{"paper":{"title":"Revisiting CUR Perturbation Analysis: A Local Tangent-Space Expansion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The Fréchet derivative of the rank-truncated CUR map is a sampling-induced oblique tangent-space projector that filters certain perturbations to first order.","cross_cats":["cs.IT","cs.NA","math.IT"],"primary_cat":"math.NA","authors_text":"Longxiu Huang","submitted_at":"2026-05-13T12:33:17Z","abstract_excerpt":"CUR decompositions approximate a matrix using selected columns, rows, and their intersection. Classical CUR theory provides exactness results for low-rank matrices and perturbation bounds controlled by the size of the noise. In this work we develop a local perturbation expansion for a fixed-index rank-truncated CUR map near an admissible rank-\\(r\\) matrix. We show that the Fr\\'echet derivative of the rank-truncated CUR map is a sampling-induced oblique tangent-space projector determined by the selected rows and columns. Consequently, the local recovery error for an underlying low-rank matrix i"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We show that the Fréchet derivative of the rank-truncated CUR map is a sampling-induced oblique tangent-space projector determined by the selected rows and columns. Consequently, the local recovery error for an underlying low-rank matrix is governed not by the full perturbation norm alone, but by the image of the perturbation under this sampling-induced tangent projector.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The underlying matrix lies near an admissible rank-r matrix with fixed selected indices, so that the rank-truncated CUR map is Fréchet differentiable at that point.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The Fréchet derivative of rank-truncated CUR is a sampling-induced oblique tangent projector, so perturbations in its kernel are removed to first order.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The Fréchet derivative of the rank-truncated CUR map is a sampling-induced oblique tangent-space projector that filters certain perturbations to first order.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"3744c29d62b1201767e0c4a277a4bd57430005d9b90edb228abc70215443335a"},"source":{"id":"2605.13437","kind":"arxiv","version":1},"verdict":{"id":"6c19e3d1-f192-4927-ba4e-4b9158d9b7e3","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:24:58.481060Z","strongest_claim":"We show that the Fréchet derivative of the rank-truncated CUR map is a sampling-induced oblique tangent-space projector determined by the selected rows and columns. Consequently, the local recovery error for an underlying low-rank matrix is governed not by the full perturbation norm alone, but by the image of the perturbation under this sampling-induced tangent projector.","one_line_summary":"The Fréchet derivative of rank-truncated CUR is a sampling-induced oblique tangent projector, so perturbations in its kernel are removed to first order.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The underlying matrix lies near an admissible rank-r matrix with fixed selected indices, so that the rank-truncated CUR map is Fréchet differentiable at that point.","pith_extraction_headline":"The Fréchet derivative of the rank-truncated CUR map is a sampling-induced oblique tangent-space projector that filters certain perturbations to first order."},"references":{"count":40,"sample":[{"doi":"","year":null,"title":"Journal of Machine Learning Research , volume=","work_id":"df288cf8-4648-4ecd-a52d-31dff1b3558e","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Proceedings of the 12th ACM SIGKDD international conference on Knowledge discovery and data mining , pages=","work_id":"9bd6a3a4-db40-4d70-befb-b256b9f20666","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Linear Algebra and its Applications , volume =","work_id":"cfbdfae1-e3fc-4237-a24f-75af64401853","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Mode-wise tensor decompositions: Multi-dimensional generalizations of","work_id":"3e567060-9217-474d-aea0-0219e93f042e","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2010,"title":"Linear Algebra and its Applications , volume=","work_id":"41433083-b1b9-4423-87fa-97a3ef35c189","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":40,"snapshot_sha256":"6817796a993fb0b2a68302027c648842f691a1fbf270c57fe0a0018370b0f711","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"}