RemarkRemark: AI-adaptedProof: AI-generatedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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An arbitrary denoiser does not inherit the exact-MMSE objective

Remark

For $γ > 0$ , a global proximal selection of an extended-real penalty $h : R \to R \cup {+ \infty}$ is a map $P : R \to R$ such that, for every $z \in R$ , $P (z)$ is a global minimizer of $x \mapsto \frac{1}{2} ∣ x - z ∣^{2} + γ h (x)$ . The potential $h_{σ, γ}$ in the exact-MMSE stationarity result is constructed from the conditional-mean map $D_{σ}$ . Replacing $D_{σ}$ in PnP-ISTA by an arbitrary map ${\tilde{D}}_{σ}$ does not in general give a global proximal representation or inherit that stationarity conclusion: for every $γ > 0$ , the map $z \mapsto - z$ is not a global proximal selection of any such $h$ that is finite somewhere and nowhere $- \infty$ . This does not assert that a particular other denoiser has no different objective.

Remark 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

Facts & Assumptions

Given: The exact-MMSE potential construction, the general PnP-ISTA input-map convention, and the global-proximal-selection counterexample claimed in the Remark.

[F1]

Tweedie identity, positive Jacobian, and the MMSE proximal potential: for the exact Gaussian posterior-mean map, the constructed extended-real potential has that map as its unique global proximal minimizer.

[F2]

Plug-and-play ISTA with a denoiser: PnP-ISTA permits a deterministic map ${\tilde{D}}_{σ}$ in the denoising step.

[F3]

The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ : on $R$ , the norm is the absolute value and the squared Euclidean distance is the square of the scalar difference.

Verification

Given: A step parameter $γ > 0$ and the global-proximal-selection definition in the Remark.

1.1givenF3algebra

Suppose an extended-real penalty $h$ that is finite somewhere and nowhere $- \infty$ had $P (z) = - z$ as a global minimizer of $\frac{1}{2} ∣ x - z ∣^{2} + γ h (x)$ for every $z$ . Its objective has a finite competitor for every input, so each claimed global minimizer has finite value. For distinct $s, t \in R$ , compare the minimizer at input $s$ with competitor $P (t)$ and the minimizer at input $t$ with competitor $P (s)$ , then add the inequalities. The finite penalty values cancel and expansion gives $(P (s) - P (t)) (s - t) \geq 0$ .

2.1F1F2step 1.1algebra∎

Substituting $P (s) - P (t) = - (s - t)$ into step 1.1 yields $- (s - t)^{2} \geq 0$ , which is false for $s \neq t$ . Thus no such $h$ exists for this map, so the exact-MMSE construction and stationarity result cannot be transferred to an arbitrary denoiser without a separate proof.

Depends on

Used by

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

Dependency tree · two levels

39 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

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)