Learning Permutation from Structure Without Supervision

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.

Many machine learning problems involve recovering the ordering of elements: sorting a list, reassembling a jigsaw puzzle, or finding a short route between cities. Such orderings can be learned without labeled answers by checking whether the reordered output has the expected structure. Standard methods rely on a single parameter that controls how decisively the system commits to a single ordering, which is gradually annealed during training. This is problematic because some assignments become clear early while others remain genuinely ambiguous: tightening too quickly locks in mistakes, while tightening too slowly prevents convergence. We propose giving each assignment its own certainty level, automatically adjusted by its current uncertainty. Across sorting, jigsaw reconstruction, and route planning, this yields more accurate solutions and more stable training, with the largest improvements on the largest and most ambiguous problems.