Scalable Learning from Probability Measures with Mean Measure Quantization

TL;DR

This paper introduces a quantization method for probability measures in statistical learning, reducing computational costs of optimal transport while maintaining accuracy, demonstrated through theoretical guarantees and experiments.

Contribution

It proposes a novel quantization approach for probability measures that ensures consistency and convergence guarantees, improving efficiency in OT-based learning tasks.

Findings

Achieves comparable performance to individual quantization methods.

Substantially reduces runtime in OT computations.

Provides theoretical guarantees for convergence and consistency.

Abstract

We consider statistical learning problems in which data are observed as a set of probability measures. Optimal transport (OT) is a popular tool to compare and manipulate such objects, but its computational cost becomes prohibitive when the measures have large support. We study a quantization-based approach in which all input measures are approximated by $K$-point discrete measures sharing a common support. We establish consistency of the resulting quantized measures. We further derive convergence guarantees for several OT-based downstream tasks computed from the quantized measures. Numerical experiments on synthetic and real datasets demonstrate that the proposed approach achieves performance comparable to individual quantization while substantially reducing runtime.