Conformal Embedding Flows: Tractable Density Estimation on Learned Manifolds

Abstract

Normalizing flows are generative models that provide tractable density estimation by transforming a simple distribution into a complex one. However, flows cannot directly model data supported on an unknown low-dimensional manifold. We propose Conformal Embedding Flows, which learn low-dimensional manifolds with tractable densities. We argue that composing a standard flow with a trainable conformal embedding is the most natural way to model manifold-supported data. To this end, we present a series of conformal building blocks and demonstrate experimentally that flows can model manifold-supported distributions without sacrificing tractable likelihoods.