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We present a novel kernel over the space of probability measures based on thedual formulation of optimal regularized transport. We propose an Hilbertianembedding of the space of probabilities using their Sinkhorn potentials, whichare solutions of the dual entropic relaxed optimal transport between theprobabilities and a reference measure $\mathcal{U}$. We prove that thisconstruction enables to obtain a valid kernel, by using the Hilbert norms. Weprove that the kernel enjoys theoretical properties such as universality andsome invariances, while still being computationally feasible. Moreover weprovide theoretical guarantees on the behaviour of a Gaussian process based onthis kernel. The empirical performances are compared with other traditionalchoices of kernels for processes indexed on distributions.

Gaussian Processes on Distributions based on Regularized Optimal Transport