{"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-in-high-dimensions","title":"Phase retrieval in high dimensions: Statistical and computational phase transitions","arxiv_id":"2006.05228","date":"2020-06-09","proceeding":"NeurIPS 2020 12","authors":["Antoine Maillard","Bruno Loureiro","Florent Krzakala","Lenka Zdeborová"],"abstract":"We consider the phase retrieval problem of reconstructing a $n$-dimensional real or complex signal $\\mathbf{X}^{\\star}$ from $m$ (possibly noisy) observations $Y_\\mu = | \\sum_{i=1}^n \\Phi_{\\mu i} X^{\\star}_i/\\sqrt{n}|$, for a large class of correlated real and complex random sensing matrices $\\mathbf{\\Phi}$, in a high-dimensional setting where $m,n\\to\\infty$ while $\\alpha = m/n=\\Theta(1)$. First, we derive sharp asymptotics for the lowest possible estimation error achievable statistically and we unveil the existence of sharp phase transitions for the weak- and full-recovery thresholds as a function of the singular values of the matrix $\\mathbf{\\Phi}$. This is achieved by providing a rigorous proof of a result first obtained by the replica method from statistical mechanics. In particular, the information-theoretic transition to perfect recovery for full-rank matrices appears at $\\alpha=1$ (real case) and $\\alpha=2$ (complex case). Secondly, we analyze the performance of the best-known polynomial time algorithm for this problem -- approximate message-passing -- establishing the existence of a statistical-to-algorithmic gap depending, again, on the spectral properties of $\\mathbf{\\Phi}$. Our work provides an extensive classification of the statistical and algorithmic thresholds in high-dimensional phase retrieval for a broad class of random matrices.","url_abs":"https://arxiv.org/abs/2006.05228v2","url_pdf":"https://arxiv.org/pdf/2006.05228v2.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-in-high-dimensions","repo_url":"https://github.com/sphinxteam/PhaseRetrieval_demo","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}}],"tasks":[{"task_slug":"retrieval","task_name":"Retrieval"},{"task_slug":"high","task_name":"Vocal Bursts Intensity Prediction"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2006.05228","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}