Group sparse optimization for inpainting of random fields on the sphere

Abstract

We propose a group sparse optimization model for inpainting of a square-integrable isotropic random field on the unit sphere, where the field is represented by spherical harmonics with random complex coefficients. In the proposed optimization model, the variable is an infinite-dimensional complex vector and the objective function is a real-valued function defined by a hybrid of the $\ell_2$ norm and non-Liptchitz $\ell_p (0<p<1)$ norm that preserves rotational invariance property and group structure of the random complex coefficients. We show that the infinite-dimensional optimization problem is equivalent to a convexly-constrained finite-dimensional optimization problem. Moreover, we propose a smoothing penalty algorithm to solve the finite-dimensional problem via unconstrained optimization problems. We provide an approximation error bound of the inpainted random field defined by a scaled KKT point of the constrained optimization problem in the square-integrable space on the sphere with probability measure. Finally, we conduct numerical experiments on band-limited random fields on the sphere and images from Earth topography data to show the promising performance of the smoothing penalty algorithm for inpainting of random fields on the sphere.