{"paper":{"title":"Solving Systems of Quadratic Equations via Exponential-type Gradient Descent Algorithm","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT"],"primary_cat":"math.NA","authors_text":"Meng Huang, Zhiqiang Xu","submitted_at":"2018-06-04T00:37:29Z","abstract_excerpt":"We consider the rank minimization problem from quadratic measurements, i.e., recovering a rank $r$ matrix $X \\in \\mathbb{R}^{n \\times r}$ from $m$ scalar measurements $y_i=a_i^{\\top} XX^{\\top} a_i,\\;a_i\\in \\mathbb{R}^n,\\;i=1,\\ldots,m$. Such problem arises in a variety of applications such as quadratic regression and quantum state tomography. We present a novel algorithm, which is termed exponential-type gradient descent algorithm, to minimize a non-convex objective function $f(U)=\\frac{1}{4m}\\sum_{i=1}^m(y_i-a_i^{\\top} UU^{\\top} a_i)^2$. This algorithm starts with a careful initialization, and"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.00904","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"}