{"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/phase-retrieval-using-alternating","title":"Phase Retrieval using Alternating Minimization","arxiv_id":"1306.0160","date":"2013-06-02","proceeding":"NeurIPS 2013 12","authors":["Praneeth Netrapalli","Prateek Jain","Sujay Sanghavi"],"abstract":"Phase retrieval problems involve solving linear equations, but with missing\nsign (or phase, for complex numbers) information. More than four decades after\nit was first proposed, the seminal error reduction algorithm of (Gerchberg and\nSaxton 1972) and (Fienup 1982) is still the popular choice for solving many\nvariants of this problem. The algorithm is based on alternating minimization;\ni.e. it alternates between estimating the missing phase information, and the\ncandidate solution. Despite its wide usage in practice, no global convergence\nguarantees for this algorithm are known. In this paper, we show that a\n(resampling) variant of this approach converges geometrically to the solution\nof one such problem -- finding a vector $\\mathbf{x}$ from\n$\\mathbf{y},\\mathbf{A}$, where $\\mathbf{y} =\n\\left|\\mathbf{A}^{\\top}\\mathbf{x}\\right|$ and $|\\mathbf{z}|$ denotes a vector\nof element-wise magnitudes of $\\mathbf{z}$ -- under the assumption that\n$\\mathbf{A}$ is Gaussian.\n  Empirically, we demonstrate that alternating minimization performs similar to\nrecently proposed convex techniques for this problem (which are based on\n\"lifting\" to a convex matrix problem) in sample complexity and robustness to\nnoise. However, it is much more efficient and can scale to large problems.\nAnalytically, for a resampling version of alternating minimization, we show\ngeometric convergence to the solution, and sample complexity that is off by log\nfactors from obvious lower bounds. We also establish close to optimal scaling\nfor the case when the unknown vector is sparse. Our work represents the first\ntheoretical guarantee for alternating minimization (albeit with resampling) for\nany variant of phase retrieval problems in the non-convex setting.","url_abs":"http://arxiv.org/abs/1306.0160v2","url_pdf":"http://arxiv.org/pdf/1306.0160v2.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":"phase-retrieval-using-alternating","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=1306.0160","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}