2021/10/20 by Andi Han, Han, Andi, Bamdev Mishra +5
Computer Science · Engineering · Mathematics · #Differential Geometry (math.DG) #FOS: Computer and information sciences #FOS: Mathematics #Functional Analysis (math.FA) #Machine Learning (stat.ML) #Optimization and Control (math.OC) #Sparse and Compressive Sensing Techniques #Statistics Theory (math.ST) #Tensor decomposition and applications #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2110.10464
openalex publication_date 2021/10/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Learning with symmetric positive definite (SPD) matrices has many applications in machine learning. Consequently, understanding the Riemannian geometry of SPD matrices has attracted much attention lately. A particular Riemannian geometry of interest is the recently proposed Bures-Wasserstein (BW) geometry which builds on the Wasserstein distance between the Gaussian densities. In this paper, we propose a novel generalization of the BW geometry, which we call the GBW geometry. The proposed generalization is parameterized by a symmetric positive definite matrix M such that when M = I, we recover the BW geometry. We provide a rigorous treatment to study various differential geometric notions on the proposed novel generalized geometry which makes it amenable to various machine learning applications. We also present experiments that illustrate the efficacy of the proposed GBW geometry over the BW geometry.