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Procrustes Wasserstein Metric: A Modified Benamou-Brenier Approach with Applications to Latent Gaussian Distributions
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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.
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Dynamical Optimal Transport with $\mathfrak{so}(d)$-Invariance: From Theory to Computation
A modified Benamou–Brenier action with Euclidean invariance equals the static Procrustes–Wasserstein distance, and for Gaussians this distance is the distance between square-root spectra.
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