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. Index Terms-Compressed sensing, Dantzig selector, 1 minimization, Gross' golfing scheme, LASSO, operator Bernstein inequalities, random matrices, sparse regression, (weak) restricted isometries.
One of the seminal manuscripts about the theory of CS was introduced in REF , in which the CS mechanism selects sensing vectors independently from a determined probability distribution without the requirement of the restricted isometry property and a given random model for the signal.
A Probabilistic and RIPless Theory of Compressed Sensing
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Mathematics Computer Science