Two-dimensional density estimation using smooth invertible transformations

We investigate the problem of estimating a smooth invertible transformation f when observing independent samples X₁,…,X_n∼P°f where P is a known measure. We focus on the two-dimensional case where P and f are defined on R². 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°f has a density with respect to Lebesgue measure on R² and demonstrate improvements over kernel density estimation on three examples.