Symmetric Space Learning for Combinatorial Generalization

Abstract

Combinatorial generalization (CG)—generalizing to unseen combinations of known semantic factors—remains a fundamental challenge in machine learning. While symmetry-based methods are promising, they learn from observed data and thus fail at what we term symmetry generalization: extending learned symmetries to novel data. We address this by proposing a novel framework that endows the latent space with the structure of a symmetric space. This class of manifolds provides a principled geometric foundation for extending learned symmetries. Our method operates in two steps: first, it imposes this structure by learning the underlying algebraic properties via the Cartan decomposition of a learnable Lie algebra. Second, it uses geodesic symmetry as a powerful self-supervisory signal to ensure this learned structure extrapolates from observed samples to unseen ones. A detailed analysis on a synthetic dataset validates our geometric claims, and experiments on standard CG benchmarks show our method significantly outperforms existing approaches.