Latent Space Oddity: on the Curvature of Deep Generative Models

TL;DR

This paper analyzes the geometric structure of deep generative models' latent spaces, showing how a Riemannian metric improves understanding of distances and sampling, and proposes an architecture with better variance estimates.

Contribution

It introduces a stochastic Riemannian metric to characterize latent space distortion and proposes a new generator architecture with enhanced variance estimation.

Findings

Distances and interpolants are improved under the Riemannian metric

Current generators have poor variance estimates

Proposed architecture shows vastly improved variance estimates

Abstract

Deep generative models provide a systematic way to learn nonlinear data distributions, through a set of latent variables and a nonlinear "generator" function that maps latent points into the input space. The nonlinearity of the generator imply that the latent space gives a distorted view of the input space. Under mild conditions, we show that this distortion can be characterized by a stochastic Riemannian metric, and demonstrate that distances and interpolants are significantly improved under this metric. This in turn improves probability distributions, sampling algorithms and clustering in the latent space. Our geometric analysis further reveals that current generators provide poor variance estimates and we propose a new generator architecture with vastly improved variance estimates. Results are demonstrated on convolutional and fully connected variational autoencoders, but the formalism easily generalize to other deep generative models.