A Specialized Semismooth Newton Method for Kernel-Based Optimal Transport
CoRR(2023)
摘要
Kernel-based optimal transport (OT) estimators offer an alternative,
functional estimation procedure to address OT problems from samples. Recent
works suggest that these estimators are more statistically efficient than
plug-in (linear programming-based) OT estimators when comparing probability
measures in high-dimensions . Unfortunately, that
statistical benefit comes at a very steep computational price: because their
computation relies on the short-step interior-point method (SSIPM), which comes
with a large iteration count in practice, these estimators quickly become
intractable w.r.t. sample size n. To scale these estimators to larger n, we
propose a nonsmooth fixed-point model for the kernel-based OT problem, and show
that it can be efficiently solved via a specialized semismooth Newton (SSN)
method: We show, exploring the problem's structure, that the per-iteration cost
of performing one SSN step can be significantly reduced in practice. We prove
that our SSN method achieves a global convergence rate of O(1/√(k)), and
a local quadratic convergence rate under standard regularity conditions. We
show substantial speedups over SSIPM on both synthetic and real datasets.
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关键词
specialized semismooth newton method,transport,optimal
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