Mapping between two Gaussians using optimal transport and the KL-divergence

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Suppose you have two multivariate Gaussian distributions \( S\) and \(T\), parameterized as \( N(\mu_S, \Sigma_S)\) and \( N(\mu_T, \Sigma_T)\). How do you linearly transform \( x \sim S\) so that the transformed vectors have distribution \( T\)? Is there an optimal way to do this? The field of optimal transport (OT) provides an answer. If we choose the transport cost as the type-2 Wasserstein distance \( W^2_2\) between probability measures, then we apply the following linear function:

## Mapping between two Gaussians using optimal transport and the KL-divergence

## Mapping between two Gaussians using optimal…

## Mapping between two Gaussians using optimal transport and the KL-divergence

Suppose you have two multivariate Gaussian distributions \( S\) and \(T\), parameterized as \( N(\mu_S, \Sigma_S)\) and \( N(\mu_T, \Sigma_T)\). How do you linearly transform \( x \sim S\) so that the transformed vectors have distribution \( T\)? Is there an optimal way to do this? The field of optimal transport (OT) provides an answer. If we choose the transport cost as the type-2 Wasserstein distance \( W^2_2\) between probability measures, then we apply the following linear function: