{"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/multi-reference-alignment-in-high-dimensions","title":"Multi-reference alignment in high dimensions: sample complexity and phase transition","arxiv_id":"2007.11482","date":"2020-07-22","proceeding":null,"authors":["Elad Romanov","Tamir Bendory","Or Ordentlich"],"abstract":"Multi-reference alignment entails estimating a signal in $\\mathbb{R}^L$ from its circularly-shifted and noisy copies. This problem has been studied thoroughly in recent years, focusing on the finite-dimensional setting (fixed $L$). Motivated by single-particle cryo-electron microscopy, we analyze the sample complexity of the problem in the high-dimensional regime $L\\to\\infty$. Our analysis uncovers a phase transition phenomenon governed by the parameter $\\alpha = L/(\\sigma^2\\log L)$, where $\\sigma^2$ is the variance of the noise. When $\\alpha>2$, the impact of the unknown circular shifts on the sample complexity is minor. Namely, the number of measurements required to achieve a desired accuracy $\\varepsilon$ approaches $\\sigma^2/\\varepsilon$ for small $\\varepsilon$; this is the sample complexity of estimating a signal in additive white Gaussian noise, which does not involve shifts. In sharp contrast, when $\\alpha\\leq 2$, the problem is significantly harder and the sample complexity grows substantially quicker with $\\sigma^2$.","url_abs":"https://arxiv.org/abs/2007.11482v3","url_pdf":"https://arxiv.org/pdf/2007.11482v3.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":"multi-reference-alignment-in-high-dimensions","repo_url":"https://github.com/TamirBendory/high-dimensional-mra-bounds","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[{"task_slug":"high","task_name":"Vocal Bursts Intensity Prediction"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}