DS*: Tighter Lifting-Free Convex Relaxations for Quadratic Matching Problems

TL;DR

This paper introduces a new lifting-free convex relaxation for quadratic permutation problems that is as tight as existing methods, offering improved practical applicability and superior performance in image arrangement and multi-graph matching tasks.

Contribution

The authors present a novel lifting-free convex relaxation that matches the tightness of existing lifted relaxations, enhancing computational efficiency and applicability.

Findings

Our relaxation is at least as tight as existing lifted relaxations.

Experimental results show superior performance over existing methods.

Effective in image arrangement and multi-graph matching problems.

Abstract

In this work we study convex relaxations of quadratic optimisation problems over permutation matrices. While existing semidefinite programming approaches can achieve remarkably tight relaxations, they have the strong disadvantage that they lift the original $n {\times} n$-dimensional variable to an $n^2 {\times} n^2$-dimensional variable, which limits their practical applicability. In contrast, here we present a lifting-free convex relaxation that is provably at least as tight as existing (lifting-free) convex relaxations. We demonstrate experimentally that our approach is superior to existing convex and non-convex methods for various problems, including image arrangement and multi-graph matching.