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Streaming Sliced Optimal Transport

Khai Nguyen

TL;DR

This work tackles the challenge of estimating the sliced Wasserstein distance from streaming data under tight memory constraints. It introduces Stream-SW, built upon streaming estimators for the 1D Wasserstein distance via quantile sketches (KKL) and projections onto random directions, to produce a streaming counterpart to SW with probabilistic error bounds and favorable space/time complexity. The paper provides theoretical guarantees for CDF/quantile estimation, 1D OT maps, and the overall Stream-SW distance, and demonstrates superior accuracy and downstream performance relative to random subsampling across Gaussian mixtures, streaming point-cloud tasks, gradient flows, and change-point detection. The work enables scalable, single-pass distribution comparison in streaming settings with practical impact for IoT, real-time analytics, and online learning pipelines.

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 \emph{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 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

Streaming Sliced Optimal Transport

TL;DR

This work tackles the challenge of estimating the sliced Wasserstein distance from streaming data under tight memory constraints. It introduces Stream-SW, built upon streaming estimators for the 1D Wasserstein distance via quantile sketches (KKL) and projections onto random directions, to produce a streaming counterpart to SW with probabilistic error bounds and favorable space/time complexity. The paper provides theoretical guarantees for CDF/quantile estimation, 1D OT maps, and the overall Stream-SW distance, and demonstrates superior accuracy and downstream performance relative to random subsampling across Gaussian mixtures, streaming point-cloud tasks, gradient flows, and change-point detection. The work enables scalable, single-pass distribution comparison in streaming settings with practical impact for IoT, real-time analytics, and online learning pipelines.

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 \emph{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 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
Paper Structure (27 sections, 6 theorems, 83 equations, 9 figures, 3 tables)

This paper contains 27 sections, 6 theorems, 83 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Streaming Sliced Optimal Transport
  4. Quantile Sketch
  5. Streaming CDF and Quantile Function Estimation
  6. Streaming CDF estimation.
  7. Streaming quantile function estimation.
  8. Streaming one-dimensional Wasserstein distance.
  9. Streaming one-dimensional transport map.
  10. Streaming Sliced Wasserstein
  11. Computational complexities.
  12. Experiments
  13. Approximation Error Analysis
  14. Streaming Point-cloud Classification
  15. Streaming Gradient Flow
  16. ...and 12 more sections

Key Result

Proposition 1

Given $\mathcal{S}_{\mu,k}$ be the quantile sketch with initial size $k>0$ constructed from streaming samples $x_1,\ldots,x_n$ from $\mu \in \mathcal{P}(\mathbb{R})$, the following probability bound holds: $\forall \, x\in \mathbb{R}$, where $C>0$ is a constant. When $\mu$ has compact support with diameter $R >0$, the following probability bound holds: where $C_{R}>0$ is a constant which depends

Figures (9)

  • Figure 1: The figure shows the compacting process of a KKL sketch.
  • Figure 2: The figure shows the computational procedure of Stream-SW. In the figure, we denotes $\widetilde{W}_p^p(\mu_n,\nu_m;\mathcal{S}_{\mu_n,k_1},\mathcal{S}_{\nu_m,k_2})$ as $\widetilde{W}_p^p(\mu_n,\nu_m)$ and $\widehat{\widetilde{SW}}_p^p(\mu_n,\nu_m;k_1,k_2,L)$ as $\widehat{\widetilde{SW}}_p^p(\mu_n,\nu_m;L)$.
  • Figure 3: Relative errors and number of samples when comparing mixtures of Gaussian distributions.
  • Figure 4: Gradient flows from Full SW, SW with random sampling, and Stream-SW in turn.
  • Figure 5: Relative errors and number of samples when comparing Gaussian distributions
  • ...and 4 more figures

Theorems & Definitions (10)

  • Proposition 1
  • Proposition 2
  • Definition 1
  • Proposition 3
  • Proposition 4
  • Definition 2
  • Corollary 1
  • Theorem 1
  • Definition 3
  • Definition 4