{"paper":{"title":"Complete Dictionary Recovery over the Sphere","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CV","cs.LG","math.IT","math.OC","stat.ML"],"primary_cat":"cs.IT","authors_text":"John Wright, Ju Sun, Qing Qu","submitted_at":"2015-04-26T04:57:19Z","abstract_excerpt":"We consider the problem of recovering a complete (i.e., square and invertible) matrix $\\mathbf A_0$, from $\\mathbf Y \\in \\mathbb R^{n \\times p}$ with $\\mathbf Y = \\mathbf A_0 \\mathbf X_0$, provided $\\mathbf X_0$ is sufficiently sparse. This recovery problem is central to the theoretical understanding of dictionary learning, which seeks a sparse representation for a collection of input signals, and finds numerous applications in modern signal processing and machine learning. We give the first efficient algorithm that provably recovers $\\mathbf A_0$ when $\\mathbf X_0$ has $O(n)$ nonzeros per col"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.06785","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}