Categorical and geometric methods in statistical, manifold, and machine learning

2025/05/06 by Hông Vân Lê, Lê, Hông Vân, Hà Quang Minh +6 · 1 voice
Computer Science · Mathematics · Physics and Astronomy · #Category Theory (math.CT) #Differential Geometry (math.DG) #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Morphological variations and asymmetry #Statistical Mechanics and Entropy #Statistics Theory (math.ST) #Topological and Geometric Data Analysis #cs.LG #math.CT #math.DG #math.ST #stat.ML

paper · pdf · doi:10.48550/arxiv.2505.03862

openalex publication_date 2025/05/06 · arxiv published 2025/05/06 · arxiv updated 2025/05/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We present and discuss applications of the category of probabilistic morphisms, initially developed in \citeLe2023, as well as some geometric methods to several classes of problems in statistical, machine and manifold learning which shall be, along with many other topics, considered in depth in the forthcoming book \citeLMPT2024.

Citations

• Probabilistic morphisms and Bayesian supervised learning
• Supervised learning with probabilistic morphisms and kernel mean embeddings
• The Modern Mathematics of Deep Learning
• Reconstruction of a Riemannian manifold from noisy intrinsic distances
• On the Bures-Wasserstein distance between positive definite matrices
• Reconstruction and interpolation of manifolds I: The geometric Whitney problem
• Geodesic Exponential Kernels: When Curvature and Linearity Conflict
• A graph discretization of the Laplace-Beltrami operator
• Towards a theoretical foundation for Laplacian-based manifold methods
• Laplacian Eigenmaps for Dimensionality Reduction and Data Representation
• Symplectic Geometry
• Metric spaces and positive definite functions

Discussions

• www.arxiv.org/abs/2505.03862 [bsky, 0 points, 0 comments]

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