Manifold Density Estimation via Generalized Dequantization

TL;DR

This paper introduces a novel density estimation method for data on manifolds, extending traditional Euclidean approaches by leveraging generalized dequantization and normalizing flows to model complex geometric structures.

Contribution

It proposes a new approach combining dequantization and normalizing flows for density estimation on manifolds like spheres and tori, addressing non-Euclidean data structures.

Findings

Effective density modeling on various manifolds

Application to spheres, tori, and orthogonal groups

Demonstrates improved modeling of non-Euclidean data

Abstract

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.