Title: Reparameterization through Coverings and Topological Weight Priors
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
Keywords: topology, Bayesian inference, generative models

Abstract: We generalise the reparameterization trick 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, in case of particular measure preservation property holding for the covering, one can establish an inequality on KL-divergence between pushforward (PF) densities on the base latent manifold, 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 sometimes a particular case of what we propose to call reparameterization through a covering. Covering maps need not be global diffeomorphisms (although Lie-exp maps, in general, need not either, but, to date only smooth ones were considered in this context, to the best of our knowledge), which makes many non-trivial topologies tamable to our proposed technique, that we detail on a particular such example. 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.