A Specialized Semismooth Newton Method for Kernel-Based Optimal Transport

Tianyi Lin, marco cuturi, Michael Jordan

Keywords: kernal-based optimal transport estimation, nonsmooth equation model, specialized semi-smooth Newton method

TL;DR: We show that kernel-based OT estimation can be efficiently solved via specialized semismooth Newton method.

Abstract: Kernel-based optimal transport (OT) estimation is an alternative to the standard plug-in OT estimation. Recent works suggested that kernel-based OT estimators are more statistically efficient than plug-in OT estimators when comparing probability measures in high-dimensions~\citep{Vacher-2021-Dimension}. However, the computation of these estimators relies on the short-step interior-point method for which the required number of iterations is known to be \textit{large} in practice. In this paper, we propose a nonsmooth equation model for kernel-based OT estimation and show that it can be efficiently solved via a specialized semismooth Newton (SSN) method. Indeed, by exploring the special problem structure, the per-iteration cost of performing one SSN step can be significantly reduced in practice. We also prove that our algorithm can achieve a global convergence rate of $O(1/\sqrt{k})$ and a local quadratic convergence rate under some standard regularity conditions. Finally, we demonstrate the effectiveness of our algorithm by conducing the experiments on both synthetic and real datasets.

Submission Number: 10920