Measure transfer via stochastic slicing and matching

TL;DR

This paper introduces a stochastic slicing-and-matching scheme for measure transfer, providing an almost sure convergence proof and demonstrating its effectiveness through image morphing applications.

Contribution

It offers the first convergence proof for stochastic slicing-and-matching schemes in measure transfer, interpreting them as stochastic gradient descent in Wasserstein space.

Findings

Convergence is proven for the proposed scheme.

Numerical experiments show effective image morphing.

Scheme benefits from closed-form solutions of 1D optimal transport.

Abstract

This paper studies iterative schemes for measure transfer and approximation problems, which are defined through a slicing-and-matching procedure. Similar to the sliced Wasserstein distance, these schemes benefit from the availability of closed-form solutions for the one-dimensional optimal transport problem and the associated computational advantages. While such schemes have already been successfully utilized in data science applications, not too many results on their convergence are available. The main contribution of this paper is an almost sure convergence proof for stochastic slicing-and-matching schemes. The proof builds on an interpretation as a stochastic gradient descent scheme on the Wasserstein space. Numerical examples on step-wise image morphing are demonstrated as well.