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Measure transfer via stochastic slicing and matching

Shiying Li, Caroline Moosmueller

TL;DR

This paper studies iterative schemes for measure transfer and approximation problems, which are defined through a slicing-and-matching procedure, and presents an almost sure convergence proof for stochastic slicing- and- matching schemes.

Abstract

This paper studies iterative schemes for measure transfer and approximation problems, which are defined through a slicing-and-matching procedure. Similar to the sliced Wasserstein distance, these schemes benefit from the availability of closed-form solutions for the one-dimensional optimal transport problem and the associated computational advantages. While such schemes have already been successfully utilized in data science applications, not too many results on their convergence are available. The main contribution of this paper is an almost sure convergence proof for stochastic slicing-and-matching schemes. The proof builds on an interpretation as a stochastic gradient descent scheme on the Wasserstein space. Numerical examples on step-wise image morphing are demonstrated as well.

Measure transfer via stochastic slicing and matching

TL;DR

This paper studies iterative schemes for measure transfer and approximation problems, which are defined through a slicing-and-matching procedure, and presents an almost sure convergence proof for stochastic slicing- and- matching schemes.

Abstract

This paper studies iterative schemes for measure transfer and approximation problems, which are defined through a slicing-and-matching procedure. Similar to the sliced Wasserstein distance, these schemes benefit from the availability of closed-form solutions for the one-dimensional optimal transport problem and the associated computational advantages. While such schemes have already been successfully utilized in data science applications, not too many results on their convergence are available. The main contribution of this paper is an almost sure convergence proof for stochastic slicing-and-matching schemes. The proof builds on an interpretation as a stochastic gradient descent scheme on the Wasserstein space. Numerical examples on step-wise image morphing are demonstrated as well.
Paper Structure (25 sections, 16 theorems, 71 equations, 2 figures)

This paper contains 25 sections, 16 theorems, 71 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Sliced Wasserstein distances and iterative approximation schemes
  3. Stochastic iterative approximation schemes
  4. Main contribution
  5. Relation to gradient flows
  6. Structure of the paper
  7. Preliminaries
  8. Optimal transport preliminaries
  9. Sliced Wasserstein distances
  10. Slice-matching maps
  11. Iterative schemes via slice-matching maps
  12. Convergence of the stochastic iterative scheme
  13. Stochastic gradient descent interpretation
  14. Stochastic single-slice matching
  15. Stochastic $j$-slice matching
  16. ...and 10 more sections

Key Result

Theorem 1

Consider two measures $\sigma_0,\mu$ over $\mathbb{R}^n$ and let $P_k \overset{\textrm{i.i.d}}{\sim} u_n, k\geq 0$, where $u_n$ is the Haar probability measure on $O(n)$. Define $\sigma_k$ by the iteration intro:randomized_iter using $\sigma_0,\mu$ and $P_k$. Then under some technical assumptions we

Figures (2)

  • Figure 1: Matrix-slice matching iteration with randomization as described in \ref{['subsec:Ex-Matrix']}. Top panel: Morphing the digit $5$ ($\sigma_0$, image top left) into the digit $1$ ($\mu$, image bottom right). Snapshots of inbetween iterations are shown. Bottom panel: The relative error $\frac{SW_2(\sigma_k,\mu)}{SW_2(\sigma_0,\mu)}$ is plotted against the iteration variable $k=0,\ldots,20$. The average relative error and standard deviation are computed across 15 trials with randomly chosen orthogonal matrices $P_k$. Grey lines indicate the relative error for each trial.
  • Figure 2: Single-slice matching iteration with randomization as described in \ref{['subsec:Ex-Single']}. Top panel: Morphing the digit $5$ ($\sigma_0$, image top left) into the digit $1$ ($\mu$, image bottom right). Snapshots of inbetween iterations are shown. Bottom panel: The relative error $\frac{SW_2(\sigma_k,\mu)}{SW_2(\sigma_0,\mu)}$ is plotted against the iteration variable $k=0,\ldots,30$. The average relative error and standard deviation are computed across 15 trials with randomly chosen orthogonal matrices $\theta_k$. Grey lines indicate the relative error for each trial.

Theorems & Definitions (50)

  • Theorem 1: Special case of \ref{['mainthm-Pj']}; informal version
  • Lemma 2.1: bonnotte13thesis
  • proof
  • Definition 1: Definition from pitie2007automated
  • Definition 2
  • Remark 1
  • Remark 2
  • Definition 3
  • Remark 3
  • Lemma 3.1
  • ...and 40 more