{"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/sample-efficient-algorithms-for-recovering","title":"Sample-Efficient Algorithms for Recovering Structured Signals from Magnitude-Only Measurements","arxiv_id":"1705.06412","date":"2017-05-18","proceeding":null,"authors":["Gauri Jagatap","Chinmay Hegde"],"abstract":"We consider the problem of recovering a signal $\\mathbf{x}^* \\in\n\\mathbf{R}^n$, from magnitude-only measurements $y_i =\n|\\left\\langle\\mathbf{a}_i,\\mathbf{x}^*\\right\\rangle|$ for $i=[m]$. Also called\nthe phase retrieval, this is a fundamental challenge in bio-,astronomical\nimaging and speech processing. The problem above is ill-posed; additional\nassumptions on the signal and/or the measurements are necessary. In this paper\nwe first study the case where the signal $\\mathbf{x}^*$ is $s$-sparse. We\ndevelop a novel algorithm that we call Compressive Phase Retrieval with\nAlternating Minimization, or CoPRAM. Our algorithm is simple; it combines the\nclassical alternating minimization approach for phase retrieval with the CoSaMP\nalgorithm for sparse recovery. Despite its simplicity, we prove that CoPRAM\nachieves a sample complexity of $O(s^2\\log n)$ with Gaussian measurements\n$\\mathbf{a}_i$, matching the best known existing results; moreover, it\ndemonstrates linear convergence in theory and practice. Additionally, it\nrequires no extra tuning parameters other than signal sparsity $s$ and is\nrobust to noise. When the sorted coefficients of the sparse signal exhibit a\npower law decay, we show that CoPRAM achieves a sample complexity of $O(s\\log\nn)$, which is close to the information-theoretic limit. We also consider the\ncase where the signal $\\mathbf{x}^*$ arises from structured sparsity models. We\nspecifically examine the case of block-sparse signals with uniform block size\nof $b$ and block sparsity $k=s/b$. For this problem, we design a recovery\nalgorithm Block CoPRAM that further reduces the sample complexity to $O(ks\\log\nn)$. For sufficiently large block lengths of $b=\\Theta(s)$, this bound equates\nto $O(s\\log n)$. To our knowledge, this constitutes the first end-to-end\nalgorithm for phase retrieval where the Gaussian sample complexity has a\nsub-quadratic dependence on the signal sparsity level.","url_abs":"http://arxiv.org/abs/1705.06412v2","url_pdf":"http://arxiv.org/pdf/1705.06412v2.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":"sample-efficient-algorithms-for-recovering","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":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}