Latent Variables on Spheres for Sampling and Inference

Variational inference is a fundamental problem in Variational AutoEncoder (VAE). The optimization with lower bound of marginal log-likelihood results in the distribution of latent variables approximate to a given prior probability, which is the dilemma of employing VAE to solve real-world problems. By virtue of high-dimensional geometry, we propose a very simple algorithm completely different from existing ones to alleviate the variational inference in VAE. We analyze the unique characteristics of random variables on spheres in high dimensions and prove that Wasserstein distance between two arbitrary data sets randomly drawn from a sphere are nearly identical when the dimension is sufficiently large. Based on our theory, a novel algorithm for distribution-robust sampling is devised. Moreover, we reform the latent space of VAE by constraining latent variables on the sphere, thus freeing VAE from the approximate optimization of posterior probability via variational inference. The new algorithm is named Spherical AutoEncoder (SAE). Extensive experiments by sampling and inference tasks validate our theoretical analysis and the superiority of SAE.