Sinkhorn Algorithm for Lifted Assignment Problems

Yam Kushinsky, Haggai Maron, Nadav Dym, Yaron Lipman

Published 2017-07-23Version 1

Recently, Sinkhorn's algorithm was applied for solving regularized linear programs emerging from optimal transport very efficiently. Sinkhorn's algorithm is an efficient method of projecting a positive matrix onto the polytope of doubly-stochastic matrices. It is based on alternating closed-form Bregman projections on the larger polytopes of row-stochastic and column-stochastic matrices. In this paper we generalize the Sinkhorn projection algorithm to higher dimensional polytopes originated from well-known lifted linear program relaxations of the Markov Random Field (MRF) energy minimization problem and the Quadratic Assignment Problem (QAP). We derive a closed-form projection on one-sided local polytopes which can be seen as a high-dimensional, generalized version of the row/column-stochastic polytopes. We then use these projections to devise a provably convergent algorithm to solve regularized linear program relaxations of MRF and QAP. Furthermore, as the regularization is decreased both the solution and the optimal energy value converge to that of the respective linear program. The resulting algorithm is considerably more scalable than standard linear solvers and is able to solve significantly larger linear programs.