  Given data, deep generative models, such as variational autoencoders (VAE)
and generative adversarial networks (GAN), train a lower dimensional latent
representation of the data space. The linear Euclidean geometry of data space
pulls back to a nonlinear Riemannian geometry on the latent space. The latent
space thus provides a low-dimensional nonlinear representation of data and
classical linear statistical techniques are no longer applicable. In this paper
we show how statistics of data in their latent space representation can be
performed using techniques from the field of nonlinear manifold statistics.
Nonlinear manifold statistics provide generalizations of Euclidean statistical
notions including means, principal component analysis, and maximum likelihood
fits of parametric probability distributions. We develop new techniques for
maximum likelihood inference in latent space, and adress the computational
complexity of using geometric algorithms with high-dimensional data by training
a separate neural network to approximate the Riemannian metric and cometric
tensor capturing the shape of the learned data manifold.
