Tractable Density Estimation on Learned Manifolds with Conformal Embedding Flows

Normalizing flows are generative models that provide tractable density\nestimation via an invertible transformation from a simple base distribution to\na complex target distribution. However, this technique cannot directly model\ndata supported on an unknown low-dimensional manifold, a common occurrence in\nreal-world domains such as image data. Recent attempts to remedy this\nlimitation have introduced geometric complications that defeat a central\nbenefit of normalizing flows: exact density estimation. We recover this benefit\nwith Conformal Embedding Flows, a framework for designing flows that learn\nmanifolds with tractable densities. We argue that composing a standard flow\nwith a trainable conformal embedding is the most natural way to model\nmanifold-supported data. To this end, we present a series of conformal building\nblocks and apply them in experiments with synthetic and real-world data to\ndemonstrate that flows can model manifold-supported distributions without\nsacrificing tractable likelihoods.\n