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CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)
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Gaussian measurement noise need not remain Gaussian at later PnP-ISTA inputs

Statement refuted

Assume the Axiom of Choice. Let ε have the standard normal law, let Y=θ+ε, and let g(x)=(x−Y)2/2. For any noise parameter σ>0, deterministic map D~σ:R→R, step γ>0, and initialization x0∈R, define zt=xt−1−γ(xt−1−Y) and xt=D~σ(zt). The false claim is that Gaussian measurement noise forces zt−θ to have a Gaussian law at every iteration t≥1, for every such denoiser, step, and initialization.

Facts & Assumptions

Given: AC, a standard normal random variable ε, Y=θ+ε, the scalar least-squares function g(x)=(x−Y)2/2, and the PnP-ISTA recurrence in the Statement.

[A1]

The Axiom of Choice: the Axiom of Choice is assumed.

[F1]

Standard normal and normal laws: under AC, N(0,1) is the probability law with density ϕ(u)=exp⁡(−u2/2)/2π, which is strictly positive for every real u.

[F2]

A positive-variance normal law has its location-scale density: under AC, every positive-variance normal law has a strictly positive Lebesgue density on all of R; a zero-variance normal law is a point mass.

[F3]

Plug-and-play ISTA with a denoiser: the gradient step is zt=xt−1−γ∇g(xt−1) and the next iterate is xt=D~σ(zt).

Counterexample

Given: The probability model and recurrence in the Statement. Choose θ=0, x0=0, σ=1, γ=1/2, and the deterministic plug-in map D~1(z)=z2.

1.1F3givenalgebra

Since ∇g(x)=x−ε, [F3] gives z1=ε/2, x1=ε2/4, and z2=x1−12(x1−ε)=ε2/8+ε/2=(ε+2)2/8−1/2. Thus z2≥−1/2 almost surely.

2.1A1F1step 1.1givenalgebra

The event −1/2<ε<1/2 has positive probability by [A1, F1] and gives −7/32<z2<9/32. The event −5/2<ε<−3/2 also has positive probability and gives −1/2≤z2<−15/32. These disjoint output intervals show that z2 is not almost surely constant.

3.1A1F2step 1.1step 2.1∎

A nondegenerate Gaussian law has positive probability below −1/2 by [A1, F2], whereas z2 has none by step 1.1; a degenerate Gaussian is a point mass, whereas step 2.1 shows z2 is not. Therefore z2 is not Gaussian and the universal iterative-AWGN claim is false.

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