{"paper":{"title":"State-Dependent Lyapunov Analysis of Rank-1 Matrix Factorization","license":"http://creativecommons.org/licenses/by/4.0/","headline":"A state-dependent Lyapunov method with quadratic certificates proves global convergence for gradient descent on rank-1 matrix factorization by deriving the certificates from structural axioms rather than ad hoc constructions.","cross_cats":["cs.LG","cs.NA","math.OC"],"primary_cat":"math.NA","authors_text":"Jaehong Moon","submitted_at":"2026-04-28T22:43:16Z","abstract_excerpt":"We study gradient descent for rank-1 matrix factorization through a state-dependent Lyapunov perspective. The central object is a parameterized quadratic certificate $I(\\delta;\\,\\cdot)$ whose boundary-inward property induces a monotone state parameter $\\delta_t$, thereby certifying that the trajectory is confined to a shrinking family of level sets. For certified initializations below the critical step size, this mechanism proves convergence to global minimizers. Above the critical step size, the same monotone-state mechanism instead leads to a balanced terminal regime; for a range of post-cri"},"claims":{"count":3,"items":[{"kind":"strongest_claim","text":"In the certified regime, this mechanism yields convergence to a global minimizer; in the post-critical regime, it forces trajectories toward a terminal balanced manifold. The certificates arise from the monotonicity structure of the dynamics, rather than from ad hoc algebraic constructions.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The structural axioms of the state-dependent Lyapunov framework hold, allowing the scalar certificate to be uniquely determined by local Lagrange analysis that constrains the signal and noise blocks of rank-1 extensions.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A state-dependent Lyapunov method with quadratic certificates proves global convergence for gradient descent on rank-1 matrix factorization by deriving the certificates from structural axioms rather than ad hoc constructions.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"}],"snapshot_sha256":"a944b4e2be6a516672618c982deced613488ab5c25f4535938c6ef3062dba482"},"source":{"id":"2604.26993","kind":"arxiv","version":2},"verdict":{"id":"eef1b4e6-3c1d-4bc2-bafb-0039b7f936c5","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-07T12:20:52.236509Z","strongest_claim":"In the certified regime, this mechanism yields convergence to a global minimizer; in the post-critical regime, it forces trajectories toward a terminal balanced manifold. The certificates arise from the monotonicity structure of the dynamics, rather than from ad hoc algebraic constructions.","one_line_summary":"A state-dependent Lyapunov method with quadratic certificates proves global convergence for gradient descent on rank-1 matrix factorization by deriving the certificates from structural axioms rather than ad hoc constructions.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The structural axioms of the state-dependent Lyapunov framework hold, allowing the scalar certificate to be uniquely determined by local Lagrange analysis that constrains the signal and noise blocks of rank-1 extensions.","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.26993/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T03:33:55.293925Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"f094515197abc241b918e451fa88002d9d9a4b04cf1c954956d63e04759132b6"},"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"}