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

This work is partly supported by EADS Prize Postdoctoral Fellowship