Frame Soft Shrinkage as Proximity Operator

TL;DR

This paper demonstrates that certain frame-based soft shrinkage operators can be characterized as proximity operators within an appropriate Hilbert space, extending the understanding of their mathematical structure.

Contribution

It proves that the operator $T^ \, \text{Prox} \, T$ is a proximity operator for any proximity operator and injective operator with closed range, including frame analysis operators.

Findings

Frame shrinkage operators are proximity operators in a suitable Hilbert space.

The result applies to the commonly used soft shrinkage operator.

Provides a new perspective on the mathematical structure of frame-based denoising methods.

Abstract

Let $\mathcal H$ and $\mathcal K$ be real Hilbert spaces and $T \in \mathcal{B} (\mathcal H,\mathcal K)$ an injective operator with closed range and Moore-Penrose inverse $T^\dagger$. Based on the well-known characterization of proximity operators by Moreau, we prove that for any proximity operator $\text{Prox} \colon \mathcal K \to \mathcal K$ the operator $T^\dagger \, \text{Prox} \, T$ is a proximity operator on the linear space $\mathcal H$ equipped with a suitable norm. In particular, it follows for the frequently applied soft shrinkage operator $\text{Prox} = S_{\lambda}\colon \ell_2 \rightarrow \ell_2$ and any frame analysis operator $T\colon \mathcal H \to \ell_2$, that the frame shrinkage operator $T^\dagger\, S_\lambda\, T$ is a proximity operator in a suitable Hilbert space.