A lower bound on the estimator variance for the sparse linear model

Schmutzhard, Sebastian; Jung, Alexander; Hlawatsch, Franz; Ben-Haim, Zvika; Eldar, Yonina C.

doi:10.1109/acssc.2010.5757886

Cited by 7 publications

(16 citation statements)

References 11 publications

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“…This is an important difference from the bound given in Theorem VI.1 and, also, from the bound to be given in Theorem VIII.8. Furthermore, still for H = I and c(·) ≡ 0, the bound (34) can be shown [36], [31, p. 106] to be tighter (higher) than the bounds in Theorem VI.1 and Theorem VIII.8.…”

Section: B a Novel Crb-type Lower Variance Boundmentioning

confidence: 99%

Minimum Variance Estimation of a Sparse Vector Within the Linear Gaussian Model: An RKHS Approach

Jung¹,

Schmutzhard

Hlawatsch³

et al. 2014

IEEE Trans. Inform. Theory

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We consider minimum variance estimation within the sparse linear Gaussian model (SLGM). A sparse vector is to be estimated from a linearly transformed version embedded in Gaussian noise. Our analysis is based on the theory of reproducing kernel Hilbert spaces (RKHS). After a characterization of the RKHS associated with the SLGM, we derive novel lower bounds on the minimum variance achievable by estimators with a prescribed bias function. This includes the important case of unbiased estimation. The variance bounds are obtained via an orthogonal projection of the prescribed mean function onto a subspace of the RKHS associated with the SLGM. Furthermore, we specialize our bounds to compressed sensing measurement matrices and express them in terms of the restricted isometry and coherence parameters. For the special case of the SLGM given by the sparse signal in noise model (SSNM), we derive closed-form expressions of the minimum achievable variance (Barankin bound) and the corresponding locally minimum variance estimator. We also analyze the effects of exact and approximate sparsity information and show that the minimum achievable variance for exact sparsity is not a limiting case of that for approximate sparsity. Finally, we compare our bounds with the variance of three well-known estimators, namely, the maximum-likelihood estimator, the hard-thresholding estimator, and compressive reconstruction using the orthogonal matching pursuit.

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Section: B a Novel Crb-type Lower Variance Boundmentioning

confidence: 99%

Minimum Variance Estimation of a Sparse Vector Within the Linear Gaussian Model: An RKHS Approach

Jung¹,

Schmutzhard

Hlawatsch³

et al. 2014

IEEE Trans. Inform. Theory

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“…These matrices can be characterized by such concepts as the spark 22 which was formally defined by Donoho and Elad [5], restricted isometry property (RIP) introduced by Cand`es and Tao [6], mutual coherence (MC) [7][8][9], and null space property (NSP) [9,12]. as a merit function for sparsity can be traced back several decades in a wide range of areas from seismic traces [13], sparse-signal recovery [14], sparse-model selection (LASSO algorithm) in statistics [15] to image processing [16], and continues its growth in other areas [17][18][19].…”

Section: Introductionmentioning

confidence: 99%

An Improved Lower Bound of the Spark with Application

Gharibi¹

2012

IJDPS

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Abstract:Spark plays a great role in studying uniqueness of sparse solutions of the underdetermined linear equations. In this article, w e derive a new lower bound of spark. As an application, we obtain a new criterion fo r the uniqueness of sparse solutions of linear equations. Keywords: : :: Underdetermined, Linear equations, Sparse solution, Spark. 1.IntroductionRecent theoretical developments have generated a great deal of interest in sparse signal representation. The setup assumes a given dictionary of "elementary" signals, and models an input signal as a linear combination of dictionary elements, with the provision that the representation is sparse, i.e., involves only a few of the dictionary elements. Finding sparse representations ultimately requires solving for the sparsest solution of an underdetermined system of linear equations. Such models arise often in signal processing, image processing, and digital communications.Given an ‫ܣ‬ ‫א‬ ܴ ൈ ሺ݊ ൏ ݉ሻ full-rank matrix with no zero columns n R b∈ , the linear system b Ax = has infinitely many solutions when the system is underdetermined. Depending on the nature of source problems, we are often interested in finding a particular solution, and thus optimization methods come into a play through certain merit functions that measure the desired special structure of the solution. One of the recent interests is to find the sparsest solution of an underdetermined linear system, which has found many applications in signal and image processing which is an NP-hard discrete optimization problem [4]. The recent study in the field of compressed sensing nevertheless shows that not all cardinality minimization problems are equally hard, and there does exist a class of matrices A such that the problem (1.1) is computationally tractable. These matrices can be characterized by such concepts as the spark

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“…However, the calculation of the HCRB for general sensing matrix A is much more complicated, and therefore attention is focussed only on the simplest case of unit sensing matrix in this section [27,40]. Nevertheless, the HCRB of this simple case is still instructive for us to have a qualitative understanding of the HCRB for general cases.…”

Section: The Hammersley-chapman-robbins Boundmentioning

confidence: 99%

“…Their closed-form result is tighter than ours when β is sufficiently large, but is not as tight as ours in the low β range and fails to close the gap between maximal and nonmaximal cases. In [40], another lower bound is provided and is tighter than ours for all β > 0, but the derivation is based on the RKHS formulation of the BB which is difficult to be generalized when σ 2 e = 0. Despite that our bound is not the tightest, it still provides a correct qualitative trend of the lower bound of sparse estimation, and is able to deal with matrix perturbation without much effort.…”

Section: Theoremmentioning

confidence: 99%

On the Performance Bound of Sparse Estimation With Sensing Matrix Perturbation

Tang

Chen

2013

IEEE Trans. Signal Process.

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This paper focusses on the sparse estimation in the situation where both the the sensing matrix and the measurement vector are corrupted by additive Gaussian noises. The performance bound of sparse estimation is analyzed and discussed in depth. Two types of lower bounds, the constrained Cramér-Rao bound (CCRB) and the HammersleyChapman-Robbins bound (HCRB), are discussed. It is shown that the situation with sensing matrix perturbation is more complex than the one with only measurement noise. For the CCRB, its closed-form expression is deduced. It demonstrates a gap between the maximal and nonmaximal support cases. It is also revealed that a gap lies between the CCRB and the MSE of the oracle pseudoinverse estimator, but it approaches zero asymptotically when the problem dimensions tend to infinity. For a tighter bound, the HCRB, despite of the difficulty in obtaining a simple expression for general sensing matrix, a closed-form expression in the unit sensing matrix case is derived for a qualitative study of the performance bound. It is shown that the gap between the maximal and nonmaximal cases is eliminated for the HCRB. Numerical simulations are performed to verify the theoretical results in this paper.

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A lower bound on the estimator variance for the sparse linear model

Cited by 7 publications

References 11 publications

Minimum Variance Estimation of a Sparse Vector Within the Linear Gaussian Model: An RKHS Approach

Minimum Variance Estimation of a Sparse Vector Within the Linear Gaussian Model: An RKHS Approach

An Improved Lower Bound of the Spark with Application

On the Performance Bound of Sparse Estimation With Sensing Matrix Perturbation

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