  Density estimation is an important technique for characterizing distributions
given observations. Much existing research on density estimation has focused on
cases wherein the data lies in a Euclidean space. However, some kinds of data
are not well-modeled by supposing that their underlying geometry is Euclidean.
Instead, it can be useful to model such data as lying on a {\it manifold} with
some known structure. For instance, some kinds of data may be known to lie on
the surface of a sphere. We study the problem of estimating densities on
manifolds. We propose a method, inspired by the literature on "dequantization,"
which we interpret through the lens of a coordinate transformation of an
ambient Euclidean space and a smooth manifold of interest. Using methods from
normalizing flows, we apply this method to the dequantization of smooth
manifold structures in order to model densities on the sphere, tori, and the
orthogonal group.
