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

TL;DR

This paper demonstrates that the density of a function's gradient can be approximated by the normalized power spectrum of a phase function using stationary phase approximation, especially as a parameter approaches zero.

Contribution

It introduces a novel method for density estimation of gradients in finite dimensions via stationary phase, distinct from characteristic function approaches.

Findings

Density function approximated by power spectrum as parameter tends to zero

Stationary phase approximation effectively estimates gradient density

Clarifies relationship and differences with characteristic function methods

Abstract

We prove that the density function of the gradient of a sufficiently smooth function $S : \Omega \subset \mathbb{R}^d \rightarrow \mathbb{R}$, obtained via a random variable transformation of a uniformly distributed random variable, is increasingly closely approximated by the normalized power spectrum of $\phi=\exp\left(\frac{iS}{\tau}\right)$ as the free parameter $\tau \rightarrow 0$. 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.