Learning Permutation from Structure Without Supervision

2026-05-25

AI Summary

This work addresses the challenges of uneven assignment uncertainty and training instability in recovering hidden permutations from unordered data under unsupervised settings. The authors propose an entropy-adaptive Gumbel-Sinkhorn approach that approximates permutation matrices via differentiable relaxation into doubly stochastic matrices, enhanced with a local temperature modulation mechanism. This mechanism dynamically adjusts the relaxation temperature at each position based on its uncertainty: high-confidence regions are encouraged to converge rapidly toward discrete solutions, while ambiguous regions retain sufficient exploration capacity. Empirical results demonstrate that the method significantly outperforms fixed-temperature baselines across sorting, jigsaw puzzle reconstruction, and routing tasks—particularly in large-scale, highly ambiguous scenarios—while simultaneously improving both training stability and final permutation accuracy.

Abstract

Many learning problems require uncovering a hidden ordering that reveals structure in unordered data, such as monotonicity in sorting or spatial continuity in jigsaw reconstruction. In these settings, permutations can be learned as latent operators by optimizing objectives defined directly on the reordered output, often without access to ground-truth orderings. Differentiable relaxations such as Gumbel-Sinkhorn make this approach practical by approximating permutation matrices with doubly stochastic matrices. However, learning from structure without supervision induces a non-uniform uncertainty: some assignments become confident early, while others remain ambiguous. Existing methods control this process using a single global temperature, forcing all assignments to sharpen or diffuse simultaneously and leading to instability at scale. We introduce an entropy-adaptive formulation of Gumbel-Sinkhorn that locally modulates temperature based on assignment uncertainty. This allows confident assignments to discretize early while preserving exploration where uncertainty remains. Across sorting and jigsaw reconstruction tasks and in routing-style settings, adaptive entropy control improves training stability and final permutation quality relative to fixed-temperature baselines, particularly as problem size and assignment ambiguity increase.

Problem

• permutation learning
• unsupervised learning
• assignment uncertainty
• temperature control
• differentiable relaxation

Innovation

• entropy-adaptive
• Gumbel-Sinkhorn
• unsupervised permutation learning
• local temperature modulation
• doubly stochastic matrices