Year: 2015
Type: preprint
Abstract: This work aims to explore the regularity properties of the smoothed-TV regularization for the functions is of the class Hölder continuous. Over some compact and convex domain $Ω,$ we study construction of multivariate function $φ(\mathbf{x}) :Ω\subset \mathbb{R}^{3} \rightarrow \mathbb{R}_{+}$ as the optimized solution to the following convex minimization problem \begin{equation} \min_Ω \left\{F_α(\cdot, f^δ) := \frac{1}{2} \Vert \mathcal{T}(\cdot) - f^δ \Vert_{\mathcal{H}}^2 + αJ(\cdot) \right\}, \end{equation} where the penalizer $J(\cdot) : \mathcal{C}^{1}(Ω,\mathbb{R}^{3})\rightarrow \mathbb{R}_{+}$ is the smoothed total variation penalizer \begin{equation} J(\cdot) = \int_Ω \sqrt{\Vert\nabla(\cdot)\Vert_2^2 + β} d \mathbf{x}, \end{equation} for a fixed $0 < β< 1.$ We assume our target function to be Hölder continuous. With this assumption, we establish relation between total variation of our target function and its Hölder coefficient. We prove that the smoothed-TV regularization is an admissible regularization strategy by evaluating the discrepancy $\Vert\mathcal{T}φ_α - f^δ\Vert \leq τδ,$ for some fixed $τ\geq 1.$ To do so, we need to assume that the target function to be class of $\mathcal{C}^{1+}(Ω).$ From here, under the fact that the penalty $J(\cdot)$ is strongly convex, we move on to showing the convergence of $\Vertφ_α - φ^{\dagger}\Vert,$ for $φ_α$ is the optimum and $φ^{\dagger}$ is the true solution for the given minimization problem above. We demonstrate that strong convexity and $2-$convexity are actually different names for the same concept. In addition to these facts, we make us of Bregman divergence in order to be able to quantify the rate of convergence.
Source: arXiv (Cornell University)
Author: Altuntac, Erdem
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2206661735