Slicing of Radial Functions: a Dimension Walk in the Fourier Space

TL;DR

This paper develops a method to reconstruct radial functions in high-dimensional Fourier space from lower-dimensional projections, extending classical formulas to tempered distributions and positive definite functions.

Contribution

It introduces a Fourier-based approach to invert the slicing formula for radial functions, including distributions, generalizing existing integral formulas.

Findings

Reconstruction of radial functions from projections using Fourier transforms

Extension of the method to tempered distributions and positive definite functions

Application of fractional derivatives in the analysis

Abstract

Computations in high-dimensional spaces can often be realized only approximately, using a certain number of projections onto lower dimensional subspaces or sampling from distributions. In this paper, we are interested in pairs of real-valued functions $(F,f)$ on $[0,\infty)$ that are related by the projection/slicing formula $F (\| x \|) = \mathbb E_{\xi} \big[ f \big(|\langle x,\xi \rangle| \big) \big]$ for $x\in\mathbb R^d$, where the expectation value is taken over uniformly distributed directions in $\mathbb R^d$. While it is known that $F$ can be obtained from $f$ by an Abel-like integral formula, we construct conversely $f$ from given $F$ using their Fourier transforms. First, we consider the relation between $F$ and $f$ for radial functions $F(\| \cdot\| )$ that are Fourier transforms of $L^1$ functions. Besides $d$- and one-dimensional Fourier transforms, it relies on a rotation operator, an averaging operator and a multiplication operator to manage the walk from $d$ to one dimension in the Fourier space. Then, we generalize the results to tempered distributions, where we are mainly interested in radial regular tempered distributions. Based on Bochner's theorem, this includes positive definite functions $F(\| \cdot\| )$ and, by the theory of fractional derivatives, also functions $F$ whose derivative of order $\lfloor d/2\rfloor$ is slowly increasing and continuous.