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Convergence rate of empirical measures in the subspace robust Wasserstein distance

Dakshesh Vasan

TL;DR

This paper analyzes the convergence rate of empirical measures under the subspace robust Wasserstein distance in a separable Hilbert space. It establishes a near-optimal rate: the expected $1$-dimensional SRW distance between any measure on the unit ball and its empirical measure from $n$ i.i.d. samples scales as $\frac{\sqrt{\log\log n}}{\sqrt{\log n}}$, and proves a matching lower bound up to a $\sqrt{\log\log n}$ factor. The approach combines spectral projection onto the top eigen-directions of the second-moment operator with finite-dimensional Wasserstein bounds and SRW-structure inequalities to decouple high- and low-dimensional contributions. The results address dimension-robust convergence in SRW and have implications for robust optimal transport in high-dimensional learning tasks by clarifying how fast empirical SRW distances converge with sample size.

Abstract

We obtain an estimate for the expected subspace robust Wasserstein distance between any probability measure on the unit ball of a separable Hilbert space, and its empirical distribution from $n$ i.i.d. samples.

Convergence rate of empirical measures in the subspace robust Wasserstein distance

TL;DR

This paper analyzes the convergence rate of empirical measures under the subspace robust Wasserstein distance in a separable Hilbert space. It establishes a near-optimal rate: the expected -dimensional SRW distance between any measure on the unit ball and its empirical measure from i.i.d. samples scales as , and proves a matching lower bound up to a factor. The approach combines spectral projection onto the top eigen-directions of the second-moment operator with finite-dimensional Wasserstein bounds and SRW-structure inequalities to decouple high- and low-dimensional contributions. The results address dimension-robust convergence in SRW and have implications for robust optimal transport in high-dimensional learning tasks by clarifying how fast empirical SRW distances converge with sample size.

Abstract

We obtain an estimate for the expected subspace robust Wasserstein distance between any probability measure on the unit ball of a separable Hilbert space, and its empirical distribution from i.i.d. samples.

Paper Structure

This paper contains 15 sections, 10 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Contribution
  4. Main result
  5. Some definitions and notations
  6. Some definitions and notations
  7. Organization of this paper
  8. Some useful preliminary results
  9. Mean Absolute Deviation Estimate for binomial random variable
  10. Covering and Packing Number of Closed Unit Ball
  11. Bound on Expected Sample Covariance
  12. Some results on Wasserstein distance
  13. Some results on subspace robust Wasserstein distance
  14. Proof of upper bound in Theorem 1.1
  15. Proof of lower bound in Theorem 1.1

Key Result

Theorem 1.1

Suppose that $\mathcal{H}$ is a separable infinite dimensional Hilbert space. Let $\mathcal{P}$ be the collection of all probability measures on the closed unit ball $\{x \in \mathcal{H}: \left\lVert x \right\rVert \leq 1\}$. Given any $\mu \in \mathcal{P}$ and $n \geq 3$; let $\mu_n = \frac{1}{n}\u where $c, C>0$ are universal constants.

Theorems & Definitions (22)

  • Remark 1
  • Remark 2
  • Remark 3
  • proof
  • Theorem 1.1
  • Remark 4
  • Remark 5
  • Proposition 2.1
  • proof
  • Lemma 2.2: Berend, Theorem 1
  • ...and 12 more