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.

翻译：标准概率稀疏编码假设拉普拉斯先验、从潜变量到观测变量的线性映射以及高斯观测分布。本文推导了标准稀疏编码参数的全熵学习目标。该新颖变分目标具有以下特性：(A) 与MAP近似不同，它采用非平凡后验近似进行概率推理；(B) 与以往非平凡近似不同，该目标完全可解析求解；(C) 该目标支持一种新颖的正则化退火形式。该目标通过首先证明标准ELBO目标收敛为熵之和推导得出，这一结果与近期生成模型（采用高斯先验）的类似结论吻合。随后证明ELBO等于熵的条件具有解析解，从而得到完全解析的目标函数。通过数值实验验证了基于熵的ELBO学习的可行性。我们研究了多种后验近似方法，包括具有相关潜变量的高斯近似和深度摊销近似。此外，我们数值研究了基于熵的退火方法，该方法能改进学习效果。然而，我们的主要贡献在于理论层面，具体包括两点：(1) 针对非平凡后验近似，我们（据作者所知）首次为标准概率稀疏编码提供了解析ELBO目标函数；(2) 我们首次论证了近期发现的ELBO向熵和收敛的现象如何用于学习。