Explicit measurements with almost optimal thresholds for compressed sensing

Parvaresh, Farzad; Hassibi, Babak

doi:10.1109/icassp.2008.4518494

Cited by 32 publications

(51 citation statements)

References 12 publications

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“…For instance, it is known that if x is exactly k-sparse then based on Reed-Solomon codes [11] one can efficiently reconstruct x with O(k) noiseless measurements (e.g. [12]) via algorithms with decoding time-complexity O(n log(n)), or via codes such as in [13], [14] with O(k) noiseless measurements with decoding time-complexity O(n). In the regime where k = θ(n), [15] show that O(k) = O(n) measurements suffice to reconstruct x. Noise/Approximate sparsity: If the length-n source vector is the sum of any exactly k-sparse vector x and a "random" source noise vector z (and possibly y = A(x + z) also has a "random" noise vector e added to it), then as long as the noise variances are not "too much larger" than the signal power, the work of [16] demonstrates that O(k) measurements suffice (though the algorithms require time exponential in n).…”

Section: Number Of Measurementsmentioning

confidence: 99%

SHO-FA: Robust compressive sensing with order-optimal complexity, measurements, and bits

Bakshi

Jaggi

Cai

et al. 2012

2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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Suppose x is any exactly k-sparse vector in n . We present a class of "sparse" matrices A, and a corresponding algorithm that we call SHO-FA (for Short and Fast 1 ) that, with high probability over A, can reconstruct x from Ax. The SHO-FA algorithm is related to the Invertible Bloom Lookup Tables (IBLTs) recently introduced by Goodrich et al., with two important distinctions -SHO-FA relies on linear measurements, and is robust to noise and approximate sparsity. The SHO-FA algorithm is the first to simultaneously have the following properties: (a) it requires only O(k) measurements, (b) the bitprecision of each measurement and each arithmetic operation is O (log(n) + P ) (here 2 −P corresponds to the desired relative error in the reconstruction of x), (c) the computational complexity of decoding is O(k) arithmetic operations, and (d) if the reconstruction goal is simply to recover a single component of x instead of all of x, with high probability over A this can be done in constant time. All constants above are independent of all problem parameters other than the desired probability of success. For a wide range of parameters these properties are information-theoretically order-optimal. In addition, our SHO-FA algorithm is robust to random noise, and (random) approximate sparsity for a large range of k. In particular, suppose the measured vector equals A(x + z) + e, where z and e correspond respectively to the source tail and measurement noise. Under reasonable statistical assumptions on z and e our decoding algorithm reconstructs x with an estimation error of O(||z||1 + (log k) 2 ||e||1). The SHO-FA algorithm works with high probability over A, z, and e, and still requires only O(k) steps and O(k) measurements over O(log(n))-bit numbers. This is in contrast to most existing algorithms which focus on the "worst-case" z model, where it is known Ω(k log(n/k)) measurements over O(log(n))-bit numbers are necessary.

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Section: Number Of Measurementsmentioning

confidence: 99%

SHO-FA: Robust compressive sensing with order-optimal complexity, measurements, and bits

Bakshi

Jaggi

Cai

et al. 2012

2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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“…(25) Proof 1 Follows by combing (20), (24), and recalling on the definition of γ introduced below (15). (1) with the null-space uniformly distributed in the Grassmanian.…”

Section: Casementioning

confidence: 99%

“…A particular way of solving (1) which will be the subject of this paper is l1-norm relaxation [5]. (More on different algorithms the interested reader can find in excellent references [1,4,20,18,19].) l1-norm relaxation proposes solving the following problem min x 1 subject to Ax = y.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we will be interested in the so-called blocksparse compressed sensing problems [25,20,14,2] (more on a related group of problems and various application the interested reader can find in [28,3,6,9,27,29] and references therein). To introduce block-sparse signals and facilitate the subsequent exposition we will assume that integers N and d are chosen such that n = N d is an integer and it represents the total number of blocks that x consists of.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Strong thresholds for ℓ<inf>2</inf>/ℓ<inf>1</inf>-optimization in block-sparse compressed sensing

Stojnic

2009

2009 IEEE International Conference on Acoustics, Speech and Signal Processing

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It has been known for a while that l1-norm relaxation can in certain cases solve an under-determined system of linear equations. Recently, [5,10] proved (in a large dimensional and statistical context) that if the number of equations (measurements in the compressed sensing terminology) in the system is proportional to the length of the unknown vector then there is a sparsity (number of non-zero elements of the unknown vector) also proportional to the length of the unknown vector such that l1-norm relaxation succeeds in solving the system. In this paper we determine sharp lower bounds on the values of allowable sparsity for any given number (proportional to the length of the unknown vector) of equations for the case of the so-called block-sparse unknown vectors considered in [25].

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“…are legitimate choices for CS, carefully designed matrices can lead to additional benefits for the sparse compression/recovery. A few examples are faster encoding time and recovery algorithms for sparse matrices and in particular expander graphs [6,7,8,9], higher recovery thresholds for measurement matrices based on Reed-Solomon codes [11,14] etc.…”

Section: Introductionmentioning

confidence: 99%

Subspace expanders and matrix rank minimization

Oymak

Khajehnejad

Hassibi

2011

2011 IEEE International Symposium on Information Theory Proceedings

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Matrix rank minimization (RM) problems recently gained extensive attention due to numerous applications in machine learning, system identification and graphical models. In RM problem, one aims to find the matrix with the lowest rank that satisfies a set of linear constraints. The existing algorithms include nuclear norm minimization (NNM) and singular value thresholding. Thus far, most of the attention has been on i.i.d. Gaussian measurement operators. In this work, we introduce a new class of measurement operators, and a novel recovery algorithm, which is notably faster than NNM.The proposed operators are based on what we refer to as subspace expanders, which are inspired by the well known expander graphs based measurement matrices in compressed sensing. We show that given an n × n PSD matrix of rank r, it can be uniquely recovered from a minimal sampling of O(nr) measurements using the proposed structures, and the recovery algorithm can be cast as matrix inversion after a few initial processing steps.

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Explicit measurements with almost optimal thresholds for compressed sensing

Cited by 32 publications

References 12 publications

SHO-FA: Robust compressive sensing with order-optimal complexity, measurements, and bits

SHO-FA: Robust compressive sensing with order-optimal complexity, measurements, and bits

Strong thresholds for ℓ<inf>2</inf>/ℓ<inf>1</inf>-optimization in block-sparse compressed sensing

Subspace expanders and matrix rank minimization

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Explicit measurements with almost optimal thresholds for compressed sensing

Cited by 32 publications

References 12 publications

SHO-FA: Robust compressive sensing with order-optimal complexity, measurements, and bits

SHO-FA: Robust compressive sensing with order-optimal complexity, measurements, and bits

Strong thresholds for &#x2113;<inf>2</inf>/&#x2113;<inf>1</inf>-optimization in block-sparse compressed sensing

Subspace expanders and matrix rank minimization

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Strong thresholds for ℓ<inf>2</inf>/ℓ<inf>1</inf>-optimization in block-sparse compressed sensing