Convergence analysis in convex regularization depending on the smoothness degree of the penalizer
Provides theoretical convergence guarantees for inverse problems with smooth regularizers, but the analysis is incremental as it extends known results to a specific smoothness class.
The paper analyzes convergence of regularized solutions for least squares minimization with smooth convex regularizers, showing that for twice-differentiable regularizers the discrepancy can be bounded using the Hessian Lipschitz constant, while for once-differentiable regularizers a different parameter choice yields convergence.
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 $Ω$ of the Hilbert space $\mathcal{H},$ the regularizer is implicitly defined as $ J(\cdot) : \mathcal{C}^{k}(Ω, \mathcal{H}) \rightarrow \mathbb{R}_{+}$ where $k \in \{1,2\}.$ So the cost functional associated with some given linear, compact and injective forward operator $\mathcal{T} :Ω\subset \mathcal{H} \rightarrow \mathcal{H},$ \begin{align} F_α(\cdot , f^δ) := \frac{1}{2} \Vert \mathcal{T}( \cdot ) - f^δ\Vert_{\mathcal{H}}^2 + αJ(\cdot) , \nonumber \end{align} where $f^δ$ is the given perturbed data with its perturbation amount $δ$ in it. Convergence of the regularized optimum solution $φ_{α(δ)} \in \mbox{argmin} F_α(φ, f^δ)$ to the true solution $φ^{\dagger}$ 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}(Ω, \mathcal{H}) \rightarrow \mathbb{R}_{+}.$ In both cases, we define such a regularization parameter that is in cooperation with the condition \begin{align} α(δ, f^δ) \in \{ α> 0 \mbox{ }\vert \mbox{ }\Vert\mathcal{T}φ_α^δ - f^δ\Vert \leq τδ\} , \nonumber \end{align} for some fixed $τ\geq 1.$ In the case of $k = 2,$ we are able to evaluate the discrepancy $\Vert\mathcal{T}φ_{α(δ)} - f^δ\Vert\leq τδ$ with the Hessian Lipschitz constant $L_H$ of the functional $F_α(\cdot , f^δ).$