Commutator-Induced Uncertainty in VAEs

2026-05-22

AI Summary

This work addresses the challenge that variational autoencoders (VAEs) struggle to effectively model the intrinsic non-commutative structure of data, as conventional symmetry constraints may suppress such critical properties. The authors propose a Lie group VAE framework trained in two stages: first, an unconstrained phase diagnoses algebraic non-commutativity in the latent space and sequence sensitivity in the decoder; second, a deformation-stability constraint explicitly aligns these behaviors. This approach is the first to link non-commutativity diagnostics with generative behavior, integrating Lie group geometry, Baker–Campbell–Hausdorff deviation metrics, and sequence-swapping reconstruction tests. Evaluated on dSprites, 3DShapes, 3DCars, and CelebA, the method significantly outperforms baselines, achieving higher reconstruction fidelity, clearer sequential dependency in latent composition, and more stable generation.

Abstract

Variational autoencoders (VAEs) often struggle to represent non-commutative structure in learned latent spaces. Symmetry-aware VAEs commonly address this issue by enforcing commutativity through algebraic regularization, which is appropriate for commutative transformation groups but can suppress meaningful non-commutative structure when it is intrinsic to the data. We argue that non-commutativity should instead be explicitly diagnosed and reflected in reconstruction behavior. We introduce a Lie Group VAE framework that combines geometric and algebraic perspectives on uncertainty while separating discrete generative factors from continuous geometric transformations. In a first phase, the model is trained without structural constraints while algebraic non-commutativity is measured through finite Baker-Campbell-Hausdorff deviations and decoder order sensitivity is measured through reconstruction order-swap tests. These diagnostics reveal a scale mismatch between latent non-commutativity and reconstruction behavior under unconstrained training. In a second phase, we introduce a deformation-stability constraint with a data-driven calibration constant that aligns decoder sensitivity with algebraic non-commutativity. We evaluate the framework on dSprites, 3DShapes, 3DCars, and CelebA against generic and symmetry-aware baselines, including beta-VAE, CLG-VAE, and CFASL. Across synthetic benchmarks, the method improves reconstruction quality and yields decoder-level behavior more consistent with latent non-commutative structure. Qualitative analyses show clearer order-dependent latent compositions and more stable reconstructions. On CelebA, the model yields more faithful reconstructions and factor-specific latent traversals than CFASL, while also exhibiting meaningful order-dependent interactions between learned latent directions.

Problem

• non-commutativity
• variational autoencoders
• latent space
• symmetry-aware learning
• reconstruction behavior

Innovation

• non-commutativity
• Lie Group VAE
• Baker-Campbell-Hausdorff deviation
• order-sensitive reconstruction
• deformation-stability constraint