{"paper":{"title":"Nonasymptotic Convergence Rates for Plug-and-Play Methods With MMSE Denoisers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The MMSE denoiser under Gaussian noise corresponds to a 1-weakly convex regularizer given by an upper Moreau envelope of the negative log-marginal density, which yields the first sublinear convergence rates for plug-and-play proximal grad","cross_cats":["eess.SP","stat.ML"],"primary_cat":"math.OC","authors_text":"Henry Pritchard, Rahul Parhi","submitted_at":"2025-10-31T06:12:49Z","abstract_excerpt":"It is known that the minimum-mean-squared-error (MMSE) denoiser under Gaussian noise can be written as a proximal operator, which suffices for asymptotic convergence of plug-and-play (PnP) methods but does not reveal the structure of the induced regularizer or give convergence rates. We show that the MMSE denoiser corresponds to a regularizer that can be written explicitly as an upper Moreau envelope of the negative log-marginal density, which in turn implies that the regularizer is 1-weakly convex. Using this property, we derive (to the best of our knowledge) the first sublinear convergence g"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We show that the MMSE denoiser corresponds to a regularizer that can be written explicitly as an upper Moreau envelope of the negative log-marginal density, which in turn implies that the regularizer is 1-weakly convex. Using this property, we derive the first sublinear convergence guarantee for PnP proximal gradient descent with an MMSE denoiser.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The derivation assumes the noise is exactly Gaussian and that the MMSE denoiser is applied without any additional approximation or clipping; if the actual noise deviates from this model or if the denoiser is replaced by a learned network that only approximates the MMSE operator, the weak-convexity and rate guarantees no longer hold directly (see abstract and the proximal-operator equivalence stated in the introduction).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"MMSE denoisers correspond to 1-weakly convex regularizers via upper Moreau envelopes of negative log-marginals, enabling the first sublinear convergence rates for PnP proximal gradient descent.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The MMSE denoiser under Gaussian noise corresponds to a 1-weakly convex regularizer given by an upper Moreau envelope of the negative log-marginal density, which yields the first sublinear convergence rates for plug-and-play proximal grad","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"954b757783bb55ac136cca34254f8fe324fc9626448e07548e837d984c17e2a1"},"source":{"id":"2510.27211","kind":"arxiv","version":7},"verdict":{"id":"88d7ccfa-6a77-443f-a86b-389c918d8860","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-18T03:34:56.233495Z","strongest_claim":"We show that the MMSE denoiser corresponds to a regularizer that can be written explicitly as an upper Moreau envelope of the negative log-marginal density, which in turn implies that the regularizer is 1-weakly convex. Using this property, we derive the first sublinear convergence guarantee for PnP proximal gradient descent with an MMSE denoiser.","one_line_summary":"MMSE denoisers correspond to 1-weakly convex regularizers via upper Moreau envelopes of negative log-marginals, enabling the first sublinear convergence rates for PnP proximal gradient descent.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The derivation assumes the noise is exactly Gaussian and that the MMSE denoiser is applied without any additional approximation or clipping; if the actual noise deviates from this model or if the denoiser is replaced by a learned network that only approximates the MMSE operator, the weak-convexity and rate guarantees no longer hold directly (see abstract and the proximal-operator equivalence stated in the introduction).","pith_extraction_headline":"The MMSE denoiser under Gaussian noise corresponds to a 1-weakly convex regularizer given by an upper Moreau envelope of the negative log-marginal density, which yields the first sublinear convergence rates for plug-and-play proximal grad"},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2510.27211/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"06041956e969190d6742826d1ed79fa4dd6dd408be486674d710b036b3b67cbd"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}