A Probabilistic and RIPless Theory of Compressed Sensing

• Published in: IEEE transactions on information theory Vol. 57; no. 11; pp. 7235 - 7254
• Main Authors: Candes, E. J., Plan, Y.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.11.2011; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: (weak) restricted isometries; Access methods and protocols, osi model; Applied sciences; Coherence; Compressed sensing; Convolution; Dantzig selector; Detection; Detection, estimation, filtering, equalization, prediction; Discrete Fourier transforms; ell_1 minimization; Exact sciences and technology; Fourier analysis; Gaussian; Gross' golfing scheme; Incoherence; Information theory; Information, signal and communications theory; LASSO; Mathematical analysis; Measurement; Noise; operator Bernstein inequalities; Probability distribution; random matrices; Sampling, quantization; Signal and communications theory; Signal, noise; sparse regression; Strategy; Telecommunications; Telecommunications and information theory; Teleprocessing networks. Isdn; Upper bound; Vectors (mathematics); operator Bernstein inequalities; Isometry; Gross' golfing scheme; Probabilistic approach; random matrices; Transmission protocol; sparse regression; Probability distribution; Frequency measurement; Dantzig selector; Bernstein polynomial; (weak) restricted isometries; Isotropy; ℓ; minimization; LASSO; Signal processing; Routing protocols; Signal reconstruction; Compressed sensing
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2011.2161794

This paper introduces a simple and very general theory of compressive sensing. In this theory, the sensing mechanism simply selects sensing vectors independently at random from a probability distribution F ; it includes all standard models-e.g., Gaussian, frequency measurements-discussed in the literature, but also provides a framework for new measurement strategies as well. We prove that if the probability distribution F obeys a simple incoherence property and an isotropy property, one can faithfully recover approximately sparse signals from a minimal number of noisy measurements. The novelty is that our recovery results do not require the restricted isometry property (RIP) to hold near the sparsity level in question, nor a random model for the signal. As an example, the paper shows that a signal with s nonzero entries can be faithfully recovered from about s log n Fourier coefficients that are contaminated with noise.