Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information

Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information Emmanuel J. Candès Justin Romberg Terence Tao

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Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information 2006 IEEE Transactions on Information Theory / arXiv 10.1109/TIT.2005.862083 math/0409186 candes2006-robust-uncertainty candes2006-robust-uncertainty.txt, the project's staged copy 7cf556ff0a002f792fc187c72b7e0d28b746662e arxiv-open information-theory

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0 spans located by searching the stored text for the quote's exact words, allowing any whitespace between them method Quotation withheld because this record is not cleared for redistribution. exact signal reconstruction from incomplete Fourier samples Quotation withheld because this record is not cleared for redistribution. mathematical-proof mathematical-proof Quotation withheld because this record is not cleared for redistribution. mechanisms compressed sensing: a sparse signal can be reconstructed exactly from far fewer measurements than the Nyquist rate requires, by l1-norm (convex) minimization Quotation withheld because this record is not cleared for redistribution. exact recovery holds only under a sparsity condition on the signal what this limits: every exact-recovery guarantee: it holds PROVIDED THAT f IS SPARSE Quotation withheld because this record is not cleared for redistribution. The sufficient condition proved is not sharp; the numerical experiments only show it is within a constant factor of the achievable bound. what this limits: the sharpness of the sufficient condition: OUR METHODS DO NOT QUITE ATTAIN THIS THEORETICAL UPPER BOUND FOR CORRECT RECOVERY, [and] OUR NUMERICAL EXPERIMENTS SUGGEST THAT THEY DO COME WITHIN A CONSTANT FACTOR - the gap is measured, not proved Quotation withheld because this record is not cleared for redistribution. The particular dual polynomial construction used in the proof is a known source of looseness. what this limits: the tightness of the analysis, with the responsible step named: ONE SOURCE OF SLACK IN THE THEORETICAL ANALYSIS IS THE WAY IN WHICH WE CHOOSE THE POLYNOMIAL P(t) Quotation withheld because this record is not cleared for redistribution. Robustness to measurement error is not established; under perturbation exact recovery no longer holds and only preliminary calculations are offered. what this limits: the noiseless case only: under measurement perturbation, ONE CANNOT EXPECT EXACT RECOVERY Quotation withheld because this record is not cleared for redistribution. The deterministic uniqueness theorem depends on the signal length being prime. what this limits: the deterministic uniqueness theorem: IF N IS NOT PRIME, THE LEMMA (AND HENCE THE THEOREM) FAILS, ESSENTIALLY BECAUSE OF THE PRESENCE OF NON-TRIVIAL SUBGROUPS Quotation withheld because this record is not cleared for redistribution. The robust uncertainty principle underlying the analysis is stated but not developed here. what this limits: the robust uncertainty principle as used here: BECAUSE OF SPACE LIMITATION, WE ARE UNABLE TO BELABORATE ON THIS FACT AND ITS IMPLICATIONS ANY FURTHER Quotation withheld because this record is not cleared for redistribution. The guarantee does not survive measurement error, and the paper says so before anyone else can: under a small perturbation 'one cannot expect exact recovery', and the question becomes how far the minimiser is from the true object rather than whether it equals it. Quotation withheld because this record is not cleared for redistribution. recovering a sparse signal from few frequency samples by solving a convex l1-minimization (linear-programming) problem Quotation withheld because this record is not cleared for redistribution. exact recovery exact recovery Quotation withheld because this record is not cleared for redistribution. sparsity sparsity Quotation withheld because this record is not cleared for redistribution. PRIME is a property of an integer here, with no relation to a priming prime: the deterministic uniqueness theorem holds only when the signal length N is a prime number, because Z_N then has no non-trivial subgroups Quotation withheld because this record is not cleared for redistribution. exact recovery - a property of the NOISELESS problem only; under perturbation the paper says plainly that one cannot expect it, and asks instead how far the minimiser lies from the true object Quotation withheld because this record is not cleared for redistribution. builds-on → convex optimization / linear programming (mathematics) Quotation withheld because this record is not cleared for redistribution. builds-on → the literature on l1 minimisation for sparse representation in overcomplete dictionaries [7, 8, 9, 12, 14] Quotation withheld because this record is not cleared for redistribution. extends → sampling-pattern results on exact recovery from consecutive Fourier coefficients [10, 11, 21] Quotation withheld because this record is not cleared for redistribution. exact recovery —operationalised-as→ convex optimization exact recovery convex optimization strength of the relation: 0.9 · note: The paper's central move stated in one clause - '…recovery may be obtained by solving a CONVEX OPTIMIZATION…' - and immediately bounded: the condition on the support size is sharp EXCEPT FOR THE LOGARITHMIC FACTOR. The guarantee and the size of the remaining gap are given together. Audit 2026-09-28: DOUBTFUL - read against its full span and kept, but exact recovery 'may be obtained by solving' a convex optimization problem; operationalised-as is loose. Quotation withheld because this record is not cleared for redistribution. Exact reconstruction is possible by solving a convex optimization problem. Quotation withheld because this record is not cleared for redistribution. The recovery solves an optimization problem that finds the object of minimal complexity. Quotation withheld because this record is not cleared for redistribution. Fourier coefficients Fourier coefficients about this quote: QUOTE RECOVERED FROM THE PAPER'S OWN CHARACTERS. The recorded quote was LETTERS-ONLY IDENTICAL to this stretch of the staged text but not an exact substring of it - spaces, hyphens, ligature glyphs or mathematical operators differed, so it could never anchor. This is the file's own text between the same two letters: not produced by a rule about the damage, copied out of the file. Round-trip checked (the recovered span is letters-only equal to the original) and length-checked. The text as originally recorded is kept in evidence as extracted. Quotation withheld because this record is not cleared for redistribution. exact recovery exact recovery Quotation withheld because this record is not cleared for redistribution. ℓ1 minimization ℓ1 minimization Quotation withheld because this record is not cleared for redistribution. regression regression Quotation withheld because this record is not cleared for redistribution. sample-size sample-size Quotation withheld because this record is not cleared for redistribution. variance variance about this quote: QUOTE RECOVERED FROM THE PAPER'S OWN CHARACTERS. The recorded quote was LETTERS-ONLY IDENTICAL to this stretch of the staged text but not an exact substring of it - spaces, hyphens, ligature glyphs or mathematical operators differed, so it could never anchor. This is the file's own text between the same two letters: not produced by a rule about the damage, copied out of the file. Round-trip checked (the recovered span is letters-only equal to the original) and length-checked. The text as originally recorded is kept in evidence as extracted. Quotation withheld because this record is not cleared for redistribution.

The full text of this article is not redistributable, so it is not included here. The evidence sentences are withheld with it; the tag layer and its grounding status remain. The article itself is at https://doi.org/10.1109/TIT.2005.862083.