Incorporating Domain Knowledge in Matching Problems via Harmonic Analysis

TL;DR

This paper introduces a novel method that leverages harmonic analysis and prior knowledge to accelerate solving quadratic assignment problems, common in machine learning and computer vision, by exploiting permutation algebraic structures.

Contribution

The paper proposes a new approach that learns parameters from previous QAP instances and uses harmonic analysis to improve solution efficiency.

Findings

Outperforms existing methods in practical domains

Utilizes algebraic structure of permutations for optimization

Accelerates solution of quadratic assignment problems

Abstract

Matching one set of objects to another is a ubiquitous task in machine learning and computer vision that often reduces to some form of the quadratic assignment problem (QAP). The QAP is known to be notoriously hard, both in theory and in practice. Here, we investigate if this difficulty can be mitigated when some additional piece of information is available: (a) that all QAP instances of interest come from the same application, and (b) the correct solution for a set of such QAP instances is given. We propose a new approach to accelerate the solution of QAPs based on learning parameters for a modified objective function from prior QAP instances. A key feature of our approach is that it takes advantage of the algebraic structure of permutations, in conjunction with special methods for optimizing functions over the symmetric group Sn in Fourier space. Experiments show that in practical domains the new method can outperform existing approaches.