Dmytro Velychko

Dmytro Velychko, Simon Damm, Asja Fischer et al.

Standard probabilistic sparse coding assumes a Laplace prior, a linear mapping from latents to observables, and Gaussian observable distributions. We here derive a solely entropy-based learning objective for the parameters of standard sparse coding. The novel variational objective has the following features: (A) unlike MAP approximations, it uses non-trivial posterior approximations for probabilistic inference; (B) unlike for previous non-trivial approximations, the novel objective is fully analytical; and (C) the objective allows for a novel principled form of annealing. The objective is derived by first showing that the standard ELBO objective converges to a sum of entropies, which matches similar recent results for generative models with Gaussian priors. The conditions under which the ELBO becomes equal to entropies are then shown to have analytical solutions, which leads to the fully analytical objective. Numerical experiments are used to demonstrate the feasibility of learning with such entropy-based ELBOs. We investigate different posterior approximations including Gaussians with correlated latents and deep amortized approximations. Furthermore, we numerically investigate entropy-based annealing which results in improved learning. Our main contributions are theoretical, however, and they are twofold: (1) for non-trivial posterior approximations, we provide the (to the knowledge of the authors) first analytical ELBO objective for standard probabilistic sparse coding; and (2) we provide the first demonstration on how a recently shown convergence of the ELBO to entropy sums can be used for learning.

Jan Warnken, Dmytro Velychko, Simon Damm et al.

The evidence lower bound (ELBO) is one of the most central objectives for probabilistic unsupervised learning. For the ELBOs of several generative models and model classes, we here prove convergence to entropy sums. As one result, we provide a list of generative models for which entropy convergence has been shown, so far, along with the corresponding expressions for entropy sums. Our considerations include very prominent generative models such as probabilistic PCA, sigmoid belief nets or Gaussian mixture models. However, we treat more models and entire model classes such as general mixtures of exponential family distributions. Our main contributions are the proofs for the individual models. For each given model we show that the conditions stated in Theorem 1 or Theorem 2 of [arXiv:2209.03077] are fulfilled such that by virtue of the theorems the given model's ELBO is equal to an entropy sum at all stationary points. The equality of the ELBO at stationary points applies under realistic conditions: for finite numbers of data points, for model/data mismatches, at any stationary point including saddle points etc, and it applies for any well behaved family of variational distributions.

Simon Damm, Dennis Forster, Dmytro Velychko et al.

The central objective function of a variational autoencoder (VAE) is its variational lower bound (the ELBO). Here we show that for standard (i.e., Gaussian) VAEs the ELBO converges to a value given by the sum of three entropies: the (negative) entropy of the prior distribution, the expected (negative) entropy of the observable distribution, and the average entropy of the variational distributions (the latter is already part of the ELBO). Our derived analytical results are exact and apply for small as well as for intricate deep networks for encoder and decoder. Furthermore, they apply for finitely and infinitely many data points and at any stationary point (including local maxima and saddle points). The result implies that the ELBO can for standard VAEs often be computed in closed-form at stationary points while the original ELBO requires numerical approximations of integrals. As a main contribution, we provide the proof that the ELBO for VAEs is at stationary points equal to entropy sums. Numerical experiments then show that the obtained analytical results are sufficiently precise also in those vicinities of stationary points that are reached in practice. Furthermore, we discuss how the novel entropy form of the ELBO can be used to analyze and understand learning behavior. More generally, we believe that our contributions can be useful for future theoretical and practical studies on VAE learning as they provide novel information on those points in parameters space that optimization of VAEs converges to.