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Graph Matching: Relax at Your Own Risk.

Vince Lyzinski, Donniell E Fishkind, Marcelo Fiori

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    Graph matching, aligning graphs to minimize edge disagreements, is computationally difficult. A new indefinite relaxation method, combined with convex relaxation, significantly improves optimal permutation discovery for graph matching problems.

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    Area of Science:

    • Graph theory
    • Combinatorics
    • Computer Vision
    • Computational Neuroscience (Connectomics)

    Background:

    • Graph matching is a fundamental problem in aligning graph pairs, crucial for fields like computer vision and connectomics.
    • Existing heuristic methods for graph matching often lack theoretical guarantees on performance.
    • Continuous relaxation techniques enable gradient-descent algorithms but have limitations.

    Purpose of the Study:

    • To theoretically analyze the performance of different continuous relaxations for graph matching.
    • To propose and validate a hybrid approach combining indefinite and convex relaxations for improved graph matching accuracy.

    Main Methods:

    • Theoretical analysis of indefinite and convex relaxations for graph matching.
    • Proving that exact solutions to indefinite relaxations typically yield optimal permutations.
    • Demonstrating that common convex relaxations often fail to find optimal permutations.

    Main Results:

    • The study proves that indefinite relaxations, when solved exactly, almost always find the optimal permutation for graph matching.
    • Conversely, common convex relaxations are shown to frequently fail in discovering the optimal permutation.
    • Experimental validation confirms that initializing indefinite algorithms with convex optima enhances practical performance.

    Conclusions:

    • Indefinite relaxations offer superior theoretical guarantees for solving the graph matching problem compared to convex relaxations.
    • A hybrid approach, leveraging the strengths of both convex and indefinite relaxations, achieves excellent results on benchmark and real-world graph matching tasks.