Sparse recovery in Wigner-D basis expansion

TL;DR

This paper develops a new bounded orthonormal system based on Wigner-D functions for sparse recovery, establishing RIP conditions and sample complexity, with applications in spherical near-field antenna measurement.

Contribution

It introduces a preconditioning technique to handle non-uniformly bounded Wigner-D functions, enabling sparse recovery analysis with explicit sample complexity bounds.

Findings

Sample complexity scales as N^{1/6} s log^3(s) log(N).

Established RIP for the new orthonormal system.

Presented phase transition diagram for sparse recovery.

Abstract

We are concerned with the recovery of $s-$sparse Wigner-D expansions in terms of $N$ Wigner-D functions. Considered as a generalization of spherical harmonics, Wigner-D functions are eigenfunctions of Laplace-Beltrami operator and form an orthonormal system. However, since they are not uniformly bounded, the existing results on BOS do not apply. Using previously introduced preconditioning technique, a new orthonormal and bounded system is obtained for which RIP property can be established. We show that the number of sufficient samples for sparse recovery scales with ${N}^{1/6} \,s\, \log^3(s) \,\log(N)$. The phase transition diagram for this problem is also presented. We will also discuss the application of our results in the spherical near-field antenna measurement.