{"paper":{"title":"Stochastic Convergence Analysis for Large-Scale Linear Discrete Ill-posed Problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Derives stochastic error bounds and parameter-choice rules for weighted Tikhonov regularization of large-scale ill-posed problems with random noise under a polynomial eigenvalue assumption.","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Duan-Peng Ling, Wenlong Zhang","submitted_at":"2026-05-18T11:57:22Z","abstract_excerpt":"We study weighted Tikhonov regularization for large-scale linear discrete ill-posed problems with random noise. Under a polynomial upper-bound assumption on the generalized eigenvalues of the discrete forward operator, we derive stochastic error bounds for two noise models: expectation bounds for independent zero-mean bounded-variance noise, and high-probability bounds for independent sub-Gaussian noise. The analysis yields an a priori parameter-choice rule and suggests an adaptive strategy suitable for large-scale computation. Numerical experiments support the theory and show that the predict"},"claims":{"count":3,"items":[{"kind":"strongest_claim","text":"Under a polynomial upper-bound assumption on the generalized eigenvalues of the discrete forward operator, stochastic error bounds are derived for weighted Tikhonov regularization under independent zero-mean bounded-variance noise (expectation bounds) and independent sub-Gaussian noise (high-probability bounds), yielding an a priori parameter-choice rule and an adaptive strategy.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The polynomial upper-bound assumption on the generalized eigenvalues of the discrete forward operator (invoked to obtain the stochastic error bounds and parameter rule).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Derives stochastic error bounds and parameter-choice rules for weighted Tikhonov regularization of large-scale ill-posed problems with random noise under a polynomial eigenvalue assumption.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"}],"snapshot_sha256":"c06f6ec8de83d0583e193d9e60f29be9305284634fbd37d5b7d6102e7a0c8c93"},"source":{"id":"2605.18259","kind":"arxiv","version":1},"verdict":{"id":"2a5cd895-39b0-4c17-b82e-b56dab59d243","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-20T00:04:55.184704Z","strongest_claim":"Under a polynomial upper-bound assumption on the generalized eigenvalues of the discrete forward operator, stochastic error bounds are derived for weighted Tikhonov regularization under independent zero-mean bounded-variance noise (expectation bounds) and independent sub-Gaussian noise (high-probability bounds), yielding an a priori parameter-choice rule and an adaptive strategy.","one_line_summary":"Derives stochastic error bounds and parameter-choice rules for weighted Tikhonov regularization of large-scale ill-posed problems with random noise under a polynomial eigenvalue assumption.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The polynomial upper-bound assumption on the generalized eigenvalues of the discrete forward operator (invoked to obtain the stochastic error bounds and parameter rule).","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.18259/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-19T23:33:35.260162Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T23:21:58.977890Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"04a4e7a73a02401ecbe69ddf17867e574ee1f48563c250d83c52aa1db6da2fa6"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}