Variational inference with Gaussian mixture model and householder flow.

The variational auto-encoder (VAE) is a powerful and scalable deep generative model. Under the architecture of VAE, the choice of the approximate posterior distribution is one of the crucial issues, and it has a significant impact on tractability and flexibility of the VAE. Generally, latent variables are assumed to be normally distributed with a diagonal covariance matrix, however, it is not flexible enough to match the true complex posterior distribution. We introduce a novel approach to design a flexible and arbitrarily complex approximate posterior distribution. Unlike VAE, firstly, an initial density is constructed by a Gaussian mixture model, and each component has a diagonal covariance matrix. Then this relatively simple distribution is transformed into a more flexible one by applying a sequence of invertible Householder transformations until the desired complexity has been achieved. Additionally, we also give a detailed theoretical and geometric interpretation of Householder transformations. Lastly, due to this change of approximate posterior distribution, the Kullback-Leibler distance between two mixture densities is required to be calculated, but it has no closed form solution. Therefore, we redefine a new variational lower bound by virtue of its upper bound. Compared with other generative models based on similar VAE architecture, our method achieves new state-of-the-art results on benchmark datasets including MNIST, Fashion-MNIST, Omniglot and Histopathology data a more challenging medical images dataset, the experimental results show that our method can improve the flexibility of posterior distribution more effectively.