{"paper":{"title":"Necessary and sufficient conditions of solution uniqueness in $\\ell_1$ minimization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT","math.NA","math.OC"],"primary_cat":"cs.IT","authors_text":"Hui Zhang, Lizhi Cheng, Wotao Yin","submitted_at":"2012-09-04T14:10:09Z","abstract_excerpt":"This paper shows that the solutions to various convex $\\ell_1$ minimization problems are \\emph{unique} if and only if a common set of conditions are satisfied. This result applies broadly to the basis pursuit model, basis pursuit denoising model, Lasso model, as well as other $\\ell_1$ models that either minimize $f(Ax-b)$ or impose the constraint $f(Ax-b)\\leq\\sigma$, where $f$ is a strictly convex function. For these models, this paper proves that, given a solution $x^*$ and defining $I=\\supp(x^*)$ and $s=\\sign(x^*_I)$, $x^*$ is the unique solution if and only if $A_I$ has full column rank and"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.0652","kind":"arxiv","version":2},"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"}