Group sparse optimization for inpainting of random fields on the sphere.

This article focuses on a constrained group sparse optimization model for inpainting square-integrable isotropic random fields on the unit sphere, represented via spherical harmonics with random complex coefficients. The authors formulate an infinite-dimensional optimization problem involving a hybrid of the \(\ell_2\) norm and a non-Lipschitz \(\ell_p\) norm (with \(0 < p < 1\)) that preserves rotational invariance and group structure of the coefficients, and prove its equivalence to a finite-dimensional problem. They propose a smoothing penalty algorithm to solve the finite-dimensional problem, establish exact penalization results, and prove convergence of the algorithm to scaled Karush–Kuhn–Tucker (KKT) points. The paper also provides approximation error bounds for the inpainted random field in the \(L_2(\Omega \times \mathbb{S}^2)\) space and demonstrates the method's effectiveness through numerical experiments on band-limited random fields and Cosmic Microwave Background (CMB) data, showing improved recovery accuracy and sparsity compared to existing \(\ell_1\)-based and group Lasso methods.