Analytical Probability Distributions and Exact Expectation-Maximization for Deep Generative Networks

Deep Generative Networks (DGNs) with probabilistic modeling of their output and latent space are currently trained via Variational Autoencoders (VAEs). In the absence of a known analytical form for the posterior and likelihood expectation, VAEs resort to approximations, including (Amortized) Variational Inference (AVI) and Monte-Carlo sampling. We exploit the Continuous Piecewise Affine property of modern DGNs to derive their posterior and marginal distributions as well as the latter's first two moments. These findings enable us to derive an analytical Expectation-Maximization (EM) algorithm for gradient-free DGN learning. We demonstrate empirically that EM training of DGNs produces greater likelihood than VAE training. Our new framework will guide the design of new VAE AVI that better approximates the true posterior and open new avenues to apply standard statistical tools for model comparison, anomaly detection, and missing data imputation.