Reparameterization through Coverings and Topological Weight Priors

arXiv:2604.23804v2 Announce Type: replace Abstract: We generalise the reparameterization trick (RT) applied in variational autoencoders (VAEs) letting these have latent spaces of non-trivial topology - i.e. that of base manifolds covered with other ones, on which some technique for RT is available. That is possible since covering maps are measurable - moreover, this allows to establish an inequality on KL-divergence between pushforward (PF) densities on the base latent manifold, bounding it with KL-divergence between pullbacks on the cover, in some cases making the KL-term of VAE's ELBO analytically tractable, despite the topological non-triviality of the supporting latent manifold. Our development follows a route close but somewhat alternative to reparameterization on Lie groups, the latest proposal for which is to reparameterize PFs of normal densities from the Lie algebra - "through" the exponential map, seen by us as a particular case of what we propose to call reparameterization via covering (RVC). We demonstrate the working of our approach by constructing a VAE with the latent space of Klein bottle (not a Lie group) topology, which we call KleinVAE, successfully learning an appropriate artificial dataset. We discuss potential applicability of such topology-informed generative models as weight priors in Bayesian learning, particularly for convolutional vision models, where said manifold was peculiarly shown to have some relevance.

Abstract:We generalise the reparameterization trick (RT) applied in variational autoencoders (VAEs) letting these have latent spaces of non-trivial topology - i.e. that of base manifolds covered with other ones, on which some technique for RT is available. That is possible since covering maps are measurable - moreover, this allows to establish an inequality on KL-divergence between pushforward (PF) densities on the base latent manifold, bounding it with KL-divergence between pullbacks on the cover, in some cases making the KL-term of VAE's ELBO analytically tractable, despite the topological non-triviality of the supporting latent manifold. Our development follows a route close but somewhat alternative to reparameterization on Lie groups, the latest proposal for which is to reparameterize PFs of normal densities from the Lie algebra - "through" the exponential map, seen by us as a particular case of what we propose to call reparameterization via covering (RVC). We demonstrate the working of our approach by constructing a VAE with the latent space of Klein bottle (not a Lie group) topology, which we call KleinVAE, successfully learning an appropriate artificial dataset. We discuss potential applicability of such topology-informed generative models as weight priors in Bayesian learning, particularly for convolutional vision models, where said manifold was peculiarly shown to have some relevance.

Machine Learning (cs.LG)

arXiv:2604.23804 [cs.LG]

(or arXiv:2604.23804v2 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2604.23804 Focus to learn more arXiv-issued DOI via DataCite

https://doi.org/10.4310/DAT.260831104503 Focus to learn more DOI(s) linking to related resources

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Reparameterization through Coverings and Topological Weight Priors

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