Topological and Geometric Data Analysis

Neueste Paper

• Fubini Study geometry of representation drift in high dimensional data

  Arturo Tozzi · 4. Februar 2026

  High dimensional representation drift is commonly quantified using Euclidean or cosine distances, which presuppose fixed coordinates when comparing representations across time, training or preprocessing stages. While effective in many settings, these measures entangle intrinsic changes in the data w…

• TopoPrune: Robust Data Pruning via Unified Latent Space Topology

  Arjun Roy, Prajna G. Malettira, Manish Nagaraj, Kaushik Roy · 4. Februar 2026

  Geometric data pruning methods, while practical for leveraging pretrained models, are fundamentally unstable. Their reliance on extrinsic geometry renders them highly sensitive to latent space perturbations, causing performance to degrade during cross-architecture transfer or in the presence of feat…

• Discovering Data Manifold Geometry via Non-Contracting Flows

  David Vigouroux (ANITI, IMT Atlantique), Lucas Drumetz (IMT Atlantique - MEE, Lab-STICC\_OSE, ODYSSEY), Ronan Fablet (IMT Atlantique - MEE, Lab-STICC\_OSE, ODYSSEY), Fran\c{c}ois Rousseau (IMT Atlantique - ITI, LaTIM) · 4. Februar 2026

  We introduce an unsupervised approach for constructing a global reference system by learning, in the ambient space, vector fields that span the tangent spaces of an unknown data manifold. In contrast to isometric objectives, which implicitly assume manifold flatness, our method learns tangent vector…

• The Flood Complex: Large-Scale Persistent Homology on Millions of Points

  Florian Graf, Paolo Pellizzoni, Martin Uray, Stefan Huber, Roland Kwitt · 3. Februar 2026

  We consider the problem of computing persistent homology (PH) for large-scale Euclidean point cloud data, aimed at downstream machine learning tasks, where the exponential growth of the most widely-used Vietoris-Rips complex imposes serious computational limitations. Although more scalable alternati…

• Hybrid Topological and Deep Feature Fusion for Accurate MRI-Based Alzheimer's Disease Severity Classification

  Faisal Ahmed · 3. Februar 2026

  Early and accurate diagnosis of Alzheimer's disease (AD) remains a critical challenge in neuroimaging-based clinical decision support systems. In this work, we propose a novel hybrid deep learning framework that integrates Topological Data Analysis (TDA) with a DenseNet121 backbone for four-class Al…

• Topological Residual Asymmetry for Bivariate Causal Direction

  Mouad El Bouchattaoui · 3. Februar 2026

  Inferring causal direction from purely observational bivariate data is fragile: many methods commit to a direction even in ambiguous or near non-identifiable regimes. We propose Topological Residual Asymmetry (TRA), a geometry-based criterion for additive-noise models. TRA compares the shapes of two…

• A Sheaf-Theoretic and Topological Perspective on Complex Network Modeling and Attention Mechanisms in Graph Neural Models

  Chuan-Shen Hu · 30. Januar 2026

  Combinatorial and topological structures, such as graphs, simplicial complexes, and cell complexes, form the foundation of geometric and topological deep learning (GDL and TDL) architectures. These models aggregate signals over such domains, integrate local features, and generate representations for…

• Test-Time Adaptation for Anomaly Segmentation via Topology-Aware Optimal Transport Chaining

  Ali Zia, Usman Ali, Umer Ramzan, Abdul Rehman, Abdelwahed Khamis, Wei Xiang · 29. Januar 2026

  Deep topological data analysis (TDA) offers a principled framework for capturing structural invariants such as connectivity and cycles that persist across scales, making it a natural fit for anomaly segmentation (AS). Unlike thresholdbased binarisation, which produces brittle masks under distributio…

• An efficient, provably optimal algorithm for the 0-1 loss linear classification problem

  Xi He, Max A. Little · 28. Januar 2026

  Algorithms for solving the linear classification problem have a long history, dating back at least to 1936 with linear discriminant analysis. For linearly separable data, many algorithms can obtain the exact solution to the corresponding 0-1 loss classification problem efficiently, but for data whic…

• Leveraging Persistence Image to Enhance Robustness and Performance in Curvilinear Structure Segmentation

  Zhuangzhi Gao, Feixiang Zhou, He Zhao, Xiuju Chen, Xiaoxin Li, Qinkai Yu, Yitian Zhao, Alena Shantsila, Gregory Y. H. Lip, Eduard Shantsila, Yalin Zheng · 27. Januar 2026

  Segmenting curvilinear structures in medical images is essential for analyzing morphological patterns in clinical applications. Integrating topological properties, such as connectivity, improves segmentation accuracy and consistency. However, extracting and embedding such properties - especially fro…

• Understanding and Improving UMAP with Geometric and Topological Priors: The JORC-UMAP Algorithm

  Xiaobin Li, Run Zhang · 26. Januar 2026

  Nonlinear dimensionality reduction techniques, particularly UMAP, are widely used for visualizing high-dimensional data. However, UMAP's local Euclidean distance assumption often fails to capture intrinsic manifold geometry, leading to topological tearing and structural collapse. We identify UMAP's …

• Robust Barycenters of Persistence Diagrams

  Keanu Sisouk, Eloi Tanguy, Julie Delon, Julien Tierny · 22. Januar 2026

  This short paper presents a general approach for computing robust Wasserstein barycenters of persistence diagrams. The classical method consists in computing assignment arithmetic means after finding the optimal transport plans between the barycenter and the persistence diagrams. However, this proce…

• Persistent Sheaf Laplacian Analysis of Protein Stability and Solubility Changes upon Mutation

  Yiming Ren, Junjie Wee, Xi Chen, Grace Qian, Guo-Wei Wei · 21. Januar 2026

  Genetic mutations frequently disrupt protein structure, stability, and solubility, acting as primary drivers for a wide spectrum of diseases. Despite the critical importance of these molecular alterations, existing computational models often lack interpretability, and fail to integrate essential phy…

• Quasi Zigzag Persistence: A Topological Framework for Analyzing Time-Varying Data

  Tamal K. Dey, Shreyas N. Samaga · 21. Januar 2026

  In this paper, we propose Quasi Zigzag Persistent Homology (QZPH) as a framework for analyzing time-varying data by integrating multiparameter persistence and zigzag persistence. To this end, we introduce a stable topological invariant that captures both static and dynamic features at different scal…

• Inverting Self-Organizing Maps: A Unified Activation-Based Framework

  Alessandro Londei, Matteo Benati, Denise Lanzieri, Vittorio Loreto · 21. Januar 2026

  Self-Organizing Maps provide topology-preserving projections of high-dimensional data and have been widely used for visualization, clustering, and vector quantization. In this work, we show that the activation pattern of a SOM - the squared distances to its prototypes - can be inverted to recover th…

• Topology-Aware Loss for Aorta and Great Vessel Segmentation in Computed Tomography Images

  Seher Ozcelik, Sinan Unver, Ilke Ali Gurses, Rustu Turkay, Cigdem Gunduz-Demir · 21. Januar 2026

  Segmentation networks are not explicitly imposed to learn global invariants of an image, such as the shape of an object and the geometry between multiple objects, when they are trained with a standard loss function. On the other hand, incorporating such invariants into network training may help impr…

• SPHENIC: Topology-Aware Multi-View Clustering for Spatial Transcriptomics

  Chenkai Guo, Yikai Zhu, Renxiang Guan, Jinli Ma, Siwei Wang, Ke Liang, Guangdun Peng, Dayu Hu · 21. Januar 2026

  Spatial transcriptomics clustering is pivotal for identifying cell subpopulations by leveraging spatial location information. While recent graph-based methods modeling cell-cell interactions have improved clustering accuracy, they remain limited in two key aspects: (i) reliance on local aggregation …

• Topology-Guaranteed Image Segmentation: Enforcing Connectivity, Genus, and Width Constraints

  Wenxiao Li, Xue-Cheng Tai, Jun Liu · 19. Januar 2026

  Existing research highlights the crucial role of topological priors in image segmentation, particularly in preserving essential structures such as connectivity and genus. Accurately capturing these topological features often requires incorporating width-related information, including the thickness a…

• Persistent Homology via Ellipsoids

  Niklas Canova, Sara Kali\v{s}nik, Aaron Moser, Bastian Rieck, Ana \v{Z}egarac · 16. Januar 2026

  Persistent homology is one of the most popular methods in topological data analysis. An initial step in its use involves constructing a nested sequence of simplicial complexes. There is an abundance of different complexes to choose from, with \v{C}ech, Rips, alpha, and witness complexes being popula…

• Geometric Stability: The Missing Axis of Representations

  Prashant C. Raju · 15. Januar 2026

  Analysis of learned representations has a blind spot: it focuses on $similarity$, measuring how closely embeddings align with external references, but similarity reveals only what is represented, not whether that structure is robust. We introduce $geometric$ $stability$, a distinct dimension that qu…

• Dynamic Graph Structure Learning via Resistance Curvature Flow

  Chaoqun Fei, Huanjiang Liu, Tinglve Zhou, Yangyang Li, Tianyong Hao · 14. Januar 2026

  Geometric Representation Learning (GRL) aims to approximate the non-Euclidean topology of high-dimensional data through discrete graph structures, grounded in the manifold hypothesis. However, traditional static graph construction methods based on Euclidean distance often fail to capture the intrins…

• The Blueprints of Intelligence: A Functional-Topological Foundation for Perception and Representation

  Eduardo Di Santi · 13. Januar 2026

  Real-world phenomena do not generate arbitrary variability: their signals concentrate on compact, low-variability subsets of functional space, enabling rapid generalization from few examples. A small child can recognize a dog after extremely limited exposure because the perceptual manifold of "dog" …

• Shape-Aware Topological Representation for Pipeline Hyperbola Detection in GPR Data

  Meiyan Kang, Shizuo Kaji, Sang-Yun Lee, Taegon Kim, Hee-Hwan Ryu, Suyoung Choi · 13. Januar 2026

  Ground Penetrating Radar (GPR) is a widely used Non-Destructive Testing (NDT) technique for subsurface exploration, particularly in infrastructure inspection and maintenance. However, conventional interpretation methods are often limited by noise sensitivity and a lack of structural awareness. This …

• Approximating Persistent Homology for Large Datasets

  Yueqi Cao, Anthea Monod · 13. Januar 2026

  Persistent homology is an important methodology in topological data analysis which adapts theory from algebraic topology to data settings. Computing persistent homology produces persistence diagrams, which have been successfully used in diverse domains. Despite its widespread use, persistent homolog…

• An Algebraic Representation Theorem for Linear GENEOs in Geometric Machine Learning

  Francesco Conti, Patrizio Frosini, Nicola Quercioli · 8. Januar 2026

  Geometric and Topological Deep Learning are rapidly growing research areas that enhance machine learning through the use of geometric and topological structures. Within this framework, Group Equivariant Non-Expansive Operators (GENEOs) have emerged as a powerful class of operators for encoding symme…