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Kernel-based optimal transport (OT) estimators offer an alternative,functional estimation procedure to address OT problems from samples. Recentworks suggest that these estimators are more statistically efficient thanplug-in (linear programming-based) OT estimators when comparing probabilitymeasures in high-dimensions~\citep{Vacher-2021-Dimension}. Unfortunately, thatstatistical benefit comes at a very steep computational price: because theircomputation relies on the short-step interior-point method (SSIPM), which comeswith a large iteration count in practice, these estimators quickly becomeintractable w.r.t. sample size $n$. To scale these estimators to larger $n$, wepropose a nonsmooth fixed-point model for the kernel-based OT problem, and showthat it can be efficiently solved via a specialized semismooth Newton (SSN)method: We show, exploring the problem's structure, that the per-iteration costof performing one SSN step can be significantly reduced in practice. We provethat our SSN method achieves a global convergence rate of $O(1/\sqrt{k})$, anda local quadratic convergence rate under standard regularity conditions. Weshow substantial speedups over SSIPM on both synthetic and real datasets.

A Specialized Semismooth Newton Method for Kernel-Based Optimal Transport