OT4P: Unlocking Effective Orthogonal Group Path for Permutation Relaxation

Yaming Guo, Chen Zhu, Hengshu Zhu, Tieru Wu

Keywords: permutation matrix, orthogonal group, differentiable transformation, stochastic optimization

TL;DR: We present a novel differentiable transformation for relaxing permutation matrices onto the orthogonal group, which enables gradient-based (stochastic) optimization of problems involving permutation matrices.

Abstract: Optimization over permutations is typically an NP-hard problem that arises extensively in ranking, matching, tracking, etc. Birkhoff polytope-based relaxation methods have made significant advancements, particularly in penalty-free optimization and probabilistic inference. Relaxation onto the orthogonal group offers unique potential advantages such as a lower representation dimension and preservation of inner products; however, equally effective approaches remain unexplored. To bridge the gap, we present a temperature-controlled differentiable transformation that maps unconstrained vector space to the orthogonal group, where the temperature, in the limit, concentrates orthogonal matrices near permutation matrices. This transformation naturally implements a parameterization for the relaxation of permutation matrices, allowing for gradient-based optimization of problems involving permutations. Additionally, by deriving a re-parameterized gradient estimator, this transformation also provides efficient stochastic optimization over the latent permutations. Extensive experiments involving the optimization over permutation matrices validate the effectiveness of the proposed method.

Primary Area: Optimization (convex and non-convex, discrete, stochastic, robust)

Submission Number: 2787