Tight bounds on the mutual coherence of sensing matrices for Wigner D-functions on regular grids

TL;DR

This paper derives analytical lower bounds on the mutual coherence of sensing matrices used in function approximation on the rotation group, leveraging quantum angular momentum analysis and Wigner D-functions, with implications for sampling pattern design.

Contribution

It introduces a novel connection between mutual coherence analysis and quantum angular momentum theory, providing explicit lower bounds for regular sampling patterns.

Findings

Lower bounds on mutual coherence are larger than the Welch bound.

Theoretical bounds are validated through numerical experiments.

Algorithms are proposed to achieve the derived lower bounds.

Abstract

Many practical sampling patterns for function approximation on the rotation group utilizes regular samples on the parameter axes. In this paper, we relate the mutual coherence analysis for sensing matrices that correspond to a class of regular patterns to angular momentum analysis in quantum mechanics and provide simple lower bounds for it. The products of Wigner d-functions, which appear in coherence analysis, arise in angular momentum analysis in quantum mechanics. We first represent the product as a linear combination of a single Wigner d-function and angular momentum coefficients, otherwise known as the Wigner 3j symbols. Using combinatorial identities, we show that under certain conditions on the bandwidth and number of samples, the inner product of the columns of the sensing matrix at zero orders, which is equal to the inner product of two Legendre polynomials, dominates the mutual coherence term and fixes a lower bound for it. In other words, for a class of regular sampling patterns, we provide a lower bound for the inner product of the columns of the sensing matrix that can be analytically computed. We verify numerically our theoretical results and show that the lower bound for the mutual coherence is larger than Welch bound. Besides, we provide algorithms that can achieve the lower bound for spherical harmonics.