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 ${\mathbf{T}}^{\mathrm{†}}{S}_{\gamma }\mathbf{T}\mathbf{x}$, 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, ${\mathbf{T}}^{\mathrm{†}}$ denotes the Moore-Penrose inverse of 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 ${\Vert \mathbf{T}\cdot \Vert }_1$ if T is an orthogonal (square) matrix. It is well-known that for rectangular frame matrices T with $L>N$, the proximity operator of ${\Vert \mathbf{T}\cdot \Vert }_1$ does not have a closed representation and needs to be computed iteratively. We show that the frame soft shrinkage operator ${\mathbf{T}}^{\mathrm{†}}{S}_{\gamma }\mathbf{T}$ is a proximity operator as well, thereby motivating its application as a replacement of the exact proximity operator of ${\Vert \mathbf{T}\cdot \Vert }_1$. 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 Φ which leads to the proximity operator ${\mathrm{prox}}_{\mathrm{\Phi }}={\mathbf{T}}^{\mathrm{†}}{S}_{\gamma }\mathbf{T}$ and show that ${\mathbf{T}}^{\mathrm{†}}{S}_{\gamma }\mathbf{T}$ behaves similarly as ${\mathrm{\text{prox}}}_{{\Vert \mathbf{T}\cdot \Vert }_1}$.

在本文中，我们展示了常用的框架软收缩算子，它映射给定的向量 $\mathbf{X}\in {\mathbb{电阻}}^N$ 到向量上 ${\mathbf{吨}}^{\mathrm{†}}{秒}_{\gamma }\mathbf{吨}\mathbf{X}$, 已经是一个邻近算子，因此可以直接用于相应的分裂算法中。在我们的设置中，帧变换矩阵$\mathbf{吨}\in {\mathbb{电阻}}^{升\times N}$ 和 $升\ge N$有满秩N，${\mathbf{吨}}^{\mathrm{†}}$表示T的 Moore-Penrose 逆，并且${秒}_{\gamma }$ 是具有阈值参数的常用软收缩算子 $\gamma >0$. 我们的结果概括了已知的断言，即${\mathbf{吨}}^{⁎}{秒}_{\gamma }\mathbf{吨}$ 是接近算子 ${\Vert \mathbf{吨}\cdot \Vert }_1$如果T是正交（方）矩阵。这是公知的，对于矩形框架矩阵Ť与$升>N$, 的邻近算子 ${\Vert \mathbf{吨}\cdot \Vert }_1$没有封闭表示，需要迭代计算。我们证明了框架软收缩算子${\mathbf{吨}}^{\mathrm{†}}{秒}_{\gamma }\mathbf{吨}$ 也是一个接近算子，从而激发了它的应用，作为替代精确接近算子的 ${\Vert \mathbf{吨}\cdot \Vert }_1$. 我们进一步解释了为什么框架软收缩算子的使用在各种应用中仍然可以提供良好的结果。特别地，我们提供了凸函数 Φ 的次微分的一些性质，这导致了邻近算子${\mathrm{近似值}}_{\mathrm{\Phi }}={\mathbf{吨}}^{\mathrm{†}}{秒}_{\gamma }\mathbf{吨}$ 并表明 ${\mathbf{吨}}^{\mathrm{†}}{秒}_{\gamma }\mathbf{吨}$ 行为类似于 ${\mathrm{\text{近似值}}}_{{\Vert \mathbf{吨}\cdot \Vert }_1}$.