Frame Soft Shrinkage Operators are Proximity Operators

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 $\textrm{prox}_{\|{\mathbf T} \cdot\|_{1}}$.