Convergence analysis in convex regularization depending on the smoothness degree of the penalizer

TL;DR

This paper investigates the convergence of solutions in convex regularization problems, analyzing how the smoothness degree of the penalizer affects convergence rates and stability.

Contribution

It provides a convergence analysis for regularized solutions depending on whether the regularizer is once or twice continuously differentiable, incorporating the Hessian Lipschitz constant for the second case.

Findings

Convergence depends on the smoothness degree of the regularizer.

For twice differentiable regularizers, the discrepancy can be estimated using the Hessian Lipschitz constant.

Regularization parameters are chosen to satisfy a discrepancy principle.

Abstract

The problem of minimization of the least squares functional with a smooth, lower semi-continuous, convex regularizer $J(\cdot)$ is considered to be solved. Over some compact and convex subset $\Omega$ of the Hilbert space $\mathcal{H},$ the regularizer is implicitly defined as $ J(\cdot) : \mathcal{C}^{k}(\Omega , \mathcal{H}) \rightarrow \mathbb{R}_{+}$ where $k \in \{1,2\}.$ So the cost functional associated with some given linear, compact and injective forward operator $\mathcal{T} :\Omega \subset \mathcal{H} \rightarrow \mathcal{H},$ \begin{align} F_{\alpha}(\cdot , f^{\delta}) := \frac{1}{2} \Vert \mathcal{T}( \cdot ) - f^{\delta}\Vert_{\mathcal{H}}^2 + \alpha J(\cdot) , \nonumber \end{align} where $f^{\delta}$ is the given perturbed data with its perturbation amount $\delta$ in it. Convergence of the regularized optimum solution $\varphi_{\alpha(\delta)} \in \mbox{argmin} F_{\alpha}(\varphi , f^{\delta})$ to the true solution $\varphi^{\dagger}$ is analysed depending on the smoothness degree of the regularizer, \textit{i.e.} the cases $k \in \{1,2\}$ in $ J(\cdot) : \mathcal{C}^{k}(\Omega , \mathcal{H}) \rightarrow \mathbb{R}_{+}.$ In both cases, we define such a regularization parameter that is in cooperation with the condition \begin{align} \alpha(\delta , f^{\delta}) \in \{ \alpha > 0 \mbox{ }\vert \mbox{ }\Vert\mathcal{T}\varphi_{\alpha}^{\delta} - f^{\delta}\Vert \leq \tau\delta \} , \nonumber \end{align} for some fixed $\tau \geq 1.$ In the case of $k = 2,$ we are able to evaluate the discrepancy $\Vert\mathcal{T}\varphi_{\alpha(\delta)} - f^{\delta}\Vert\leq \tau\delta$ with the Hessian Lipschitz constant $L_H$ of the functional $F_{\alpha}(\cdot , f^{\delta}).$