Density Estimation on Low-Dimensional Manifolds: An Inflation-Deflation Approach
Authors: Christian Horvat, Jean-Pascal Pfister; Volume 24, Issue 61, Pages 1-37, 2023.
Abstract
Normalizing flows (NFs) are density estimators that use neural networks. However, NFs have a limitation: the density’s support must be diffeomorphic to a Euclidean space. In this paper, we propose a new method to overcome this limitation while still maintaining universality. Our method involves inflating the data manifold by adding noise in the normal space, training an NF on this inflated manifold, and then deflating the learned density. We provide sufficient conditions on the manifold and the specific noise choice to ensure the estimator’s exactness. Our method has the same computational complexity as NFs and does not require computing an inverse flow. Additionally, we demonstrate through theory and empirical examples that Gaussian noise can be a good approximation for noise in the normal space. This means our method can be used to approximate arbitrary densities on unknown manifolds, as long as the manifold dimension is known.
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