Zero-SNR Analyticity of the Scalar MMSE Is Equivalent to Gaussianity
cs.IT · cs.LG
Quick verdict
Proves MMSE analytic at zero SNR iff input Gaussian (sub-Gaussian class) via MGF zeros and Borel analysis; consistent at spot-check, needs full refereeing.
Summary
This paper proves that for real inputs X satisfying a square-exponential moment condition E e^{βX²}<∞, the scalar MMSE in the Gaussian channel Y=√s·X+Z is real-analytic at zero SNR if and only if X is Gaussian (constants included as degenerate). The proof route is novel: the posterior mean is written via Q(s,z)=E e^{zX−sX²/2}, a backward heat flow applied to the moment-generating function M(z); Hadamard factorization forces every non-Gaussian law in the class to have a complex zero z0 of M; each zero cluster creates poles of F=Qz²/Q, which an exact double-Borel identity transfers to singularities of the Borel transform of the zero-SNR series at the action ξ=z0²/2. Simple zeros give coefficients ∼(2n−1)!!·z0^{−2n} with nonzero prefactor; zeros of multiplicity m split along Hermite roots, giving scale n^{−m/2}e^{r_m√(2n)}; an Abel-double-cover continuation plus a dichotomy for ±z0 clusters shows a singularity survives cancellation. Hence the formal zero-SNR expansion is Gevrey-1 and divergent for every non-Gaussian input in the class. Rational-MMSE rigidity and mutual-information analyticity rigidity follow as corollaries. Strengths include worked examples (binary input; the cosh² double zero) and honest attribution of classical ingredients. Limitations: the tail condition may not be minimal (left open by the author); the delicate saddle and continuation constants (Lemma 6.1, eqs. 120/131) are verified only in outline; the author discloses AI-assisted drafting of Section 7's exposition; and the work is a single-author, non-peer-reviewed preprint.
Review complete
Assessment at a glance
4
risk flags identified
● high 0
● medium 2
● low 2
5 strengths · 5 weaknesses · 5 author questions
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Key claims under review
- Under E e^{βX²}<∞ for some β>0, the scalar mmse_X(s) is analytic at s=0 if and only if X is Gaussian, constants counting as degenerate Gaussians (Theorem 1.1).
- Every non-Gaussian law in the class has a nonzero complex MGF zero, and the induced zero clusters create poles of F=Qz²/Q that become finite, non-cancelable singularities of the Borel transform of the zero-SNR series at the actions ξ=z0²/2 (Lemmas 2.1/7.1, Propositions 4.2/7.3).
- Simple zeros give coefficients ∼−(M′(z0)/z0)e^{−z0M″(z0)/(2M′(z0))}(2n−1)!!z0^{−2n}; zeros of multiplicity m≥2 split along Hermite roots and contribute scale n^{−m/2}e^{r_m√(2n)} with nonzero constant (Propositions 5.1/6.2), so the formal zero-SNR expansion is Gevrey-1 and divergent for every non-Gaussian input in the class.
- Rational-MMSE rigidity (Corollary 1.2) and mutual-information analyticity rigidity (Corollary 1.3) hold under the same tail condition, the latter with I_X(s)=½log(1+σ²s) in the Gaussian case.
Strengths
Complete characterization with clean, properly scoped statement
Theorem 1.1 resolves a rigidity question left implicit in Guo–Wu–Shamai–Verdú [6], who showed infinite right-differentiability at zero but a divergent Taylor series for binary input; the iff form (constants as degenerate Gaussians) is sharp within the stated class, and Corollaries 1.2–1.3 follow via short correct arguments (removable singularity for rational MMSE; I–MMSE identity with the correct ½ factor in eq. 162).
New quantitative bridge from MGF zeros to Borel singularities
Propositions 5.1 and 6.2 give explicit large-order asymptotics — (2n−1)!!z0^{−2n} with nonzero prefactor e^{−z0M″(z0)/(2M′(z0))} for simple zeros, and n^{−m/2}e^{r_m√(2n)} with Hermite-root splitting for multiplicity m — upgrading the known binary nonanalyticity into a full complex-action spectrum ξ=z0²/2 (Example 5.3: ξ0=−π²/8).
Noncancellation handled honestly
Verifiability aids are unusually good for a proof paper
Supporting estimates are done properly, not asserted
Weaknesses
Central asymptotic constants verified only in outline
AI-assisted drafting of the most delicate continuation section
Scope confined to square-exponential tails
Novelty positioning is narrative, not tabulated
No external verification artifacts
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Scope & Claims
The paper states and proves one central theorem plus two corollaries. Theorem 1.1: for real inputs X with E e^{βX²}<∞ for some β>0, the scalar MMSE mmse_X(s) of the channel Y=√s·X+Z is analytic at s=0 if and only if X is Gaussian (constants as degenerate case; Gaussian case gives mmse=σ²/(1+σ²s), eq. 5). Corollary 1.2: if the MMSE equals a rational function of s, the input is Gaussian. Corollary 1.3: mutual information is analytic at zero iff the input is Gaussian (then I_X(s)=½log(1+σ²s)). The positioning is honest: Guo–Wu–Shamai–Verdú [6] already knew all moments give infinite right-differentiability and that symmetric binary input is nonanalytic; Mansanarez–Poly–Swan [15] study the (different) MMSE conjecture. The genuinely new content is twofold: the complete characterization, and a quantitative structure — the paper shows the MGF zero set acts as a 'non-Gaussianity spectrum' via the action map z↦z²/2, with explicit large-order coefficient asymptotics (Propositions 5.1 and 6.2).
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Frequently asked questions
What does this paper prove about MMSE at zero SNR?
For input distributions with Gaussian-type tails (finite E e^{βX²} for some β>0), the minimum mean-square error curve of the scalar Gaussian channel has a convergent Taylor series at SNR=0 exactly when the input is Gaussian (constants counting as degenerate Gaussians). For every other input in that class, the zero-SNR expansion diverges — it is Gevrey-1, with coefficients controlled by complex zeros of the input's moment-generating function.
How does the proof work?
The posterior mean is written through a function Q solving a backward heat equation applied to the moment-generating function M(z)=E e^{zX}. For non-Gaussian inputs, M must have a complex zero; each zero becomes a pole that an exact double-Borel identity transfers to a singularity of the Borel transform of the zero-SNR series at the point z0²/2. A continuation and noncancellation argument shows such a singularity always survives, which forces the Taylor series to diverge.
Does this paper settle the MMSE conjecture?
No. The MMSE conjecture asks whether the full MMSE curve determines the input law up to translation and reflection. This paper asks a different question — whether analyticity at the single endpoint s=0 alone forces Gaussianity — and cites the conjecture (Mansanarez–Poly–Swan, 2024) only as related work.
Is the paper peer reviewed?
It is a September 2026 arXiv preprint (arXiv:2609.15048) by a single author, who discloses AI assistance (OpenAI Codex) for language editing and parts of the Section 7 exposition. This review found the argument internally consistent at every step it could recompute, but the result has not yet been through journal peer review.
About this review
This is an AI-generated preliminary review by Evolve.Pub, produced from the openly available text of the paper. It is not peer review and does not represent the authors' views. The canonical record of the paper is at the publisher link above.
- Mathematical notation was extracted with artifacts (sub/superscripts flattened); displayed constants such as (120), (123), (131) could not be fully re-derived here, only spot-checked for internal consistency — the checks performed (double-Borel coefficient identity (46), Example 5.3 action −π²/8, Example 6.3 residues and constant (136), prefactor 2^{(1−m)/2} in (131)) all pass.
- Figure 1 (dependency graph) survives only as flattened text; node-edge relations were reconstructed from surrounding prose.
- Full line-by-line refereeing of Lemma 6.1 and Lemma 7.1 is beyond a text-only review; 'internal consistency' here does not certify correctness.
- No external literature search was performed (network disabled); novelty positioning rests on the paper's own citations.
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