Parameter Selection for HOTV Regularization
This work simplifies parameter selection for practitioners using HOTV regularization, though it is an incremental improvement over existing TV-based methods.
The authors address the problem of selecting the regularization parameter λ for higher-order total variation (HOTV) methods in inverse problems. They theoretically derive a scaling that allows λ to be chosen for one order and applied to all orders, with numerical validation.
Popular methods for finding regularized solutions to inverse problems include sparsity promoting $\ell_1$ regularization techniques, one in particular which is the well known total variation (TV) regularization. More recently, several higher order (HO) methods similar to TV have been proposed, which we generally refer to as HOTV methods. In this letter, we investigate problem of the often debated selection of $λ$, the parameter used to carefully balance the interplay between data fitting and regularization terms. We theoretically argue for a scaling of the parameter that works for all orders for HOTV methods, based off of a single selection of the parameter for any one of the orders. We also provide numerical results which justify our theoretical findings.