{"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/reshaped-wirtinger-flow-and-incremental","title":"Reshaped Wirtinger Flow and Incremental Algorithm for Solving Quadratic System of Equations","arxiv_id":"1605.07719","date":"2016-05-25","proceeding":null,"authors":["Huishuai Zhang","Yi Zhou","Yingbin Liang","Yuejie Chi"],"abstract":"We study the phase retrieval problem, which solves quadratic system of\nequations, i.e., recovers a vector $\\boldsymbol{x}\\in \\mathbb{R}^n$ from its\nmagnitude measurements $y_i=|\\langle \\boldsymbol{a}_i, \\boldsymbol{x}\\rangle|,\ni=1,..., m$. We develop a gradient-like algorithm (referred to as RWF\nrepresenting reshaped Wirtinger flow) by minimizing a nonconvex nonsmooth loss\nfunction. In comparison with existing nonconvex Wirtinger flow (WF) algorithm\n\\cite{candes2015phase}, although the loss function becomes nonsmooth, it\ninvolves only the second power of variable and hence reduces the complexity. We\nshow that for random Gaussian measurements, RWF enjoys geometric convergence to\na global optimal point as long as the number $m$ of measurements is on the\norder of $n$, the dimension of the unknown $\\boldsymbol{x}$. This improves the\nsample complexity of WF, and achieves the same sample complexity as truncated\nWirtinger flow (TWF) \\cite{chen2015solving}, but without truncation in gradient\nloop. Furthermore, RWF costs less computationally than WF, and runs faster\nnumerically than both WF and TWF. We further develop the incremental\n(stochastic) reshaped Wirtinger flow (IRWF) and show that IRWF converges\nlinearly to the true signal. We further establish performance guarantee of an\nexisting Kaczmarz method for the phase retrieval problem based on its\nconnection to IRWF. We also empirically demonstrate that IRWF outperforms\nexisting ITWF algorithm (stochastic version of TWF) as well as other batch\nalgorithms.","url_abs":"http://arxiv.org/abs/1605.07719v2","url_pdf":"http://arxiv.org/pdf/1605.07719v2.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":"reshaped-wirtinger-flow-and-incremental","repo_url":"https://github.com/soominkwon/Reshaped-Wirtinger-Flow","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[{"task_slug":"retrieval","task_name":"Retrieval"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}