A Probabilistic and RIPless Theory of Compressed Sensing

作者：Candes Emmanuel J^*; Plan Yaniv

来源：IEEE Transactions on Information Theory, 2011, 57(11): 7235-7254.

DOI：10.1109/TIT.2011.2161794

• Compressed sensing
• Dantzig selector
• l(1) minimization
• Gross' golfing scheme
• LASSO
• operator Bernstein inequalities
• random matrices
• sparse regression
• (weak) restricted isometries

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.

• 出版日期2011-11