Tractable Density Estimation on Learned Manifolds with Conformal Embedding Flows. (arXiv:2106.05275v1 [stat.ML])

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



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