Gradient density estimation in arbitrary finite dimensions using the method of stationary phase

Abstract

We prove that the density function of the gradient of a sufficiently smooth function S : R^d R, obtained via a random variable transformation of a uniformly distributed random variable, is increasingly closely approximated by the normalized power spectrum of =(iS) as the free parameter 0. The result is shown using the stationary phase approximation and standard integration techniques and requires proper ordering of limits. We highlight a relationship with the well-known characteristic function approach to density estimation, and detail why our result is distinct from this approach.

Tasks

Reproductions