Broadly speaking, we study an evolution of mass from a source distribution to a target, defined by a (continuous-time) gradient descent of a functional acting like a ‘distance’ to the target. We are interested in the well-posedness of such an evolution (characterized by a PDE) and the criterion under which it converges to the target. Our study takes the source and target to be Gaussian distributions, as it yields closed-form expressions. We show convergence when the source is not degenerate (i.e. is supported on the whole Euclidean space), and provide counter-examples when it is degenerate (i.e. supported on a strict subspace). In the case where the covariances of the source and target distributions commute, we give quantitative convergence rates, being exponential when the source and measures have the same support but dropping to sublinear (O(1/t)) when the target is concentrated on a strict subspace of the source’s support. The first section of the presentation will be devoted to the exposition of the theory of optimal transport and how it allows to generalize the notion of gradient descent to the space of probability measures, as well as the definition and properties of the considered functional, which is built on an entropic version of optimal transport. The rest of the presentation will deal with the results of our paper.
Localisation
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