A Study of Lagrangean Decompositions and Dual Ascent Solvers for Graph Matching

Abstract

We study the quadratic assignment problem, in computer vision also known as graph matching. Two leading solvers for this problem optimize the Lagrange decomposition duals with sub-gradient and dual ascent (also known as message passing) updates. We explore this direction further and propose several additional Lagrangean relaxations of the graph matching problem along with corresponding algorithms, which are all based on a common dual ascent framework. Our extensive empirical evaluation gives several theoretical insights and suggests a new state-of-the-art anytime solver for the considered problem. Our improvement over state-of-the-art is particularly visible on a new dataset with large-scale sparse problem instances containing more than 500 graph nodes each.

MAP Inference via Block-Coordinate Frank-Wolfe Algorithm

“…Additionally, we tested state-of-the-art versions of message passing (MP) solvers, when applicable. MP is a popular method for MAP-MRF problems, and has recently been applied to the graph matching problem [38]. All solvers optimize the same underlying linear programming relaxation.…”

HiPPI: Higher-Order Projected Power Iterations for Scalable Multi-Matching

“…The QAP is a popular formalism for graph matching problems, where the first-order terms (on the diagonal of W ) account for node matching costs, and the second-order terms (on the off-diagonal of W ) account for edge matching costs. Existing methods that tackle the QAP/graph matching include spectral relaxations [28,15], linear relaxations [43,42], convex relaxations [55,38,34,19,1,24,18,6], path-following methods [54,56,23], kernel density estimation [46], branchand-bound methods [5] and many more, as described in the survey papers [36,29]. Also, tensor-based approaches for higher-order graph matching have been considered [17,33].…”

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

“…Whilst there exist many different ways to tackle GM problems (e.g. [50,12,58,26,30,47,48]), in the following we focus on convex-to-concave path-following (PF) approaches, as they are most relevant in our context. The idea of PF methods is to approximate the NP-hard GM problem by solving a sequence of continuous optimisation problems.…”