Entropic Gromov-Wasserstein between Gaussian Distributions
arXiv:2108.10961
Abstract
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.
52 pages, 3 figures. Khang Le, Dung Le, Huy Nguyen contributed equally to this work
References in corpus (5)
- Multilevel Clustering via Wasserstein Means
- Entropic Optimal Transport between Unbalanced Gaussian Measures has a Closed Form
- The Unbalanced Gromov Wasserstein Distance: Conic Formulation and Relaxation
- Graph Optimal Transport for Cross-Domain Alignment
- Gromov-Wasserstein Distances between Gaussian Distributions