Semidefinite Programming Relaxations of the Simplified Wasserstein Barycenter Problem: An ADMM Approach
Loading...
Date
2023-05-04
Authors
Cheng, Jiahui
Journal Title
Journal ISSN
Volume Title
Publisher
University of Waterloo
Abstract
The Simplified Wasserstein Barycenter problem, the problem of picking k points each chosen from a distinct set of n points as to minimize the sum of distances to their barycenter, finds applications in various areas of data science. Despite the simple formulation, it is a hard computational problem. The difficulty comes in the lack of efficient algorithms for approximating the solution. In this thesis, I propose a doubly non-negative relaxation to this problem and apply the alternating direction method of multipliers (ADMM) with intermediate update of multipliers, to efficiently compute tight lower and upper bounds on its optimal value for certain input data distributions. Our empirics show that generically the gap between upper and lower bounds is zero, though problems with symmetries exhibit positive gaps.
Description
Keywords
continuous optimization, semidefinite programming relaxation, Wasserstein Barycenters, alternating direction method of multipliers, doubly non-negative relaxation, Peaceman-Rachford splitting method, projection method, splitting method, convex relaxation, simplified Wasserstein Barycenters