[Paper Review] Incorporating Domain Knowledge in Matching Problems via Harmonic Analysis

Deepti Pachauri, Maxwell D. Collins+2 more|arXiv (Cornell University)|Jun 27, 2012

Machine Learning and Algorithms4 citations

TL;DR

This paper proposes a novel method to accelerate solving quadratic assignment problems (QAPs) in matching tasks by leveraging domain-specific knowledge through harmonic analysis on the symmetric group. By learning parameters for a modified objective function using Fourier-domain optimization over permutations, the approach significantly improves efficiency and accuracy on practical matching problems compared to existing methods.

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.

Motivation & Objective

• To address the computational hardness of quadratic assignment problems (QAPs) in matching tasks across machine learning and computer vision.
• To exploit domain-specific knowledge—specifically, prior solutions to similar QAP instances—to improve solution efficiency.
• To develop a learning-based method that adapts the objective function for QAPs using algebraic structure of permutations.
• To accelerate convergence and improve accuracy in matching problems by optimizing over the symmetric group Sn in Fourier space.
• To demonstrate superior performance over existing QAP solvers in practical applications with known solution patterns.

Proposed method

• The method learns parameters for a modified QAP objective function from a set of previously solved QAP instances from the same application domain.
• It leverages the algebraic structure of the symmetric group Sn to represent and optimize over permutations.
• Fourier analysis on the symmetric group is used to represent and manipulate functions over permutations efficiently.
• The optimization is performed in the Fourier domain, enabling faster computation of function values and gradients.
• The learned parameters are used to guide the search for optimal matchings, reducing the need for exhaustive enumeration.
• The approach is validated on real-world matching problems where prior solutions are available for training.

Experimental results

Research questions

• RQ1 Can prior solutions to similar QAP instances be used to accelerate solving new instances of the same problem?
• RQ2 How can the algebraic structure of permutations be exploited to improve optimization efficiency in matching problems?
• RQ3 To what extent does learning a modified objective function via harmonic analysis improve solution quality and speed?
• RQ4 Can Fourier-domain optimization over the symmetric group Sn outperform traditional QAP solvers in practical settings?
• RQ5 How robust is the method to variations in the training set of prior QAP solutions?

Key findings

• The proposed method significantly outperforms standard QAP solvers in terms of solution quality and computational speed on practical matching tasks.
• By learning from prior instances, the method reduces the search space and accelerates convergence to optimal or near-optimal matchings.
• Fourier-domain optimization over the symmetric group enables efficient computation and parameter learning, even for large permutation spaces.
• The approach achieves state-of-the-art performance on benchmark matching problems where domain-specific knowledge is available.
• Experiments on real-world datasets show consistent improvements in accuracy and runtime over baseline methods.
• The method is particularly effective when multiple prior solutions from the same application domain are available for training.