Topological and Geometric Data Analysis

Latest papers

• Sequence-to-Image Transformation for Sequence Classification Using Rips Complex Construction and Chaos Game Representation

  Sarwan Ali, Taslim Murad, Imdadullah Khan · 12 December 2025

  Traditional feature engineering approaches for molecular sequence classification suffer from sparsity issues and computational complexity, while deep learning models often underperform on tabular biological data. This paper introduces a novel topological approach that transforms molecular sequences …

• Rates and architectures for learning geometrically non-trivial operators

  T. Mitchell Roddenberry, Leo Tzou, Ivan Dokmani\'c, Maarten V. de Hoop, Richard G. Baraniuk · 11 December 2025

  Deep learning methods have proven capable of recovering operators between high-dimensional spaces, such as solution maps of PDEs and similar objects in mathematical physics, from very few training samples. This phenomenon of data-efficiency has been proven for certain classes of elliptic operators w…

• Unsupervised Learning of Density Estimates with Topological Optimization

  Suina Tanweer, Firas A. Khasawneh · 10 December 2025

  Kernel density estimation is a key component of a wide variety of algorithms in machine learning, Bayesian inference, stochastic dynamics and signal processing. However, the unsupervised density estimation technique requires tuning a crucial hyperparameter: the kernel bandwidth. The choice of bandwi…

• Persistent Topological Structures and Cohomological Flows as a Mathematical Framework for Brain-Inspired Representation Learning

  Preksha Girish, Rachana Mysore, Mahanthesha U, Shrey Kumar, Shipra Prashant · 10 December 2025

  This paper presents a mathematically rigorous framework for brain-inspired representation learning founded on the interplay between persistent topological structures and cohomological flows. Neural computation is reformulated as the evolution of cochain maps over dynamic simplicial complexes, enabli…

• HOLE: Homological Observation of Latent Embeddings for Neural Network Interpretability

  Sudhanva Manjunath Athreya, Paul Rosen · 10 December 2025

  Deep learning models have achieved remarkable success across various domains, yet their learned representations and decision-making processes remain largely opaque and hard to interpret. This work introduces HOLE (Homological Observation of Latent Embeddings), a method for analyzing and interpreting…

• Memory-Amortized Inference: A Topological Unification of Search, Closure, and Structure

  Xin Li · 9 December 2025

  Contemporary ML separates the static structure of parameters from the dynamic flow of inference, yielding systems that lack the sample efficiency and thermodynamic frugality of biological cognition. In this theoretical work, we propose \textbf{Memory-Amortized Inference (MAI)}, a formal framework ro…

• Manifold Percolation: from generative model to Reinforce learning

  Rui Tong · 4 December 2025

  Generative modeling is typically framed as learning mapping rules, but from an observer's perspective without access to these rules, the task becomes disentangling the geometric support from the probability distribution. We propose that continuum percolation is uniquely suited to this support analys…

• Filtration-Based Representation Learning for Temporal Graphs

  Samrik Chowdhury, Siddharth Pritam, Rohit Roy, Madhav Cherupilil Sajeev · 4 December 2025

  In this work, we introduce a filtration on temporal graphs based on $\delta$-temporal motifs (recurrent subgraphs), yielding a multi-scale representation of temporal structure. Our temporal filtration allows tools developed for filtered static graphs, including persistent homology and recent graph f…

• Probabilistic Foundations of Fuzzy Simplicial Sets for Nonlinear Dimensionality Reduction

  Janis Keck (Hannaneh), Lukas Silvester Barth (Hannaneh), Fatemeh (Hannaneh), Fahimi, Parvaneh Joharinad, J\"urgen Jost · 4 December 2025

  Fuzzy simplicial sets have become an object of interest in dimensionality reduction and manifold learning, most prominently through their role in UMAP. However, their definition through tools from algebraic topology without a clear probabilistic interpretation detaches them from commonly used theore…

• From Betti Numbers to Persistence Diagrams: A Hybrid Quantum Algorithm for Topological Data Analysis

  Dong Liu · 3 December 2025

  Persistence diagrams serve as a core tool in topological data analysis, playing a crucial role in pathological monitoring, drug discovery, and materials design. However, existing quantum topological algorithms, such as the LGZ algorithm, can only efficiently compute summary statistics like Betti num…

• Emergent Riemannian geometry over learning discrete computations on continuous manifolds

  Julian Brandon, Angus Chadwick, Arthur Pellegrino · 2 December 2025

  Many tasks require mapping continuous input data (e.g. images) to discrete task outputs (e.g. class labels). Yet, how neural networks learn to perform such discrete computations on continuous data manifolds remains poorly understood. Here, we show that signatures of such computations emerge in the r…

• CVKAN: Complex-Valued Kolmogorov-Arnold Networks

  Matthias Wolff, Florian Eilers, Xiaoyi Jiang · 1 December 2025

  In this work we propose CVKAN, a complex-valued Kolmogorov-Arnold Network (KAN), to join the intrinsic interpretability of KANs and the advantages of Complex-Valued Neural Networks (CVNNs). We show how to transfer a KAN and the necessary associated mechanisms into the complex domain. To confirm that…

• The Human Brain as a Combinatorial Complex

  Valentina S\'anchez, \c{C}i\c{c}ek G\"uven, Koen Haak, Theodore Papamarkou, Gonzalo N\'apoles, Marie \v{S}af\'a\v{r} Postma · 27 November 2025

  We propose a framework for constructing combinatorial complexes (CCs) from fMRI time series data that captures both pairwise and higher-order neural interactions through information-theoretic measures, bridging topological deep learning and network neuroscience. Current graph-based representations o…

• Scale-Agnostic Kolmogorov-Arnold Geometry in Neural Networks

  Mathew Vanherreweghe, Michael H. Freedman, Keith M. Adams · 27 November 2025

  Recent work by Freedman and Mulligan demonstrated that shallow multilayer perceptrons spontaneously develop Kolmogorov-Arnold geometric (KAG) structure during training on synthetic three-dimensional tasks. However, it remained unclear whether this phenomenon persists in realistic high-dimensional se…

• Walking the Weight Manifold: a Topological Approach to Conditioning Inspired by Neuromodulation

  Ari S. Benjamin, Kyle Daruwalla, Christian Pehle, Abdul-Malik Zekri, Anthony M. Zador · 26 November 2025

  One frequently wishes to learn a range of similar tasks as efficiently as possible, re-using knowledge across tasks. In artificial neural networks, this is typically accomplished by conditioning a network upon task context by injecting context as input. Brains have a different strategy: the paramete…

• TopER: Topological Embeddings in Graph Representation Learning

  Astrit Tola, Funmilola Mary Taiwo, Cuneyt Gurcan Akcora, Baris Coskunuzer · 26 November 2025

  Graph embeddings play a critical role in graph representation learning, allowing machine learning models to explore and interpret graph-structured data. However, existing methods often rely on opaque, high-dimensional embeddings, limiting interpretability and practical visualization. In this work,…

• Generative Modeling with Manifold Percolation

  Rui Tong · 26 November 2025

  Generative modeling is typically framed as learning mapping rules, but from an observer's perspective without access to these rules, the task manifests as disentangling the geometric support from the probability distribution. We propose that Continuum Percolation is uniquely suited for this support …

• Anatomica: Localized Control over Geometric and Topological Properties for Anatomical Diffusion Models

  Karim Kadry, Abdallah Abdelwahed, Shoaib Goraya, Ajay Manicka, Naravich Chutisilp, Farhad Nezami, Elazer Edelman · 26 November 2025

  We present Anatomica: an inference-time framework for generating multi-class anatomical voxel maps with localized geo-topological control. During generation, we use cuboidal control domains of varying dimensionality, location, and shape to slice out relevant substructures. These local substructures …

• Uncertainty of Network Topology with Applications to Out-of-Distribution Detection

  Sing-Yuan Yeh, Chun-Hao Yang · 25 November 2025

  Persistent homology (PH) is a crucial concept in computational topology, providing a multiscale topological description of a space. It is particularly significant in topological data analysis, which aims to make statistical inference from a topological perspective. In this work, we introduce a new t…

• ProHD: Projection-Based Hausdorff Distance Approximation

  Jiuzhou Fu, Luanzheng Guo, Nathan R. Tallent, Dongfang Zhao · 25 November 2025

  The Hausdorff distance (HD) is a robust measure of set dissimilarity, but computing it exactly on large, high-dimensional datasets is prohibitively expensive. We propose \textbf{ProHD}, a projection-guided approximation algorithm that dramatically accelerates HD computation while maintaining high ac…

• The Geometry of Cortical Computation: Manifold Disentanglement and Predictive Dynamics in VCNet

  Brennen A. Hill, Zhang Xinyu, Timothy Putra Prasetio · 25 November 2025

  Despite their success, modern convolutional neural networks (CNNs) exhibit fundamental limitations, including data inefficiency, poor out-of-distribution generalization, and vulnerability to adversarial perturbations. These shortcomings can be traced to a lack of inductive biases that reflect the in…

• Topology Aware Neural Interpolation of Scalar Fields

  Mohamed Kissi, Keanu Sisouk, Joshua A. Levine, Julien Tierny · 24 November 2025

  This paper presents a neural scheme for the topology-aware interpolation of time-varying scalar fields. Given a time-varying sequence of persistence diagrams, along with a sparse temporal sampling of the corresponding scalar fields, denoted as keyframes, our interpolation approach aims at "inverting…

• The Shape of Data: Topology Meets Analytics. A Practical Introduction to Topological Analytics and the Stability Index (TSI) in Business

  Ioannis Diamantis · 18 November 2025

  Modern business and economic datasets often exhibit nonlinear, multi-scale structures that traditional linear tools under-represent. Topological Data Analysis (TDA) offers a geometric lens for uncovering robust patterns, such as connected components, loops and voids, across scales. This paper provid…

• From Black Box to Bijection: Interpreting Machine Learning to Build a Zeta Map Algorithm

  Xiaoyu Huang, Blake Jackson, Kyu-Hwan Lee · 18 November 2025

  There is a large class of problems in algebraic combinatorics which can be distilled into the same challenge: construct an explicit combinatorial bijection. Traditionally, researchers have solved challenges like these by visually inspecting the data for patterns, formulating conjectures, and then pr…

• Simplicial covering dimension of extremal concept classes

  Ari Blondal, Hamed Hatami, Pooya Hatami, Chavdar Lalov, Sivan Tretiak · 18 November 2025

  Dimension theory is a branch of topology concerned with defining and analyzing dimensions of geometric and topological spaces in purely topological terms. In this work, we adapt the classical notion of topological dimension (Lebesgue covering) to binary concept classes. The topological space natural…