Procrustes Wasserstein Metric: A Modified Benamou-Brenier Approach with Applications to Latent Gaussian Distributions
This work addresses the need for invariant metrics in optimal transport for applications like latent variable modeling, but it appears incremental as it modifies an existing approach.
The paper tackles the problem of defining a Wasserstein-type distance with global invariance to isometries, showing that for Gaussian distributions it reduces to measuring Euclidean distance between ordered eigenvalues, with a direct application in recovering latent Gaussian distributions.
We introduce a modified Benamou-Brenier type approach leading to a Wasserstein type distance that allows global invariance, specifically, isometries, and we show that the problem can be summarized to orthogonal transformations. This distance is defined by penalizing the action with a costless movement of the particle that does not change the direction and speed of its trajectory. We show that for Gaussian distribution resume to measuring the Euclidean distance between their ordered vector of eigenvalues and we show a direct application in recovering Latent Gaussian distributions.