Group sparse optimization problems for random fields on the unit sphere

Abstract

In this thesis, we consider two types of group sparse optimization problems for random fields on the unit sphere, where the fields can be expanded by spherical harmonics with group structured complex Fourier coefficients. One is the group sparse representation for an isotropic and Gaussian random field using unconstrained optimization. The other one is the inpainting of the random fields using constrained optimization. We first consider the group sparse representation for an isotropic and Gaussian random field on the unit sphere, which is solved via an unconstrained optimization problem with non-Lipschitz regularizer over an infinite-dimensional space. The regularizer not only gives a group sparse solution, but also preserves the isotropy of the regularized random field represented by the solution. We present the first order and second order necessary optimality conditions for local minimizers of the optimization problem. We also derive two lower bounds for the l2 norm of nonzero groups of stationary points, which are used to prove that the infinite-dimensional optimization problem can be reduced to a finite-dimensional problem. We show that with an appropriate choice of the regularization parameter, the global minimizers of the finite-dimensional problem have desirable group sparsity. Then, we propose an iteratively reweighted algorithm for the finite-dimensional problem and prove its convergence. Moreover, we prove that the approximation error of the regularized random field represented by the solution of the finite-dimensional problem from the observed field can be arbitrarily small with an appropriate choice of the regularization parameter and the truncated spherical polynomial degree. Finally, numerical experiments on the Cosmic Microwave Background (CMB) data are presented to show the efficiency of the non-Lipschitz regularization. We then consider the inpainting of random fields on the unit sphere, which is solved by a constrained optimization problem over an infinite-dimensional space. The feasible set of the problem is convex while the objective function is non-Lipschitz, nonsmooth and nonconvex. We show that under some assumptions, the infinite-dimensional constrained optimization problem can be reduced to a finite-dimensional constrained optimization problem. Then we consider a penalty formulation for the finite-dimensional constrained optimization problem. We study the existence of exact penalty parameters concerning local minimizers, first order stationary points and ε-minimizers under some assumptions. Moreover, we present a smoothing penalty algorithm whose subproblems are solved via a nonmonotone proximal gradient method and prove the convergence of the algorithm to a scaled Karush-Kuhn-Tucker (KKT) point of the finite-dimensional constrained problem. Finally, we conduct numerical experiments on band-limited random fields to show the promising performance of the penalty method.