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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.23449 |
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| _version_ | 1866917523565314048 |
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| author | Dehdarirad, Tahereh Felsberg, Michael Eilertsen, Gabriel Xiong, Ziliang |
| author_facet | Dehdarirad, Tahereh Felsberg, Michael Eilertsen, Gabriel Xiong, Ziliang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_23449 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Commutator-Induced Uncertainty in VAEs Dehdarirad, Tahereh Felsberg, Michael Eilertsen, Gabriel Xiong, Ziliang Machine Learning Computer Vision and Pattern Recognition Algebraic Geometry 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. |
| title | Commutator-Induced Uncertainty in VAEs |
| topic | Machine Learning Computer Vision and Pattern Recognition Algebraic Geometry |
| url | https://arxiv.org/abs/2605.23449 |