Reparameterization through Coverings and Topological Weight Priors

Variational autoencoders (VAEs) learn low-dimensional latent representations of high-dimensional data. When the data lies on a manifold with non-Euclidean topology, the standard Gaussian prior introduces a topological mismatch that degrades reconstruction quality and prevents faithful representation. We present a constructive mathematical framework that resolves this mismatch for all manifolds that admit a product covering space. These are manifolds expressible as products of elementary factors (circles, intervals, or lines) or as quotients of such products by a finite symmetry group. The class includes cylinders, tori, Möbius strips, Klein bottles, and real projective spaces. Factorized distributions over the elementary factors yield product topologies with closed-form, decoupled KL divergences, so that each latent factor can be shaped independently while keeping training tractable. We catalogue reparametrizable encoder-prior pairs for periodic, bounded, and unbounded supports, and provide coordinate transformations that allow standard neural networks to output non-Euclidean parameters with smooth gradients. For quotient manifolds, the decoder receives group-invariant features of the covering-space coordinates, so that identified points produce identical outputs. Anchor constraints fix the coordinate system relative to the data or create soft topological holes. Experiments on synthetic manifolds and real-image datasets (rotated and cyclically shifted MNIST) confirm that a topology-matched prior aligns KL regularization with the data manifold. The resulting topology-aware models outperform the Gaussian baseline at all practically relevant regularization strengths. The code is available at https://github.com/JvHulst/VAE-Topology.

Despite their ubiquity, variational autoencoders (VAEs) inherently suffer from posterior collapse, a failure mode in which latent variables are effectively ignored. This failure arises because explicit prior imposition drives optimization toward loss landscape regions corresponding to uninformative latent representations. Here, we introduce Entropic Autoencoders (EAEs), a framework in which reconstruction loss is the only explicit objective, and entropy generates the latent variables' prior implicitly through a free energy-minimizing ensemble of encoders. This ensemble biases learning toward high-volume regions of near-optimal solutions, while decoder updates direct the search trajectories toward informative latent representations. We demonstrate that EAEs mitigate posterior collapse by learning non-Gaussian, multimodal latent distributions that yield diverse, data-consistent generations and preserve different forms of underlying structure in the data. As a proof-of-concept, we show that an EAE captures a superposition of the known low-dimensional dynamics of a reaction-diffusion process. Then, we show that an EAE identifies implicit categorical distinctions in MNIST latent representations, and displays a hierarchical understanding of facial structure on the CelebA dataset, from an "all-human" face to individual-dependent features.

Variational Autoencoders (VAEs) frequently suffer from posterior collapse, a failure mode in which the approximate posterior converges to the prior, rendering the latent code uninformative. Despite extensive research, a unified account of why collapse occurs has remained an open question. We identify and formalize two logically independent but coupled causes. \emph{Gradient imbalance} occurs when the decoder's reconstruction signal vanishes faster than the $\mathbb{KL}$ regularization pressure as the posterior widens. \emph{Information gap} occurs when the stochastic sampling step discards a substantial fraction of the encoder's computed representation, attenuating decoder sensitivity and making collapse inexpensive. Both causes share the same collapse trajectory, and we show that the information gap is algebraically equivalent to mismatch between the aggregate posterior and the prior, unifying two pathologies. Subsequently, we introduce $λ$-VAE, which resolves both causes through a single modification to the reparameterization step: the sampling noise is scaled by per-dimension exponent, while the $\mathbb{KL}$ penalty retains the original posterior variance. This asymmetry shifts the stable training attractor away from the degenerate collapsed state, driving all latent dimensions toward the same equilibrium -- a mechanism we term \emph{variance equalization}. A closed-form optimal exponent per dimension follows from a net information gain objective, with a single hyperparameter controlling the reconstruction-generation tradeoff. We validate on standard benchmarks (Binary MNIST, Binary Omniglot, CIFAR-10, CelebA-64), showing consistent reductions in collapsed dimensions, information capacity gains of up to $2.8\times$ nats, and reconstruction quality improvements of up to $+0.33$ BPD.