Isotropic non-Lipschitz regularization for sparse representations of random fields on the sphere

Abstract

In this paper, we consider an infinite-dimensional isotropic non-Lipschitz optimization problem with ℓ 2 , p \ell _{2,p} ( 0 > p > 1 0>p>1 ) regularizer for random fields on the unit sphere with spherical harmonic representations. 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 first order and second order necessary optimality conditions for local minimizers of the optimization problem. We also derive two lower bounds for the nonzero groups of stationary points, which are used to prove that the infinite-dimensional optimization problem can be reduced to a finite-dimensional problem. Moreover, we propose an iteratively reweighted algorithm for the finite-dimensional problem and prove its convergence. Finally, numerical experiments on Cosmic Microwave Background data are presented to show the efficiency of the non-Lipschitz regularization.