{"paper":{"title":"Elastic-net regularization versus $\\ell^1$-regularization for linear inverse problems with quasi-sparse solutions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Bernd Hofmann, De-Han Chen, Jun Zou","submitted_at":"2016-04-12T12:19:57Z","abstract_excerpt":"We consider the ill-posed operator equation $Ax=y$ with an injective and bounded linear operator $A$ mapping between $\\ell^2$ and a Hilbert space $Y$, possessing the unique solution \\linebreak $x^\\dag=\\{x^\\dag_k\\}_{k=1}^\\infty$. For the cases that sparsity $x^\\dag \\in \\ell^0$ is expected but often slightly violated in practice, we investigate in comparison with the $\\ell^1$-regularization the elastic-net regularization, where the penalty is a weighted superposition of the $\\ell^1$-norm and the $\\ell^2$-norm square, under the assumption that $x^\\dag \\in \\ell^1$. There occur two positive paramet"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.03364","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"}