Symmetric Space Learning for Combinatorial Generalization

ICLR 2025 Conference Submission5940 Authors

Keywords: Generative Model, Generalization, Combinatorial Generalization, Machine Learning, Manifold Learning, Representation Learning

Abstract: Symmetries on representations within generative models have shown essential roles in predicting unobserved combinations of semantic changes, known as combinatorial generalization tasks. However, these efforts have primarily focused on learning symmetries from only training data, and thus, the extension of trained symmetries to unseen samples remains uncontrolled. A potential approach for generalizing the symmetries is leveraging geometric information on manifolds that contain functional semantic structures for unseen data, but it still falls short of supporting symmetry learning. In this paper, we address this $\textit{symmetry generalization}$ by forcing $\textit{symmetric space}$ on latent space for utilizing semantic structures from symmetry and manifold perspectives. We clarify an equivariance-based constraint that restricts symmetry generalization, and prove that: 1) enforcing the homogeneous space property of symmetric space onto the data manifold eliminates this constraint, 2) a homogeneous latent manifold induces the data manifold through diffeomorphic data-to-latent mapping, and 3) the isometry property of symmetric space extends neighbor symmetries of a point to another within the space. For practical implementation, we propose a method to align sampled points from symmetric space with their explicitly trained geodesic. We verify the method in a detailed analysis on a toy dataset and enhance combinatorial generalization on common benchmarks. This work represents the first effective effort to align symmetries with manifolds for combinatorial generalization.

Primary Area: generative models

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Submission Number: 5940