Entropic Gromov-Wasserstein between Gaussian Distributions

• URL: http://arxiv.org/abs/2108.10961v1
• Date: Tue, 24 Aug 2021 21:27:11 GMT
• Title: Entropic Gromov-Wasserstein between Gaussian Distributions
• Authors: Khang Le and Dung Le and Huy Nguyen and Dat Do and Tung Pham and Nhat Ho
• Abstract summary: We study the entropic Gromov-Wasserstein and its unbalanced version between (unbalanced) Gaussian distributions. 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.
• Score: 9.624666285528612
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 Gaussian distribution when the entropic regularization parameter is small. We further derive closed-form expressions for the covariance matrix of the barycenter.

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