Entropic Gromov-Wasserstein between Gaussian Distributions
This provides theoretical tools for aligning and averaging Gaussian distributions in machine learning, but it is incremental as it extends existing entropic Gromov-Wasserstein methods to specific cases.
The paper tackled the problem of computing the entropic Gromov-Wasserstein distance between Gaussian distributions with different dimensions, showing that optimal transport plans are Gaussian and deriving closed-form expressions for the distance and barycenter covariance matrix.
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.