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Exact Solutions for the NP-hard Wasserstein Barycenter Problem using a Doubly Nonnegative Relaxation and a Splitting Method

Woosuk L. Jung, Henry Wolkowicz

TL;DR

There is a duality gap for problems with \emph{enough} multiple optimal solutions, and that this arises from problems with highly symmetrized structure, and that this arises from problems with highly symmetrized structure is demonstrated.

Abstract

The so-called \emph{simplified} Wasserstein barycenter problem, also known as the cheapest hub problem, consists in selecting one point from each of $k$ given sets, each set consisting of $n$ points, with the aim of minimizing the sum of distances to the barycenter of the $k$ chosen points. This problem is known to be NP-hard. We compute the Wasserstein barycenter by exploiting the Euclidean distance matrix structure to obtain a facially reduced doubly nonnegative, DNN, relaxation. The facial reduction provides a natural splitting for applying the symmetric alternating directions method of multipliers (sADMM) to the DNN relaxation. The sADMM method exploits structure in the subproblems to find strong upper and lower bounds. In addition, we extend the problem to allow varying $n_j$ points for the $j$-th set. The purpose of this paper is twofold. First we want to illustrate the strength of this DNN relaxation with the natural splitting approach mentioned above. Our numerical tests then illustrate the surprising success on random problems, as we generally, efficiently, find the provable exact solution of this NP-hard problem. Comparisons with current commercial software illustrate this surprising efficiency. However, we demonstrate and prove that there is a duality gap for problems with \emph{enough} multiple optimal solutions, and that this arises from problems with highly symmetrized structure.

Exact Solutions for the NP-hard Wasserstein Barycenter Problem using a Doubly Nonnegative Relaxation and a Splitting Method

TL;DR

There is a duality gap for problems with \emph{enough} multiple optimal solutions, and that this arises from problems with highly symmetrized structure, and that this arises from problems with highly symmetrized structure is demonstrated.

Abstract

The so-called \emph{simplified} Wasserstein barycenter problem, also known as the cheapest hub problem, consists in selecting one point from each of given sets, each set consisting of points, with the aim of minimizing the sum of distances to the barycenter of the chosen points. This problem is known to be NP-hard. We compute the Wasserstein barycenter by exploiting the Euclidean distance matrix structure to obtain a facially reduced doubly nonnegative, DNN, relaxation. The facial reduction provides a natural splitting for applying the symmetric alternating directions method of multipliers (sADMM) to the DNN relaxation. The sADMM method exploits structure in the subproblems to find strong upper and lower bounds. In addition, we extend the problem to allow varying points for the -th set. The purpose of this paper is twofold. First we want to illustrate the strength of this DNN relaxation with the natural splitting approach mentioned above. Our numerical tests then illustrate the surprising success on random problems, as we generally, efficiently, find the provable exact solution of this NP-hard problem. Comparisons with current commercial software illustrate this surprising efficiency. However, we demonstrate and prove that there is a duality gap for problems with \emph{enough} multiple optimal solutions, and that this arises from problems with highly symmetrized structure.
Paper Structure (31 sections, 13 theorems, 75 equations, 6 figures, 2 tables, 1 algorithm)

This paper contains 31 sections, 13 theorems, 75 equations, 6 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Outline
  3. Notation
  4. Simplified Wasserstein barycenters, WBP
  5. Main problem and EDM connection
  6. A reformulation using a Euclidean distance matrix
  7. Facially reduced DNN relaxation
  8. Semidefinite programming (SDP) relaxation
  9. SDP reformulation
  10. Relaxing the rank-$1$ constraint
  11. The gangster constraint
  12. Doubly nonnegative (DNN) relaxation
  13. Formulating facial vector $V$ for sparsity
  14. DNN reformulation via facial reduction
  15. Characterization of optimality for DNN relaxation
  16. ...and 16 more sections

Key Result

Lemma 2.2

Suppose that we are given $k$ points $q_i\in \mathbb{R}^d, i=1,\ldots k$. Let $\bar{y}=\frac{1}{k}\sum_{i=1}^k q_i$ denote the barycenter. Then

Figures (6)

  • Figure 1: V matrix for k=25, n=25
  • Figure 1: Gurobi: size $N=kn$ versus cpu time; illustrating exponential time
  • Figure 1: Duality gap for wheel of wheels: k=3=n
  • Figure 2: sADMM : size $N=kn$ versus cpu time; illustrating linear time
  • Figure 2: No duality gap for wheel of wheels: k=6=n
  • ...and 1 more figures

Theorems & Definitions (28)

  • Lemma 2.2
  • Proof 1
  • Lemma 2.3
  • Proof 2
  • Corollary 2.4
  • Proof 3
  • Remark 2.5: difficulty of the Wasserstein barycenter problem
  • Proposition 3.1
  • Proof 4
  • Proposition 3.2
  • ...and 18 more