{"paper":{"title":"Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information","license":"","headline":"","cross_cats":["cs.NA","math.CA"],"primary_cat":"math.NA","authors_text":"Emmanuel Candes, Justin Romberg, Terence Tao","submitted_at":"2004-09-10T18:45:22Z","abstract_excerpt":"This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discrete-time signal $f \\in \\C^N$ and a randomly chosen set of frequencies $\\Omega$ of mean size $\\tau N$. Is it possible to reconstruct $f$ from the partial knowledge of its Fourier coefficients on the set $\\Omega$?\n  A typical result of this paper is as follows: for each $M > 0$, suppose that $f$ obeys $$ # \\{t, f(t) \\neq 0 \\} \\le \\alpha(M) \\cdot (\\log N)^{-1} \\cdot # \\Omega, $$ then with probability at least $1-O(N^{-M})$, $f$ can be reconstructed exactly as the solution to the $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0409186","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0409186/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}