Plug-and-play ISTA with a denoiser

Let $n\ge 1$, $\sigma ,\gamma >0$, let $g:{\mathbb{R}}^n\to \mathbb{R}$ be continuously differentiable, let ${\tilde{D}}_{\sigma }:{\mathbb{R}}^n\to {\mathbb{R}}^n$ be a supplied deterministic map used in the denoising step, and choose $x_0\in {\mathbb{R}}^n$. Define $z_t=x_{t-1}-\gamma \mathrm{\nabla }g(x_{t-1}),x_t={\tilde{D}}_{\sigma }(z_t)(t=1,2,\dots )\mathrm{.}$ The output is the sequence $(x_t{)}_{t\ge 0}$. This definition imposes no convexity, proximal representation, exact-MMSE identity, or convergence property on ${\tilde{D}}_{\sigma }$.

Here “denoiser” names the role of the plug-in map; Borel measurability and training under a signal-corruption law, as in the learned-estimator interface, are not assumed.

The diagram depicts the data flow for one update; the displayed recurrence, not the diagram, defines the iteration.

\begin{tikzpicture}[x=1.3cm,y=0.9cm,>=stealth]
  \node at (0,0) (old) {$x_{t-1}$};
  \node at (3,0) (grad) {$z_t=x_{t-1}-\gamma\nabla g(x_{t-1})$};
  \node at (7,0) (den) {$x_t=\widetilde D_\sigma(z_t)$};
  \draw[->] (old) -- (grad);
  \draw[->] (grad) -- (den);
\end{tikzpicture}

The narrower interface for Borel denoisers trained under a specified signal-corruption law