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 $S:\Omega \subset {\mathbb{R}}^d\to \mathbb{R}$, obtained via a random variable transformation of a uniformly distributed random variable, is increasingly closely approximated by the normalized power spectrum of $\varphi =\exp \left(\frac{iS}{\tau }\right)$ as the free parameter $\tau \to 0$. 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.

arXiv commentThis work is partly supported by EADS Prize Postdoctoral Fellowship