TY - JOUR
T1 - VAE-KRnet and Its Applications to Variational Bayes
AU - Wan , Xiaoliang
AU - Wei , Shuangqing
JO - Communications in Computational Physics
VL - 4
SP - 1049
EP - 1082
PY - 2022
DA - 2022/03
SN - 31
DO - http://doi.org/10.4208/cicp.OA-2021-0087
UR - https://global-sci.org/intro/article_detail/cicp/20376.html
KW - Deep learning, variational Bayes, uncertainty quantification, Bayesian inverse problems, generative modeling.
AB - <p style="text-align: justify;">In this work, we have proposed a generative model, called VAE-KRnet, for
density 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, and
KRnet is used to model the distribution of the latent variable. Using a linear model
between 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 to
approximate either data distribution or an arbitrary probability density function (PDF)
known up to a constant. VAE-KRnet is flexible in terms of dimensionality. When the
number of dimensions is relatively small, KRnet can effectively approximate the distribution in terms of the original random variable. For high-dimensional cases, we
may use VAE-KRnet to incorporate dimension reduction. One important application
of VAE-KRnet is the variational Bayes for the approximation of the posterior distribution. The variational Bayes approaches are usually based on the minimization of
the Kullback-Leibler (KL) divergence between the model and the posterior. For high-dimensional distributions, it is very challenging to construct an accurate density model
due to the curse of dimensionality, where extra assumptions are often introduced for
efficiency. For instance, the classical mean-field approach assumes mutual independence between dimensions, which often yields an underestimated variance due to
oversimplification. To alleviate this issue, we include into the loss the maximization of
the mutual information between the latent random variable and the original random
variable, which helps keep more information from the region of low density such that
the estimation of variance is improved. Numerical experiments have been presented
to demonstrate the effectiveness of our model.</p>