{"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/signal-recovery-from-a-few-linear","title":"Signal recovery from a few linear measurements of its high-order spectra","arxiv_id":"2103.01551","date":"2021-03-02","proceeding":null,"authors":["Tamir Bendory","Dan Edidin","Shay Kreymer"],"abstract":"The $q$-th order spectrum is a polynomial of degree $q$ in the entries of a signal $x\\in\\mathbb{C}^N$, which is invariant under circular shifts of the signal. For $q\\geq 3$, this polynomial determines the signal uniquely, up to a circular shift, and is called a high-order spectrum. The high-order spectra, and in particular the bispectrum ($q=3$) and the trispectrum ($q=4$), play a prominent role in various statistical signal processing and imaging applications, such as phase retrieval and single-particle reconstruction. However, the dimension of the $q$-th order spectrum is $N^{q-1}$, far exceeding the dimension of $x$, leading to increased computational load and storage requirements. In this work, we show that it is unnecessary to store and process the full high-order spectra: a signal can be characterized uniquely, up to symmetries, from only $N+1$ linear measurements of its high-order spectra. The proof relies on tools from algebraic geometry and is corroborated by numerical experiments.","url_abs":"https://arxiv.org/abs/2103.01551v2","url_pdf":"https://arxiv.org/pdf/2103.01551v2.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":"signal-recovery-from-a-few-linear","repo_url":"https://github.com/krshay/recovery-high-order-spectra","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}