Efficient Joint Object Matching via Linear Programming

TL;DR

This paper introduces scalable linear programming relaxations with theoretical guarantees for joint object matching, a complex problem in computer vision, demonstrating high accuracy under certain corruption levels and outperforming existing methods.

Contribution

The paper develops novel LP relaxations for joint object matching, including a new characterization of consistent maps and a tight polytope approximation with theoretical performance guarantees.

Findings

LP relaxation recovers ground truth with high probability below 40% corruption

Proposed LP relaxations outperform SDP relaxations in recovery and tightness

Exponential family of facet inequalities enables efficient polytope approximation

Abstract

Joint object matching, also known as multi-image matching, namely, the problem of finding consistent partial maps among all pairs of objects within a collection, is a crucial task in many areas of computer vision. This problem subsumes bipartite graph matching and graph partitioning as special cases and is NP-hard, in general. We develop scalable linear programming (LP) relaxations with theoretical performance guarantees for joint object matching. We start by proposing a new characterization of consistent partial maps; this in turn enables us to formulate joint object matching as an integer linear programming (ILP) problem. To construct strong LP relaxations, we study the facial structure of the convex hull of the feasible region of this ILP, which we refer to as the joint matching polytope. We present an exponential family of facet-defining inequalities that can be separated in strongly polynomial time, hence obtaining a partial characterization of the joint matching polytope that is both tight and cheap to compute. To analyze the theoretical performance of the proposed LP relaxations, we focus on permutation group synchronization, an important special case of joint object matching. We show that under the random corruption model for the input maps, a simple LP relaxation, that is, an LP containing only a very small fraction of the proposed facet-defining inequalities, recovers the ground truth with high probability if the corruption level is below $40\%$. Finally, via a preliminary computational study on synthetic data, we show that the proposed LP relaxations outperform a popular SDP relaxation both in terms of recovery and tightness.