengcheng Zheng, Shuai Yuan, Sheng Zheng, Qigan Sun, Jie Zou, Lik-Hang Lee, Yang Yang · 3 de marzo de 2026

  Chain-of-Thought (CoT) has been shown to significantly improve the reasoning accuracy of large language models (LLMs) on complex tasks. However, due to the autoregressive, step-by-step generation paradigm, existing CoT methods suffer from two fundamental limitations. First, the reasoning process is …

• The Geometry of Transfer: Unlocking Medical Vision Manifolds for Training-Free Model Ranking

  Jiaqi Tang, Shaoyang Zhang, Xiaoqi Wang, Jiaying Zhou, Yang Liu, Qingchao Chen · 2 de marzo 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…

• Learning Tangent Bundles and Characteristic Classes with Autoencoder Atlases

  Eduardo Paluzo-Hidalgo, Yuichi Ike · 27 de febrero de 2026

  We introduce a theoretical framework that connects multi-chart autoencoders in manifold learning with the classical theory of vector bundles and characteristic classes. Rather than viewing autoencoders as producing a single global Euclidean embedding, we treat a collection of locally trained encoder…

• Hyperbolic Busemann Neural Networks

  Ziheng Chen, Bernhard Sch\"olkopf, Nicu Sebe · 24 de febrero de 2026

  Hyperbolic spaces provide a natural geometry for representing hierarchical and tree-structured data due to their exponential volume growth. To leverage these benefits, neural networks require intrinsic and efficient components that operate directly in hyperbolic space. In this work, we lift two core…

• Exploring Singularities in point clouds with the graph Laplacian: An explicit approach

  Martin Andersson, Benny Avelin · 24 de febrero de 2026

  We develop theory and methods that use the graph Laplacian to analyze the geometry of the underlying manifold of datasets. Our theory provides theoretical guarantees and explicit bounds on the functional forms of the graph Laplacian when it acts on functions defined close to singularities of the und…

• Topological Exploration of High-Dimensional Empirical Risk Landscapes: general approach, and applications to phase retrieval

  Antoine Maillard, Tony Bonnaire, Giulio Biroli · 23 de febrero de 2026

  We consider the landscape of empirical risk minimization for high-dimensional Gaussian single-index models (generalized linear models). The objective is to recover an unknown signal $\boldsymbol{\theta}^\star \in \mathbb{R}^d$ (where $d \gg 1$) from a loss function $\hat{R}(\boldsymbol{\theta})$ tha…

• A Parameter-free Adaptive Resonance Theory-based Topological Clustering Algorithm Capable of Continual Learning

  Naoki Masuyama, Takanori Takebayashi, Yusuke Nojima, Chu Kiong Loo, Hisao Ishibuchi, Stefan Wermter · 20 de febrero de 2026

  In general, a similarity threshold (i.e., a vigilance parameter) for a node learning process in Adaptive Resonance Theory (ART)-based algorithms has a significant impact on clustering performance. In addition, an edge deletion threshold in a topological clustering algorithm plays an important role i…

• Multi-Class Boundary Extraction from Implicit Representations

  Jash Vira, Andrew Myers, Simon Ratcliffe · 19 de febrero de 2026

  Surface extraction from implicit neural representations modelling a single class surface is a well-known task. However, there exist no surface extraction methods from an implicit representation of multiple classes that guarantee topological correctness and no holes. In this work, we lay the groundwo…

• Feature-based morphological analysis of shape graph data

  Murad Hossen, Demetrio Labate, Nicolas Charon · 19 de febrero de 2026

  This paper introduces and demonstrates a computational pipeline for the statistical analysis of shape graph datasets, namely geometric networks embedded in 2D or 3D spaces. Unlike traditional abstract graphs, our purpose is not only to retrieve and distinguish variations in the connectivity structur…

• MacroGuide: Topological Guidance for Macrocycle Generation

  Alicja Maksymiuk, Alexandre Duplessis, Michael Bronstein, Alexander Tong, Fernanda Duarte, \.Ismail \.Ilkan Ceylan · 17 de febrero de 2026

  Macrocycles are ring-shaped molecules that offer a promising alternative to small-molecule drugs due to their enhanced selectivity and binding affinity against difficult targets. Despite their chemical value, they remain underexplored in generative modeling, likely owing to their scarcity in public …

• Diagnostic Benchmarks for Invariant Learning Dynamics: Empirical Validation of the Eidos Architecture

  Datorien L. Anderson · 17 de febrero de 2026

  We present the PolyShapes-Ideal (PSI) dataset, a suite of diagnostic benchmarks designed to isolate topological invariance -- the ability to maintain structural identity across affine transformations -- from the textural correlations that dominate standard vision benchmarks. Through three diagnostic…

• Spectral Convolution on Orbifolds for Geometric Deep Learning

  Tim Mangliers, Bernhard M\"ossner, Benjamin Himpel · 17 de febrero de 2026

  Geometric deep learning (GDL) deals with supervised learning on data domains that go beyond Euclidean structure, such as data with graph or manifold structure. Due to the demand that arises from application-related data, there is a need to identify further topological and geometric structures with w…

• Multi-dimensional Persistent Sheaf Laplacians for Image Analysis

  Xiang Xiang Wang, Guo-Wei Wei · 17 de febrero de 2026

  We propose a multi-dimensional persistent sheaf Laplacian (MPSL) framework on simplicial complexes for image analysis. The proposed method is motivated by the strong sensitivity of commonly used dimensionality reduction techniques, such as principal component analysis (PCA), to the choice of reduced…

• Understanding Chain-of-Thought in Large Language Models via Topological Data Analysis

  Chenghao Li, Chaoning Zhang, Yi Lu, Shuxu Chen, Xudong Wang, Jiaquan Zhang, Zhicheng Wang, Zhengxun Jin, Kuien Liu, Sung-Ho Bae, Guoqing Wang, Yang Yang, Heng Tao Shen · 16 de febrero de 2026

  With the development of large language models (LLMs), particularly with the introduction of the long reasoning chain technique, the reasoning ability of LLMs in complex problem-solving has been significantly enhanced. While acknowledging the power of long reasoning chains, we cannot help but wonder:…

• UMAP Is Spectral Clustering on the Fuzzy Nearest-Neighbor Graph

  Yang Yang · 13 de febrero de 2026

  UMAP (Uniform Manifold Approximation and Projection) is among the most widely used algorithms for non linear dimensionality reduction and data visualisation. Despite its popularity, and despite being presented through the lens of algebraic topology, the exact relationship between UMAP and classical …

• From Classical to Topological Neural Networks Under Uncertainty

  Sarah Harkins Dayton, Layal Bou Hamdan, Ioannis D. Schizas, David L. Boothe, Vasileios Maroulas · 12 de febrero de 2026

  This chapter explores neural networks, topological data analysis, and topological deep learning techniques, alongside statistical Bayesian methods, for processing images, time series, and graphs to maximize the potential of artificial intelligence in the military domain. Throughout the chapter, we h…

• Text summarization via global structure awareness

  Jiaquan Zhang, Chaoning Zhang, Shuxu Chen, Yibei Liu, Chenghao Li, Qigan Sun, Shuai Yuan, Fachrina Dewi Puspitasari, Dongshen Han, Guoqing Wang, Sung-Ho Bae, Yang Yang · 11 de febrero de 2026

  Text summarization is a fundamental task in natural language processing (NLP), and the information explosion has made long-document processing increasingly demanding, making summarization essential. Existing research mainly focuses on model improvements and sentence-level pruning, but often overlook…

• Measuring Dataset Diversity from a Geometric Perspective

  Yang Ba, Mohammad Sadeq Abolhasani, Michelle V Mancenido, Rong Pan · 11 de febrero de 2026

  Diversity can be broadly defined as the presence of meaningful variation across elements, which can be viewed from multiple perspectives, including statistical variation and geometric structural richness in the dataset. Existing diversity metrics, such as feature-space dispersion and metric-space ma…

• GHS-TDA: A Synergistic Reasoning Framework Integrating Global Hypothesis Space with Topological Data Analysis

  Jiaquan Zhang, Chaoning Zhang, Shuxu Chen, Xudong Wang, Zhenzhen Huang, Pengcheng Zheng, Shuai Yuan, Sheng Zheng, Qigan Sun, Jie Zou, Lik-Hang Lee, Yang Yang · 11 de febrero de 2026

  Chain-of-Thought (CoT) has been shown to significantly improve the reasoning accuracy of large language models (LLMs) on complex tasks. However, due to the autoregressive, step-by-step generation paradigm, existing CoT methods suffer from two fundamental limitations. First, the reasoning process is …

• Persistent Entropy as a Detector of Phase Transitions

  Matteo Rucco · 11 de febrero de 2026

  Persistent entropy (PE) is an information-theoretic summary statistic of persistence barcodes that has been widely used to detect regime changes in complex systems. Despite its empirical success, a general theoretical understanding of when and why persistent entropy reliably detects phase transition…

• What do Geometric Hallucination Detection Metrics Actually Measure?

  Eric Yeats, John Buckheit, Sarah Scullen, Brendan Kennedy, Loc Truong, Davis Brown, Bill Kay, Cliff Joslyn, Tegan Emerson, Michael J. Henry, John Emanuello, Henry Kvinge · 11 de febrero de 2026

  Hallucination remains a barrier to deploying generative models in high-consequence applications. This is especially true in cases where external ground truth is not readily available to validate model outputs. This situation has motivated the study of geometric signals in the internal state of an LL…

• RNNs perform task computations by dynamically warping neural representations

  Arthur Pellegrino, Angus Chadwick · 10 de febrero de 2026

  Analysing how neural networks represent data features in their activations can help interpret how they perform tasks. Hence, a long line of work has focused on mathematically characterising the geometry of such "neural representations." In parallel, machine learning has seen a surge of interest in u…

• Topological Signatures vs. Gradient Histograms: A Comparative Study for Medical Image Classification

  Faisal Ahmed · 10 de febrero de 2026

  This work presents a comparative evaluation of two fundamentally different feature extraction paradigms--Histogram of Oriented Gradients (HOG) and Topological Data Analysis (TDA)--for medical image classification, with a focus on retinal fundus imagery. HOG captures local structural information by m…