Two Dimensional Density Estimation using Smooth Invertible Transformations

TL;DR

This paper develops a method for estimating smooth invertible transformations in two dimensions using penalized maximum likelihood, showing improvements over kernel density estimation in practical examples.

Contribution

Introduces a flexible class of smooth invertible transformations with variational equations for optimization, advancing density estimation techniques in two dimensions.

Findings

Method outperforms kernel density estimation in examples

Flexible class of transformations enables better density modeling

Optimization via variational equations improves estimation accuracy

Abstract

We investigate the problem of estimating a smooth invertible transformation f when observing independent samples X_1, ..., X_n ~ P \circ f, where P is a known measure. We focus on the two dimensional case where P and f are defined on R^2. We present a flexible class of smooth invertible transformations in two dimensions with variational equations for optimizing over the classes, then study the problem of estimating the transformation f by penalized maximum likelihood estimation. We apply our methodology to the case when P \circ f has a density with respect to Lebesgue measure on R^2 and demonstrate improvements over kernel density estimation on three examples.