{ "schemaVersion": "1.6", "slug": "gaussian-pnp-stability", "title": "Gaussian-kernel plug-and-play iterations can be unstable: a sharp dimension threshold for irreducible unit-diagonal kernels, the role of the mixing gap, and why unit diagonals matter", "shortTitle": "Gaussian-kernel plug-and-play can be unstable, but only barely", "url": "https://evidencepress.org/releases/gaussian-pnp-stability/", "oneLine": "Answering an open question of Sinha et al. (2025): PnP-ISTA and PnP-ADMM with row-normalised Gaussian kernel denoisers are always stable in dimensions one to three but can diverge from dimension four, by at most about 1.7×10⁻⁵ per iteration; safe steps scale with the mixing gap, and symmetrised denoisers are stable.", "abstract": "Plug-and-play (PnP) methods for linear inverse problems alternate a data step with a denoiser. When the denoiser is a row-normalised kernel smoother such as Gaussian non-local means, Sinha, Kumar, Athalye and Chaudhury (IEEE Trans. Signal Process. 73, 2025, Section II-A) observed that the iteration seems to converge but that neither a proof nor a counterexample was known. This candidate answers the question for PnP-ISTA and PnP-ADMM. With irreducible unit-diagonal kernels (entries at most one) and a positive semidefinite data Hessian that preserves constants, both iterations are Schur stable in dimensions one to three for every admissible step; the proof reduces to Banerjee and Chaudhury's Theorem 1.3 by a new positivity identity. In dimension four they need not be: explicit Gaussian kernels and a fixed positive forward operator make both iterations diverge at any prescribed step 0 < t ≤ 1/2, and an exact two-parameter classification, with certificate data printed in an appendix and re-checked by a standard-library checker, locates the boundary t* ≈ 0.5313. A twelve-dimensional positive convolution example fails exactly for 0 < t < 1.9157×10⁻⁴, and without unit diagonals failure occurs in dimension three. The cause is a denoiser whose similarity graph is almost disconnected and averages unequally: steps below g/√κ_π are always safe, where g is the mixing gap, and linear scaling in g cannot be improved. Symmetric denoisers are stable exactly when no vector fixed by the denoiser is annihilated by the Hessian, extending Athalye, Chaudhury and Kumar (2023). The divergence is extremely slow, at most about 1.7×10⁻⁵ per iteration, and the affinities involved are tiny, so the results rule out a uniform guarantee rather than demonstrate divergence in an imaging pipeline. Not the first disproof of the broader Banerjee–Chaudhury conjecture. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.", "datePublished": "2026-10-27", "dateModified": "2026-10-27", "version": "1.0.0-candidate", "doi": "10.5281/zenodo.23690125", "doiUrl": "https://doi.org/10.5281/zenodo.23690125", "conceptDoi": "10.5281/zenodo.23690124", "pdfUrl": "https://github.com/ipitchford/gaussian-pnp-stability/releases/download/v1.0.0-candidate/gaussian-pnp-stability-1.0.0-candidate.pdf", "altPdfUrl": null, "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23690125", "repoUrl": "https://github.com/ipitchford/gaussian-pnp-stability", "releaseUrl": "https://github.com/ipitchford/gaussian-pnp-stability/releases/tag/v1.0.0-candidate", "markdownUrl": "https://evidencepress.org/releases/gaussian-pnp-stability/index.md", "bibtexUrl": "https://evidencepress.org/releases/gaussian-pnp-stability/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/gaussian-pnp-stability.mp3", "imageUrl": "https://evidencepress.org/assets/og/gaussian-pnp-stability.png?v=ccc9e4bced", "coverArtUrl": "https://evidencepress.org/assets/art/gaussian-pnp-stability.svg?v=427659fd8f", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/gaussian-pnp-stability.mp3", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/gaussian-pnp-stability.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed AI-assisted mathematical candidate. Every theorem has a written proof; Theorems 1(b), 3 and 4 rest on exact rational certificates whose decisive data are printed; the Section 12 growth table and Figure 1 are floating-point illustrations. Internal AI-assisted review only, including correctness audits by a different model and a targeted proof audit. External specialist review, independent reproduction and formal verification have not been established." }, "assurance": [ { "dimension": "availability", "label": "Availability and archiving", "question": "Is the evidence package publicly retrievable from an archive under a persistent identifier?", "state": "passed", "evidenceUrl": "https://zenodo.org/records/23690125", "note": "The versioned research package and manuscript are publicly archived on GitHub and Zenodo; availability does not establish correctness or peer review." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "replay_all.py reruns every exact verifier from the frozen archive and compares the records (timing and version fields excluded); check_certificates.py re-verifies Appendix C; eight negative controls are rejected. GitHub CI repeats these." }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "No unaffiliated rerun has been established." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "partial", "note": "A targeted proof audit rebuilt the Theorem 3 and 4 polynomials from the matrix definitions with its own Bernstein and root-isolation code and reproduced the isolating intervals; this is producer-side implementation diversity, not unaffiliated reimplementation." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "No proof-assistant formalisation." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "AI-assisted internal review and correctness audits are not external specialist review." }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed", "note": "No external editorial peer review." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "passed", "note": "Python 3.11; standard library for the core verifiers, checker and controls; sympy 1.13.3 for the symbolic certificates; numpy and matplotlib only for the figure." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted mathematical research", "AI-assisted internal review, correctness audits by a different model and a targeted proof audit", "Claude Code: final editor fixes, publication packaging and release" ], "humanRole": "Publication authority only; no human research contribution inferred.", "reviewStatus": "Internal AI-assisted review, not external peer review." }, "attributionBoundary": null, "problem": null, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Prepared from the final manuscript (dated 27 October 2026, 28 pages), its verifier outputs and the frozen release archive.", "grandfatheredAtSchemaVersion": null, "keywords": [ "plug-and-play", "kernel denoiser", "non-local means", "Schur stability", "PnP-ISTA", "PnP-ADMM", "mixing gap", "exact certificate" ], "keyResults": [ "Dimensions one to three are stable: for irreducible unit-diagonal kernels with entries at most one and B ⪰ 0 with Be = be, PnP-ISTA and PnP-ADMM are Schur stable for every 0 < t < 2/ρ(B) (Theorem 1(a), proof, via Banerjee–Chaudhury Theorem 1.3 and a new positivity identity).", "Dimension four can be unstable: a fixed positive forward operator and, for each 0 < t ≤ 1/2, an explicit Gaussian kernel give a real eigenvalue above one for both iterations (Theorem 1(b), exact certificate); the family is classified exactly, with boundary t* ≈ 0.5313 and ε_crit ≈ 1.69×10⁻⁵ (Theorem 4, exact certificate, Appendix C).", "Convolution forward operators are not immune: a twelve-state positive circulant example fails exactly for 0 < t < 1.9157×10⁻⁴ along ε = t² (Theorem 3, exact certificate).", "The cause is a small mixing gap g: steps with t ρ(B) < g/(√κ_π − g) are always safe (Theorem 2, proof), and linear scaling in g cannot be improved within the family, though the constant is conservative by about 8,600.", "Unit diagonals matter: without them a reversible positive kernel fails already in dimension three (Theorem 5, proof).", "Symmetric denoisers are stable exactly when ker(I − W) ∩ ker B = {0} (Theorem 6, Corollary 1, proof), extending Athalye, Chaudhury and Kumar (2023).", "Practical size: the growth is at most about 1.7×10⁻⁵ per iteration and the affinities are 10⁻⁸ to 10⁻⁶ (Section 12, numerical illustration)." ], "reviews": [], "evidencePackage": "Written proofs of every theorem; exact rational verifiers for the four-state counterexample, the twelve-state circulant constants and the three-state reentrant family; exact sympy certificates for the two-parameter classification (Theorem 4) and the certified circulant window (Theorem 3), whose decisive polynomials, Bernstein sign summaries, Sturm counts and isolating intervals are printed in Appendix C and stored in hashed coefficient files re-verified by a standard-library checker; a replay driver; eight semantic negative controls; CI.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Sinha, Kumar, Athalye and Chaudhury (2025), Linear convergence of plug-and-play algorithms with kernel denoisers. IEEE Transactions on Signal Processing 73, 2646–2659 (poses the open question answered here, Section II-A; read on arXiv:2505.15318v1).", "url": "https://doi.org/10.1109/TSP.2025.3573044", "doi": "10.1109/TSP.2025.3573044" }, { "citation": "Banerjee and Chaudhury (2025), On the Schur stability of some image reconstruction operators. arXiv:2509.06715 (Theorem 1.3 gives the n ≤ 3 result after the positivity identity; Conjecture 1.1).", "url": "https://arxiv.org/abs/2509.06715", "doi": "10.48550/arXiv.2509.06715" }, { "citation": "Athalye, Chaudhury and Kumar (2023), On the contractivity of plug-and-play operators. IEEE Signal Processing Letters 30, 1447–1451 (symmetric-denoiser contractivity; Theorem 6 extends it).", "url": "https://doi.org/10.1109/LSP.2023.3323248", "doi": "10.1109/LSP.2023.3323248" }, { "citation": "Gavaskar, Athalye and Chaudhury (2021), On plug-and-play regularization using linear denoisers. IEEE Transactions on Image Processing 30, 4802–4813 (earlier ADMM divergence example with a signed forward model).", "url": "https://doi.org/10.1109/TIP.2021.3075092", "doi": "10.1109/TIP.2021.3075092" }, { "citation": "Gavaskar and Chaudhury (2020), Plug-and-play ISTA converges with kernel denoisers. IEEE Signal Processing Letters 27, 610–614.", "url": "https://doi.org/10.1109/LSP.2020.2986643", "doi": "10.1109/LSP.2020.2986643" }, { "citation": "Liu, Xing and Sun (2021), Study on convergence of plug-and-play ISTA with adaptive-kernel denoisers. IEEE Signal Processing Letters 28, 1918–1922 (abstract only consulted).", "url": "https://doi.org/10.1109/LSP.2021.3111594", "doi": "10.1109/LSP.2021.3111594" }, { "citation": "Sreehari et al. (2016), Plug-and-play priors for bright field electron tomography and sparse interpolation. IEEE Transactions on Computational Imaging 2, 408–423 (symmetric doubly stochastic smoothing).", "url": "https://doi.org/10.1109/TCI.2016.2599778", "doi": "10.1109/TCI.2016.2599778" }, { "citation": "Chan, Wang and Elgendy (2017), Plug-and-play ADMM for image restoration: fixed-point convergence and applications. IEEE Transactions on Computational Imaging 3, 84–98.", "url": "https://doi.org/10.1109/TCI.2016.2629286", "doi": "10.1109/TCI.2016.2629286" }, { "citation": "Romano, Elad and Milanfar (2017), The little engine that could: regularization by denoising (RED). SIAM Journal on Imaging Sciences 10, 1804–1844.", "url": "https://doi.org/10.1137/16M1102884", "doi": "10.1137/16M1102884" }, { "citation": "Pesquet, Repetti, Terris and Wiaux (2021), Learning maximally monotone operators for image recovery. SIAM Journal on Imaging Sciences 14, 1206–1237.", "url": "https://doi.org/10.1137/20M1387961", "doi": "10.1137/20M1387961" }, { "citation": "Ryu et al. (2019), Plug-and-play methods provably converge with properly trained denoisers. arXiv:1905.05406.", "url": "https://arxiv.org/abs/1905.05406", "doi": "10.48550/arXiv.1905.05406" }, { "citation": "Hurault, Leclaire and Papadakis (2021), Gradient step denoiser for convergent plug-and-play. arXiv:2110.03220.", "url": "https://arxiv.org/abs/2110.03220", "doi": "10.48550/arXiv.2110.03220" }, { "citation": "Hurault, Leclaire and Papadakis (2022), Proximal denoiser for convergent plug-and-play optimization with nonconvex regularization. arXiv:2201.13256.", "url": "https://arxiv.org/abs/2201.13256", "doi": "10.48550/arXiv.2201.13256" } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:gaussian-pnp-stability", "attemptIds": [ "ep-attempt:gaussian-pnp-stability-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "communication" ], "lineageId": null, "accelerationPrimitives": [ "certificate-first", "counterexample-proxy-first", "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "counterexample", "description": "Explicit Gaussian kernels with a fixed positive forward operator make PnP-ISTA and PnP-ADMM unstable in dimension four, answering the Sinha et al. 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