Consistent Symmetry Representation over Latent Factors of Variation

Hee-Jun Jung, Hoyong Kim, Ilmin Kang, Kangil Kim

Keywords: Combinatorial Generalizaton, Disentanglement Learning, Variational Auto-Encoder, Symmetry

Abstract:

Recent symmetry-based methods on variational autoencoders have advanced disentanglement learning and combinatorial generalization, yet the appropriate symmetry representation for both tasks is under-clarified. We identify that existing methods struggle with maintaining the $\textit{consistent symmetries}$ when representing identical changes of latent factors of variation, and they cause issues in achieving equivari-
ance. We theoretically prove the limitations of three frequently used group settings: matrix multiplication with General Lie Groups, defining group action with set of vectors and vector addition, and cyclic groups modeled through surjective functions. To overcome these issues, we introduce a novel method of $\textit{conformal mapping}$ of latent vectors into a complex number space, ensuring consistent symmetries
and cyclic semantics. Through empirical validation with ground truth of factors variation for transparent analysis, this study fills two significant gaps in the literature: 1) the inductive bias to enhance disentanglement learning and combinatorial generalization simultaneously, and 2) well-represented symmetries ensure significantly high disentanglement performance without a trade-off in reconstruction error, compared to current unsupervised methods. Additionally, we introduce less guidance-dependent validation results, extending our findings to more practical use. Our research highlights the significant impact of verifying consistent symmetry and suggests required future research for advancing combinatorial generalization and disentanglement learning.

Primary Area: generative models

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Submission Number: 5896