VAE-KRnet and Its Applications to Variational Bayes (Communications in Computational Physics, 2022)
In this work, we have proposed a generative model, called VAE-KRnet, fordensity estimation or approximation, which combines the canonical variational autoencoder (VAE) with our recently developed flow-based generative model, called KRnet. VAE is used as a dimension reduction technique to capture the latent space, andKRnet is used to model the distribution of the latent variable. Using a linear modelbetween the data and the latent variable, we show that VAE-KRnet can be more effective and robust than the canonical VAE. VAE-KRnet can be used as a density model toapproximate either data distribution or an arbitrary probability density function (PDF)known up to a constant. VAE-KRnet is flexible in terms of dimensionality. When thenumber of dimensions is relatively small, KRnet can effectively approximate the distribution in terms of the original random variable. For high-dimensional cases, wemay use VAE-KRnet to incorporate dimension reduction. One important applicationof VAE-KRnet is the variational Bayes for the approximation of the posterior distribution. The variational Bayes approaches are usually based on the minimization ofthe Kullback-Leibler (KL) divergence between the model and the posterior. For high-dimensional distributions, it is very challenging to construct an accurate density modeldue to the curse of dimensionality, where extra assumptions are often introduced forefficiency. For instance, the classical mean-field approach assumes mutual independence between dimensions, which often yields an underestimated variance due tooversimplification. To alleviate this issue, we include into the loss the maximization ofthe mutual information between the latent random variable and the original randomvariable, which helps keep more information from the region of low density such thatthe estimation of variance is improved. Numerical experiments have been presentedto demonstrate the effectiveness of our model.
Keywords: Deep learning; variational Bayes; uncertainty quantification; Bayesian inverse problems; generative modeling.