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  • Richard Baraniuk, Mark Davenport, Ronald Devore, Michael Wakin, Emmanuel J R Candès, M Baraniuk +5 others
  • 2007
We give a simple technique for verifying the Restricted Isometry Property (as introduced by Candès and Tao) for random matrices that underlies Compressed Sensing. Our approach has two main ingredients: (i) concentration inequalities for random inner products that have recently provided algorithmically simple proofs of the Johnson–Lindenstrauss lemma; and(More)
Recent theoretical developments in the area of compressive sensing (CS) have the potential to significantly extend the capabilities of digital data acquisition systems such as analog-to-digital converters and digital imagers in certain applications. To date, most of the CS literature has been devoted to studying the recovery of sparse signals from a small(More)
In this paper we develop a theory of matrix completion for the extreme case of noisy 1-bit observations. Instead of observing a subset of the real-valued entries of a matrix M , we obtain a small number of binary (1-bit) measurements generated according to a probability distribution determined by the real-valued entries of M. The central question we ask is(More)
—The recently introduced theory of compressive sensing enables the recovery of sparse or compressible signals from a small set of nonadaptive, linear measurements. If properly chosen, the number of measurements can be much smaller than the number of Nyquist-rate samples. Interestingly, it has been shown that random projections are a near-optimal measurement(More)
Orthogonal Matching Pursuit (OMP) is the canonical greedy algorithm for sparse approximation. In this paper we demonstrate that the restricted isometry property (RIP) can be used for a very straightforward analysis of OMP. Our main conclusion is that the RIP of order K + 1 (with isometry constant δ < 1 3 √ K) is sufficient for OMP to exactly recover any(More)
The recently introduced theory of Compressed Sensing (CS) enables the reconstruction or approximation of sparse or compressible signals from a small set of incoherent projections; often the number of projections can be much smaller than the number of Nyquist rate samples. In this paper, we show that the CS framework is information scalable to a wide range(More)
—Compressive sensing provides a framework for recovering sparse signals of length N from M N measurements. If the measurements contain noise bounded by , then standard algorithms recover sparse signals with error at most CC. However, these algorithms perform suboptimally when the measurement noise is also sparse. This can occur in practice due to shot(More)
We show how two fundamental results in analysis related to n-widths and Compressed Sensing are intimately related to the Johnson-Lindenstrauss lemma. Our elementary approach is based on the same concentration inequalities for random inner products that have recently provided simple proofs of the Johnson-Lindenstrauss lemma. We show how these ideas lead to(More)
Suppose we can sequentially acquire arbitrary linear measurements of an n-dimensional vector x resulting in the linear model y = Ax + z, where z represents measurement noise. If the signal is known to be sparse, one would expect the following folk theorem to be true: choosing an adaptive strategy which cleverly selects the next row of A based on what has(More)