2021/08/24 by Khang Le, Dung T. Le, Le, Khang +9 · 1 citation
Mathematics · #FOS: Computer and information sciences #FOS: Mathematics #Geometric Analysis and Curvature Flows #Information Theory (cs.IT) #Machine Learning (stat.ML) #Markov Chains and Monte Carlo Methods #Point processes and geometric inequalities #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2108.10961
openalex publication_date 2021/08/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the entropic Gromov-Wasserstein and its unbalanced version between (unbalanced) Gaussian distributions with different dimensions. When the metric is the inner product, which we refer to as inner product Gromov-Wasserstein (IGW), we demonstrate that the optimal transportation plans of entropic IGW and its unbalanced variant are (unbalanced) Gaussian distributions. Via an application of von Neumann's trace inequality, we obtain closed-form expressions for the entropic IGW between these Gaussian distributions. Finally, we consider an entropic inner product Gromov-Wasserstein barycenter of multiple Gaussian distributions. We prove that the barycenter is a Gaussian distribution when the entropic regularization parameter is small. We further derive a closed-form expression for the covariance matrix of the barycenter.