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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)

- Mark A. Davenport, Petros Boufounos, Michael B. Wakin, Richard G. Baraniuk
- IEEE Journal of Selected Topics in Signal…
- 2010

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)

- Mark A. Davenport, Yaniv Plan, Ewout van den Berg, Mary Wootters
- ArXiv
- 2012

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)

- Mark A. Davenport, Michael B. Wakin
- IEEE Transactions on Information Theory
- 2010

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… (More)

- M. A. Davenport, M. F. Duarte, +5 authors Gitta Kutyniok

The consecutive numbering of the publications is determined by their chronological order. The aim of this preprint series is to make new research rapidly available for scientific discussion. Therefore, the responsibility for the contents is solely due to the authors. The publications will be distributed by the authors. In recent years, compressed sensing… (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)

The recently introduced theory of compressed sensing enables the reconstruction 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 satisfactory… (More)

- Mark A. Davenport, Marco F. Duarte, +4 authors Richard G. Baraniuk
- Computational Imaging
- 2007

The theory of compressive sensing (CS) enables the reconstruction of a sparse or compressible image or signal from a small set of linear, non-adaptive (even random) projections. However, in many applications, including object and target recognition, we are ultimately interested in making a decision about an image rather than computing a reconstruction. We… (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)