Tractable Density Estimation on Learned Manifolds with Conformal Embedding Flows

Brendan Leigh Ross, Jesse C. Cresswell

Normalizing flows are generative models that provide tractable density estimation by transforming a simple base distribution into a complex target distribution. However, this technique cannot directly model data supported on an unknown low-dimensional manifold, a common occurrence in real-world domains such as image data. Recent attempts to remedy this limitation have introduced geometric complications that defeat a central benefit of normalizing flows: exact density estimation. We recover this benefit with Conformal Embedding Flows, a framework for designing flows that learn 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 apply them in experiments with real-world and synthetic data to demonstrate that flows can model manifold-supported distributions without sacrificing tractable likelihoods.

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