Frame Soft Shrinkage Operators are Proximity Operators

TL;DR

This paper proves that the frame soft shrinkage operator is a proximity operator, enabling its direct use in splitting algorithms and providing a practical approximation for the more complex proximity operator of the frame-based L1 norm.

Contribution

It establishes that the frame soft shrinkage operator is a proximity operator, generalizing known results and facilitating efficient algorithms for frame-based sparse regularization.

Findings

Frame soft shrinkage operator is a proximity operator.

It approximates the proximity operator of the frame-based L1 norm.

The operator can be used directly in splitting algorithms.

Abstract

In this paper, we show that the commonly used frame soft shrinkage operator, that maps a given vector ${\mathbf x} \in {\mathbb R}^{N}$ onto the vector ${\mathbf T}^{\dagger} S_{\gamma} {\mathbf T} {\mathbf x}$, is already a proximity operator, which can therefore be directly used in corresponding splitting algorithms. In our setting, the frame transform matrix ${\mathbf T} \in {\mathbb R}^{L \times N}$ with $L \ge N$ has full rank $N$, ${\mathbf T}^{\dagger}$ denotes the Moore-Penrose inverse of ${\mathbf T}$, and $S_{\gamma}$ is the usual soft shrinkage operator with threshold parameter $\gamma >0$. Our result generalizes the known assertion that ${\mathbf T}^{*} S_{\gamma} {\mathbf T}$ is the proximity operator of $\| {\mathbf T} \cdot \|_{1}$ if ${\mathbf T}$ is an orthogonal (square) matrix. It is well-known that for rectangular frame matrices ${\mathbf T}$ with $L > N$, the proximity operator of $\| {\mathbf T} \cdot \|_{1}$ does not have a closed representation and needs to be computed iteratively. We show that the frame soft shrinkage operator {${\mathbf T}^{\dagger} S_{\gamma} {\mathbf T}$} is a proximity operator as well, thereby motivating its application as a replacement of the exact proximity operator of $\| {\mathbf T} \cdot \|_{1}$. We further give an explanation, why the usage of the frame soft shrinkage operator still provides good results in various applications. In particular, we provide some properties of the subdifferential of the convex functional $\Phi$ which leads to the proximity operator ${\mathbf T}^{\dagger} S_{\gamma} {\mathbf T}$ and show that ${\mathbf T}^{\dagger} S_{\gamma} {\mathbf T}$ approximates $\textrm{prox}_{\|{\mathbf T} \cdot\|_{1}}$.