Fast entropy-regularized SDP relaxations for permutation synchronization

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.