Gaussian Processes on Distributions based on Regularized Optimal Transport
This work addresses the challenge of modeling distributions in machine learning, offering a computationally feasible kernel with theoretical guarantees, though it appears incremental as it builds on existing optimal transport methods.
The authors tackled the problem of defining a kernel over probability measures by introducing a novel kernel based on regularized optimal transport, specifically using Sinkhorn potentials, and demonstrated its empirical performance compared to traditional kernels.
We present a novel kernel over the space of probability measures based on the dual formulation of optimal regularized transport. We propose an Hilbertian embedding of the space of probabilities using their Sinkhorn potentials, which are solutions of the dual entropic relaxed optimal transport between the probabilities and a reference measure $\mathcal{U}$. We prove that this construction enables to obtain a valid kernel, by using the Hilbert norms. We prove that the kernel enjoys theoretical properties such as universality and some invariances, while still being computationally feasible. Moreover we provide theoretical guarantees on the behaviour of a Gaussian process based on this kernel. The empirical performances are compared with other traditional choices of kernels for processes indexed on distributions.