Measure transfer via stochastic slicing and matching
This provides theoretical guarantees for a computational method in data science, but it is incremental as it builds on existing slicing techniques.
The paper tackles the convergence of iterative measure transfer schemes using stochastic slicing and matching, proving almost sure convergence and demonstrating results with numerical examples on image morphing.
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.