Projected Newton method for noise constrained ℓp regularization. (3rd December 2020) - Cornelis, J; Vanroose, W
Abstract: Choosing an appropriate regularization term is necessary to obtain a meaningful solution to an ill-posed linear inverse problem contaminated with measurement errors or noise. The ℓ p norm covers a wide range of choices for the regularization term since its behavior critically depends on the choice of p and since it can easily be combined with a suitable regularization matrix. We develop an efficient algorithm that simultaneously determines the regularization parameter and corresponding ℓ p regularized solution such that the discrepancy principle is satisfied. We project the problem on a low-dimensional generalized Krylov subspace and compute the Newton direction for this much smaller problem. We illustrate some interesting properties of the algorithm and compare its performance with other state-of-the-art approaches using a number of numerical experiments, with a special focus of the sparsity inducing ℓ 1 norm and edge-preserving total variation regularization.