{"paper":{"title":"Optimal Rates of Convergence for Noisy Sparse Phase Retrieval via Thresholded Wirtinger Flow","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT","math.NA","stat.ML","stat.TH"],"primary_cat":"math.ST","authors_text":"T. Tony Cai, Xiaodong Li, Zongming Ma","submitted_at":"2015-06-10T16:32:33Z","abstract_excerpt":"This paper considers the noisy sparse phase retrieval problem: recovering a sparse signal $x \\in \\mathbb{R}^p$ from noisy quadratic measurements $y_j = (a_j' x )^2 + \\epsilon_j$, $j=1, \\ldots, m$, with independent sub-exponential noise $\\epsilon_j$. The goals are to understand the effect of the sparsity of $x$ on the estimation precision and to construct a computationally feasible estimator to achieve the optimal rates. Inspired by the Wirtinger Flow [12] proposed for noiseless and non-sparse phase retrieval, a novel thresholded gradient descent algorithm is proposed and it is shown to adaptiv"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.03382","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"}