Streaming Sliced Optimal Transport

TL;DR

This paper introduces Stream-SW, a novel streaming estimator for sliced Wasserstein distance that offers improved computational efficiency and accuracy for sample streams, with applications in point cloud analysis and change detection.

Contribution

The authors develop the first streaming method for estimating sliced Wasserstein distance, reducing memory usage while maintaining theoretical approximation guarantees.

Findings

Stream-SW outperforms random subsampling in accuracy for Gaussian and Gaussian mixture distributions.

Stream-SW demonstrates lower memory consumption compared to traditional methods.

Experiments show effective application of Stream-SW in point cloud classification and change point detection.

Abstract

Sliced optimal transport (SOT), or sliced Wasserstein (SW) distance, is widely recognized for its statistical and computational scalability. In this work, we further enhance computational scalability by proposing the first method for estimating SW from sample streams, called streaming sliced Wasserstein (Stream-SW). To define Stream-SW, we first introduce a streaming estimator of the one-dimensional Wasserstein distance (1DW). Since the 1DW has a closed-form expression, given by the integral of the absolute difference between the quantile functions of the compared distributions, we leverage quantile approximation techniques for sample streams to define a streaming 1DW estimator. By applying the streaming 1DW to all projections, we obtain Stream-SW. The key advantage of Stream-SW is its low memory complexity while providing theoretical guarantees on the approximation error. We demonstrate that Stream-SW achieves a more accurate approximation of SW than random subsampling, with lower memory consumption, when comparing Gaussian distributions and mixtures of Gaussians from streaming samples. Additionally, we conduct experiments on point cloud classification, point cloud gradient flows, and streaming change point detection to further highlight the favorable performance of the proposed Stream-SW.