[论文解读] Learning Latent Permutations with Gumbel-Sinkhorn Networks

Gonzalo E. Mena, David Belanger+2 more|arXiv (Cornell University)|Feb 23, 2018

Topic Modeling被引用 93

一句话总结

论文介绍 Sinkhorn 网络和 Gumbel-Sinkhorn 分布，以实现带潜在置换的端到端学习，使用可微的 Sinkhorn 松弛来近似最大权匹配并实现重参数化梯度。

ABSTRACT

Permutations and matchings are core building blocks in a variety of latent variable models, as they allow us to align, canonicalize, and sort data. Learning in such models is difficult, however, because exact marginalization over these combinatorial objects is intractable. In response, this paper introduces a collection of new methods for end-to-end learning in such models that approximate discrete maximum-weight matching using the continuous Sinkhorn operator. Sinkhorn iteration is attractive because it functions as a simple, easy-to-implement analog of the softmax operator. With this, we can define the Gumbel-Sinkhorn method, an extension of the Gumbel-Softmax method (Jang et al. 2016, Maddison2016 et al. 2016) to distributions over latent matchings. We demonstrate the effectiveness of our method by outperforming competitive baselines on a range of qualitatively different tasks: sorting numbers, solving jigsaw puzzles, and identifying neural signals in worms.

研究动机与目标

• Motivate learning in models with latent matchings and permutations where exact marginalization is intractable.
• Introduce a differentiable approximation to permutation selection via the Sinkhorn operator.
• Extend Gumbel-Softmax ideas to permutations with Gumbel-Sinkhorn for reparameterizable inference.
• Develop permutation-equivariant network architectures that produce soft/perfect matchings for reconstruction tasks.
• Demonstrate empirical effectiveness on sorting, jigsaw puzzles, and neural signal identification in C. elegans.

提出的方法

• Define the Sinkhorn operator to map arbitrary matrices to doubly stochastic matrices as a differentiable relaxation of permutation matrices.
• Prove that the non-differentiable matching operator M(X) can be recovered as the limit of S(X/τ) as τ → 0, enabling a differentiable approximation.
• Introduce Sinkhorn networks where the final layer outputs a matrix whose rows represent unnormalized assignment scores and are turned into a soft permutation via S(·/τ).
• Extend to Gumbel-Sinkhorn: P ~ GS(X, τ) as S((X+ε)/τ) to enable reparameterized sampling for latent permutations.
• Apply variational inference with GS distributions to approximate posteriors over permutations in latent-variable models.
• Demonstrate end-to-end learning and reparameterized gradients in tasks including sorting, jigsaw puzzle reconstruction, and C. elegans neuron identification.

实验结果

研究问题

• RQ1 How can we approximate and optimize over latent permutations in end-to-end differentiable models?
• RQ2 Can the Sinkhorn operator serve as a differentiable analog to softmax for permutations and enable reparameterization-based learning?
• RQ3 Do Gumbel-based relaxations (Gumbel-Sinkhorn) provide effective variational inference for latent permutation structures?
• RQ4 Are permutation-equivariant architectures effective for reconstructing scrambled objects and learning matchings between sets?
• RQ5 What empirical gains arise in tasks requiring latent alignment: sorting, jigsaw puzzles, and neural identification in C. elegans?

主要发现

• The Sinkhorn operator provides a differentiable relaxation that approximates the maximum-weight matching in the limit of small temperature τ.
• Gumbel-Sinkhorn distributions enable reparameterized learning for latent permutations, allowing gradient-based optimization.
• Sinkhorn networks achieve strong performance on sorting numbers, solving jigsaw puzzles, and reconstructing images from scrambled pieces with competitive metrics.
• In C. elegans neural inference, Gumbel-Sinkhorn with variational inference outperforms MCMC and other baselines in permutation identification accuracy across varying known-neuron proportions and difficulty.
• Using permutation-equivariant architectures ensures reconstructions depend only on the pieces, not on their scrambled arrangement, improving consistency and learning efficiency.
• Compared to prior work, Gumbel-Sinkhorn offers a tighter relaxation and effective latent permutation modeling without requiring dense, high-parameter networks.