{"paper":{"title":"Partial $\\ell_1$ optimization in random linear systems -- phase transitions and large deviations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT","math.PR"],"primary_cat":"math.OC","authors_text":"Mihailo Stojnic","submitted_at":"2016-12-22T04:05:02Z","abstract_excerpt":"$\\ell_1$ optimization is a well known heuristic often employed for solving various forms of sparse linear problems. In this paper we look at its a variant that we refer to as the \\emph{partial} $\\ell_1$ and discuss its mathematical properties when used for solving linear under-determined systems of equations. We will focus on large random systems and discuss the phase transition (PT) phenomena and how they connect to the large deviation principles (LDP). Using a variety of probabilistic and geometric techniques that we have developed in recent years we will first present general guidelines tha"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.07435","kind":"arxiv","version":1},"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"}