Abstract
We define a novel class of distances between statistical multivariate
distributions by solving an optimal transportation problem on their marginal
densities with respect to a ground distance defined on their conditional
densities. By using the chain rule factorization of probabilities, we show how
to perform optimal transport on a ground space being an information-geometric
manifold of conditional probabilities. We prove that this new distance is a
metric whenever the chosen ground distance is a metric. Our distance
generalizes both the Wasserstein distances between point sets and a recently
introduced metric distance between statistical mixtures. As a first application
of this Chain Rule Optimal Transport (CROT) distance, we show that the ground
distance between statistical mixtures is upper bounded by this optimal
transport distance and its fast relaxed Sinkhorn distance, whenever the ground
distance is joint convex. We report on our experiments which quantify the
tightness of the CROT distance for the total variation distance, the square
root generalization of the Jensen-Shannon divergence, the Wasserstein $W_p$
metric and the Rényi divergence between mixtures.
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