Parameter Selection for HOTV Regularization

TL;DR

This paper investigates the parameter selection for HOTV regularization methods, proposing a scaling approach that applies across all orders, supported by theoretical analysis and numerical validation.

Contribution

It introduces a unified scaling strategy for the regularization parameter in HOTV methods, simplifying the parameter tuning process across different orders.

Findings

A theoretical argument for a universal parameter scaling for HOTV methods.

Numerical results validating the proposed scaling approach.

Enhanced understanding of parameter interplay in higher order regularization.

Abstract

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 $\lambda$, 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.