  This paper studies the fundamental problem of learning energy-based model
(EBM) in the latent space of the generator model. Learning such prior model
typically requires running costly Markov Chain Monte Carlo (MCMC). Instead, we
propose to use noise contrastive estimation (NCE) to discriminatively learn the
EBM through density ratio estimation between the latent prior density and
latent posterior density. However, the NCE typically fails to accurately
estimate such density ratio given large gap between two densities. To
effectively tackle this issue and learn more expressive prior models, we
develop the adaptive multi-stage density ratio estimation which breaks the
estimation into multiple stages and learn different stages of density ratio
sequentially and adaptively. The latent prior model can be gradually learned
using ratio estimated in previous stage so that the final latent space EBM
prior can be naturally formed by product of ratios in different stages. The
proposed method enables informative and much sharper prior than existing
baselines, and can be trained efficiently. Our experiments demonstrate strong
performances in image generation and reconstruction as well as anomaly
detection.
