Spherical designs and nonconvex minimization for recovery of sparse signals on the sphere

Abstract

This paper considers the use of spherical designs and nonconvex minimization for recovery of sparse signals on the unit sphere 𝕊2The available information consists of low order, potentially noisy, Fourier coefficients for 𝕊2As Fourier coefficients are integrals of the product of a function and spherical harmonics, a good cubature rule is essential for the recovery. A spherical t-design is a set of points on 𝕊2which are nodes of an equal weight cubature rule integrating exactly all spherical polynomials of degree ≤ t. We will show that a spherical t-design provides a sharp error bound for the approximation signals. Moreover, the resulting coefficient matrix has orthonormal rows. In general the l1minimization model for recovery of sparse signals on 𝕊2using spherical harmonics has infinitely many minimizers, which means that most existing sufficient conditions for sparse recovery do not hold. To induce the sparsity, we replace the l1-norm by the lq-norm (0 < q < 1) in the basis pursuit denoise model. Recovery properties and optimality conditions are discussed. Moreover, we show that the penalty method with a starting point obtained from the reweighted l1method is promising to solve the lqbasis pursuit denoise model. Numerical performance on nodes using spherical t-designs and tϵ-designs (extremal fundamental systems) are compared with tensor product nodes. We also compare the basis pursuit denoise problem with q = 1 and 0 < q < 1.