New Bounds for Restricted Isometry Constants

In this paper, the high-dimensional sparse linear regression model is considered, where the overall number of variables is larger than the number of observations. We investigate the L 1 penalized least absolute deviation method. Different from most of other methods, the L 1 penalized LAD method does not need any knowledge of standard deviation of the noises or any moment assumptions of the noises. Our analysis shows that the method achieves near oracle performance, i.e. with large probability, the L 2 norm of the estimation error is of order O( √ k log p/n). The result is true for a wide range of noise distributions, even for the Cauchy distribution. Numerical results are also presented.

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“…can be replaced by a number of similar RIP conditions, see for example [8]. We keep it here just to simplify the arguments.…”

Section: Remark 2 From the Proof Of The Theorem We Can See That The mentioning

confidence: 99%

“…General constrained L 1 minimization methods for noiseless case and Gaussian noise were studied in [7]. More results about the constrained L 1 minimization can be found in for example [9], [12], [8] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

TheL1penalized LAD estimator for high dimensional linear regression

Wang

2013

Journal of Multivariate Analysis

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131

113

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“…The sufficient condition δ 2k < √ 2 − 1 = 0.414 for the exact recovery of arbitrary k-sparse signals via the 1 optimization has further been refined in subsequent studies: For example, Foucart and Lai have improved it to δ 2k < 0.4531 [15], and Cai et al have obtained δ 2k < 0.472 [16] and δ k < 0.307 [17].…”

Section: Restricted Isometry Property (Rip)mentioning

confidence: 99%

A User's Guide to Compressed Sensing for Communications Systems

Hayashi

Nagahara

Tanaka

2013

IEICE Trans. Commun.

200

109

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SUMMARYThis survey provides a brief introduction to compressed sensing as well as several major algorithms to solve it and its various applications to communications systems. We firstly review linear simultaneous equations as ill-posed inverse problems, since the idea of compressed sensing could be best understood in the context of the linear equations. Then, we consider the problem of compressed sensing as an underdetermined linear system with a prior information that the true solution is sparse, and explain the sparse signal recovery based on 1 optimization, which plays the central role in compressed sensing, with some intuitive explanations on the optimization problem. Moreover, we introduce some important properties of the sensing matrix in order to establish the guarantee of the exact recovery of sparse signals from the underdetermined system. After summarizing several major algorithms to obtain a sparse solution focusing on the 1 optimization and the greedy approaches, we introduce applications of compressed sensing to communications systems, such as wireless channel estimation, wireless sensor network, network tomography, cognitive radio, array signal processing, multiple access scheme, and networked control.

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“…It has been shown that if        , the signal  can be perfectly recovered from noise-free measurements [8] . More conditions on the RIC values can be found in [9][10][11]. The RIP is a sufficient and necessary condition which guarantees unique and exact reconstruction of sparse signal via   -minimization.…”

Section: ⅱ Compressive Sensing Problem and Motivationmentioning

confidence: 99%