Autoencoders and Probabilistic Inference with Missing Data: An Exact Solution for The Factor Analysis Case

Latent variable models, like factor analysis and "deeper" versions such as the variational autoencoder (VAE, Kingma and Welling 2014; Rezende et al. 2014) and generative adversarial networks (GANs, Goodfellow et al. 2014), are a compelling approach to modelling structure in complex high-dimensional data such as images. One important use case of such models is when some part of the observable data is missing--such as when some part of an image is not observed. In this case we would like to use the latent variable model to "inpaint" the missing data. Another example is modelling 3-D volumetric data--we may have observations of an object from one viewpoint, and wish to make inferences about the whole 3-D object. The VAE is composed of two parts, an encoder (or recognition) network that predicts the distribution of the latent variables given the data, and a decoder (or generative) model that maps from the latent variables to the visible variables.