{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/optimal-rates-of-convergence-for-noisy-sparse","title":"Optimal Rates of Convergence for Noisy Sparse Phase Retrieval via Thresholded Wirtinger Flow","arxiv_id":"1506.03382","date":"2015-06-10","proceeding":null,"authors":["T. Tony Cai","Xiao-Dong Li","Zongming Ma"],"abstract":"This paper considers the noisy sparse phase retrieval problem: recovering a\nsparse signal $x \\in \\mathbb{R}^p$ from noisy quadratic measurements $y_j =\n(a_j' x )^2 + \\epsilon_j$, $j=1, \\ldots, m$, with independent sub-exponential\nnoise $\\epsilon_j$. The goals are to understand the effect of the sparsity of\n$x$ on the estimation precision and to construct a computationally feasible\nestimator to achieve the optimal rates. Inspired by the Wirtinger Flow [12]\nproposed for noiseless and non-sparse phase retrieval, a novel thresholded\ngradient descent algorithm is proposed and it is shown to adaptively achieve\nthe minimax optimal rates of convergence over a wide range of sparsity levels\nwhen the $a_j$'s are independent standard Gaussian random vectors, provided\nthat the sample size is sufficiently large compared to the sparsity of $x$.","url_abs":"http://arxiv.org/abs/1506.03382v1","url_pdf":"http://arxiv.org/pdf/1506.03382v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"optimal-rates-of-convergence-for-noisy-sparse","repo_url":"https://github.com/GauriJagatap/model-copram","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[{"task_slug":"retrieval","task_name":"Retrieval"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1506.03382","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}