Fast entropy-regularized SDP relaxations for permutation synchronization

TL;DR

This paper presents fast randomized algorithms for entropy-regularized SDP relaxations tailored to permutation synchronization, improving speed and accuracy in multi-image matching and 3D reconstruction tasks.

Contribution

It introduces entropy regularization to resolve non-uniqueness in SDP relaxations and develops scalable randomized solvers with effective rounding procedures for permutation synchronization.

Findings

Achieves state-of-the-art speed and accuracy on synthetic and real datasets.

Demonstrates theoretical advantages of entropy regularization in permutation synchronization.

Provides scalable algorithms with near-optimal complexity.

Abstract

We introduce fast randomized algorithms for solving semidefinite programming (SDP) relaxations of the partial permutation synchronization (PPS) problem, a core task in multi-image matching with significant relevance to 3D reconstruction. Our methods build on recent advances in entropy-regularized semidefinite programming and are tailored to the unique structure of PPS, in which the unknowns are partial permutation matrices aligning sparse and noisy pairwise correspondences across images. We prove that entropy regularization resolves optimizer non-uniqueness in standard relaxations, and we develop a randomized solver with nearly optimal scaling in the number of observed correspondences. We also develop several rounding procedures for recovering combinatorial solutions from the implicitly represented primal solution variable, maintaining cycle consistency if desired without harming computational scaling. We demonstrate that our approach achieves state-of-the-art performance on synthetic and real-world datasets in terms of speed and accuracy. Our results highlight PPS as a paradigmatic setting in which entropy-regularized SDP admits both theoretical and practical advantages over traditional low-rank or spectral techniques.