DeepAI AI Chat
Log In Sign Up

Manifold Density Estimation via Generalized Dequantization

by   James A. Brofos, et al.

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 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.


page 1

page 2

page 3

page 4


Normalizing Flows on Riemannian Manifolds

We consider the problem of density estimation on Riemannian manifolds. D...

Density estimation on low-dimensional manifolds: an inflation-deflation approach

Normalizing Flows (NFs) are universal density estimators based on Neuron...

Areas on the space of smooth probability density functions on S^2

We present symbolic and numerical methods for computing Poisson brackets...

Lie PCA: Density estimation for symmetric manifolds

We introduce an extension to local principal component analysis for lear...

Nonlinear Isometric Manifold Learning for Injective Normalizing Flows

To model manifold data using normalizing flows, we propose to employ the...

Estimating a density near an unknown manifold: a Bayesian nonparametric approach

We study the Bayesian density estimation of data living in the offset of...

General Probabilistic Surface Optimization and Log Density Estimation

In this paper we contribute a novel algorithm family, which generalizes ...