ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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A biased one-pixel CNN approximates but is not the exact MMSE denoiser

Example

Let $a, σ > 0$ and use the scalar two-point prior assigning mass $1 / 2$ to each of $- a$ and $a$ , whose exact Gaussian posterior-mean map is $D_{σ} (z) = a \tanh (a z / σ^{2})$ , where $\tanh (s) = (\exp (s) - \exp (- s)) / (\exp (s) + \exp (- s))$ . For any tolerance $η > 0$ , choose $b$ with $0 < ∣ b ∣ < η$ and define the one-pixel convolutional neural network (CNN) map ${\tilde{D}}_{σ} (z) = a \tanh (a z / σ^{2}) + b .$ This map is implemented by a $1 \times 1$ convolution, a $\tanh$ activation, output scaling, and an output bias $b$ . It satisfies $\sup_{z \in R} ∣ {\tilde{D}}_{σ} (z) - D_{σ} (z) ∣ = ∣ b ∣ < η$ , while ${\tilde{D}}_{σ} (0) \neq D_{σ} (0)$ . It is approximate and does not meet the exact-denoiser hypothesis of the stationarity theorem; this example makes no claim that it lacks some other objective.

Example references

Tweedie identity, positive Jacobian, and the MMSE proximal potential

Plug-and-play ISTA with a denoiser

Exact-MMSE PnP-ISTA has monotone objective and vanishing gradient

An arbitrary denoiser does not inherit the exact-MMSE objective

Network diagram

For this one-channel, one-pixel input, a $1 \times 1$ convolution with weight $w$ is scalar multiplication $z \mapsto w z$ . The architecture applies weight $a / σ^{2}$ , then $\tanh$ , then output scaling by $a$ and bias addition by $b$ . The arrows depict function composition, and the sequence computes exactly ${\tilde{D}}_{σ} (z) = a \tanh (a z / σ^{2}) + b$ .

\begin{tikzpicture}[x=1.45cm,y=0.8cm,>=stealth]
  \node (input) at (0,0) {$z$};
  \node[draw,rounded corners,minimum width=2.3cm,minimum height=0.9cm] (conv) at (2,0) {$1\times1$ conv: $a/\sigma^2$};
  \node[draw,rounded corners,minimum width=1.4cm,minimum height=0.9cm] (act) at (4.5,0) {$\tanh$};
  \node[draw,rounded corners,minimum width=2cm,minimum height=0.9cm] (out) at (7,0) {scale $a$, bias $b$};
  \node (result) at (9.2,0) {$\widetilde D_\sigma(z)$};
  \draw[->] (input) -- (conv);
  \draw[->] (conv) -- node[above] {$az/\sigma^2$} (act);
  \draw[->] (act) -- node[above] {$\tanh(az/\sigma^2)$} (out);
  \draw[->] (out) -- (result);
\end{tikzpicture}

Facts & Assumptions

Given: $a, σ, η > 0$ , a nonzero bias $b$ with $∣ b ∣ < η$ , and the finite two-point prior and Gaussian likelihood in the Example.

[F1]

The Dirac set function at a point: $δ_{x_{0}} (E) = 1$ when $x_{0} \in E$ and $0$ otherwise.

[F2]

A Dirac set function is a probability measure: each Dirac set function is a probability measure.

[F3]

The exponential addition formula $\exp (x + y) = \exp (x) \exp (y)$ : $\exp (x + y) = \exp (x) \exp (y)$ for all real $x, y$ .

[F4]

Tweedie identity, positive Jacobian, and the MMSE proximal potential: for a Borel probability measure not concentrated on an affine hyperplane, the Gaussian posterior-mean map is defined by the normalized likelihood integral and uniquely minimizes posterior squared-error risk.

[F5]

Exact-MMSE PnP-ISTA has monotone objective and vanishing gradient: the stationarity conclusion assumes the plug-in map is exactly the analytic posterior-mean map $D_{σ}$ .

Verification

Given: The prior, noise parameter, tolerance, bias, and network map in the Example.

1.1F1F2F3F4givenalgebra

By [F1]–[F2], $μ = \frac{1}{2} δ_{- a} + \frac{1}{2} δ_{a}$ is a Borel probability. Because $a > 0$ , the atoms are distinct, each has positive mass, and every affine hyperplane in $R$ is a singleton, so $μ$ is not concentrated on any affine hyperplane. Its posterior mean is $D_{σ} (z) = a \frac{ρ_{σ} (z - a) - ρ_{σ} (z + a)}{ρ_{σ} (z - a) + ρ_{σ} (z + a)} = a \tanh (a z / σ^{2})$ , where the second equality follows by factoring the common Gaussian term and using [F3].

2.1F4F5givenalgebra∎

For every $z$ , $∣ {\tilde{D}}_{σ} (z) - D_{σ} (z) ∣ = ∣ b ∣ < η$ , so the uniform error is $∣ b ∣$ . At $z = 0$ , $D_{σ} (0) = 0$ while ${\tilde{D}}_{σ} (0) = b \neq 0$ ; hence the network is not the exact MMSE map and [F5] does not apply. No conclusion about a different objective follows.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Xiaojian Xu, Yu Sun, Jiaming Liu, Brendt Wohlberg, and Ulugbek S. Kamilov, Provable Convergence of Plug-and-Play Priors with MMSE Denoisers, IEEE Signal Processing Letters 27 (2020), arXiv:2005.07685 (standard reference, not scraped)
Rémi Gribonval and Mila Nikolova, On Bayesian Estimation and Proximity Operators, Applied and Computational Harmonic Analysis 50 (2021), 49–72, arXiv:1807.04021 (standard reference, not scraped)
Neal Parikh and Stephen Boyd, Proximal Algorithms, Foundations and Trends in Optimization 1(3) (2014) (standard reference, not scraped)