{"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/construction-of-optimal-spectral-methods-in","title":"Construction of optimal spectral methods in phase retrieval","arxiv_id":"2012.04524","date":"2020-12-08","proceeding":null,"authors":["Antoine Maillard","Florent Krzakala","Yue M. Lu","Lenka Zdeborová"],"abstract":"We consider the phase retrieval problem, in which the observer wishes to recover a $n$-dimensional real or complex signal $\\mathbf{X}^\\star$ from the (possibly noisy) observation of $|\\mathbf{\\Phi} \\mathbf{X}^\\star|$, in which $\\mathbf{\\Phi}$ is a matrix of size $m \\times n$. We consider a \\emph{high-dimensional} setting where $n,m \\to \\infty$ with $m/n = \\mathcal{O}(1)$, and a large class of (possibly correlated) random matrices $\\mathbf{\\Phi}$ and observation channels. Spectral methods are a powerful tool to obtain approximate observations of the signal $\\mathbf{X}^\\star$ which can be then used as initialization for a subsequent algorithm, at a low computational cost. In this paper, we extend and unify previous results and approaches on spectral methods for the phase retrieval problem. More precisely, we combine the linearization of message-passing algorithms and the analysis of the \\emph{Bethe Hessian}, a classical tool of statistical physics. Using this toolbox, we show how to derive optimal spectral methods for arbitrary channel noise and right-unitarily invariant matrix $\\mathbf{\\Phi}$, in an automated manner (i.e. with no optimization over any hyperparameter or preprocessing function).","url_abs":"https://arxiv.org/abs/2012.04524v3","url_pdf":"https://arxiv.org/pdf/2012.04524v3.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":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"construction-of-optimal-spectral-methods-in","repo_url":"https://github.com/AnMaillard/Optimal_Spectral_Methods_PR","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2012.04524","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2012.04524"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-24T18:15:14+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. Samples come from repositories linked to the paper, official or community; repo_kind says which.","repos":[{"provenance":"external:paperswithcode_snapshot_2025-07-28","url":"https://github.com/AnMaillard/Optimal_Spectral_Methods_PR","reach":null}],"summary":{"ran_draft_wrong":1},"by_repo_kind":{"official":{"samples":1,"ran":1,"repositories":1}},"repo_kind_vocabulary":{"official":"The archive marks this repository official for the paper","named_in_paper":"The archive records that the paper mentions this repository; it is not marked official","listed":"In the archive's code links for this paper, not marked official and not recorded as mentioned in the paper","found_in_text":"Syntology found this repository in the paper's own text; whether it is the authors' implementation is not asserted","community":"Not in the archive's code links for this paper; a community repository Syntology harvested"},"n_pointer_only_for_licence":1,"samples":[{"code_sha256_prefix":"75f9a00d5a1af5e0","entry":"power_method","repo":"AnMaillard/Optimal_Spectral_Methods_PR","repo_kind":"official","path":"Class/Spectral.py","file_url":"https://github.com/AnMaillard/Optimal_Spectral_Methods_PR/blob/HEAD/Class/Spectral.py","link_basis":"first_harvest_node","language":"python","status":"ran_draft_wrong","verification_level":1,"contract_check":"MISDECLARED","metamorphic_tier":"deterministic","behaviour_fingerprint":true,"licence":"NONE","inline_ok":false,"mcp_get_code":{"code_sha256":"75f9a00d5a1af5e0"}}]},"arxiv_metadata":null,"syntology_extracted_results":null}