Projected Newton method for noise constrained $\ell_p$ regularization

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. A regularization term in the the $\ell_p$ norm with $p\geq 1$ covers a wide range of choices 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 $\ell_p$ 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 $\ell_1$ norm and edge-preserving total variation regularization.