{"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/a-geometric-analysis-of-phase-retrieval","title":"A Geometric Analysis of Phase Retrieval","arxiv_id":"1602.06664","date":"2016-02-22","proceeding":null,"authors":["Ju Sun","Qing Qu","John Wright"],"abstract":"Can we recover a complex signal from its Fourier magnitudes? More generally,\ngiven a set of $m$ measurements, $y_k = |\\mathbf a_k^* \\mathbf x|$ for $k = 1,\n\\dots, m$, is it possible to recover $\\mathbf x \\in \\mathbb{C}^n$ (i.e.,\nlength-$n$ complex vector)? This **generalized phase retrieval** (GPR) problem\nis a fundamental task in various disciplines, and has been the subject of much\nrecent investigation. Natural nonconvex heuristics often work remarkably well\nfor GPR in practice, but lack clear theoretical explanations. In this paper, we\ntake a step towards bridging this gap. We prove that when the measurement\nvectors $\\mathbf a_k$'s are generic (i.i.d. complex Gaussian) and the number of\nmeasurements is large enough ($m \\ge C n \\log^3 n$), with high probability, a\nnatural least-squares formulation for GPR has the following benign geometric\nstructure: (1) there are no spurious local minimizers, and all global\nminimizers are equal to the target signal $\\mathbf x$, up to a global phase;\nand (2) the objective function has a negative curvature around each saddle\npoint. This structure allows a number of iterative optimization methods to\nefficiently find a global minimizer, without special initialization. To\ncorroborate the claim, we describe and analyze a second-order trust-region\nalgorithm.","url_abs":"http://arxiv.org/abs/1602.06664v3","url_pdf":"http://arxiv.org/pdf/1602.06664v3.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":"a-geometric-analysis-of-phase-retrieval","repo_url":"https://github.com/sunju/pr_plain","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}}],"tasks":[{"task_slug":"gpr","task_name":"GPR"},{"task_slug":"retrieval","task_name":"Retrieval"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1602.06664","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}