Elsevier

Information Sciences

Volume 316, 20 September 2015, Pages 1-17
Information Sciences

Robust sparse signal reconstructions against basis mismatch and their applications

https://doi.org/10.1016/j.ins.2015.04.027Get rights and content

Abstract

Creating a proper dictionary is an essential step in sparse signal recovery to explore sparsity in a variety of applications. To utilize the sparse property of the signal of interest, a large and fine dictionary is usually desired to achieve high estimation accuracy. Unfortunately, a big dictionary requires high computational complexity. Furthermore, one can imagine that no matter how fine we grid the domain to create the dictionary, there will always be an off-grid problem, namely the parameters to be estimated do not lie on the grids. This off-grid problem is the so-called a basis mismatch, which will degrade the estimation performance and it can be formulated as the multiplicative noise to the unknown parameters or quantization error to the dictionary. To tackle this issue, in this paper, robust algorithms are developed to enhance the estimation accuracy by utilizing the robust optimization techniques such as stochastic robust and worst case optimizations. As a result, an extra 2-norm constraint on parameter of interest is introduced to increase the estimation robustness. In addition, both theoretical analysis and simulations show that the proposed robust sparsity recovery approaches are superior in performance to other recovery schemes.

Introduction

Parameter estimations have evolved in so many ways since least squares (LS) which is utilized to minimize the data error in order to obtain the estimation. One of the recent important developments in parameter estimation is sparse signal reconstruction techniques [11], [6] in which it has revolutionized our understanding on sub-sampling reconstruction from underdetermined systems. It follows from reconstruction theory that a signal from a lot of less measurements, less than that needed in Nyquist Sampling theory, can be perfectly reconstructed, provided that the signal of interest is sparse in a basis [7]. Some signals are naturally sparse like sparse acoustic channel, which we can take advantage of directly to improve the estimation performance. However, some signals are not sparse in one basis, but in another basis such as image, which is sparse in wavelet domain [19]. For some other applications, like those based on the ideas of [27], [34], a basis needs to be artificially created in order to make a sparse representation of a signal.

Consider a linear underdetermined system, denoted by zN×1=HN×MxM×1, which has less measurements than the parameters to be estimated, namely NM. The recovery theory states that this system can be solved by using as less as O(Klog(M/K)) measurements, provided that x has only K nonzero components with the remaining components being zeros, called a K-sparse signal. In general, the solution can be obtained by solving the following convex optimization in noiseless case, for a detailed discussion, see [11], [6]. That is,minimizex1subject toz=Hx.

The key idea behind a sparse signal reconstruction is that the sparsity is exploited by 1-norm to enable one to use less measurements. The matrix H is the basis at which the x is sparse. As we can see, the basis plays an essential role in reconstructions and creating one is not an easy task. For example, in frequency estimation, since signal of multiple sinusoidals is sparse in frequency domain, a basis matrix by constructing a J-point discrete Fourier transform (DFT) matrix is then created to perform the estimation. If the frequencies to be estimated lie on the DFT grid, of course, a perfect reconstruction can be obtained. However, that is not always the case because some frequencies are off the grid, which causes the mismatch in basis. In turn, it degrades the estimation performance. One way to remedy this issue is to increase J to create a bigger basis matrix. But, at the same time it substantially increases the computational complexity in solving the problem. One can surely imagine that no matter how large J gets, there are always some frequencies that will not be on the gird. In other words, the basis mismatch problem is always present. Another illustrated example is 2-D positioning. To estimate the positions of targets, a dictionary/basis can be constructed via discretizing the target space over two dimensions to utilize the sparsity recovery concept. In doing so, we will run into the same issue we have before in frequency estimation. No matter how fine the grid is, off-grid targets always exist in the estimation problem. Actually, off-grid is a common problem in many other applications as well (see, for example, [10], [35]). Without a doubt, this mismatch problem degrades the reconstruction performance unless it is handled.

These are a few studies on the sparse signal recovery, where the sensing matrix is assumed to be unknown or erroneous. In [14], the authors concluded that the signal reconstruction is robust to small perturbations and established the linear relationship between the recovery error and perturbation level. However, it is shown in [10], [8] that the recovery performance still suffers from large perturbations. Recently, the authors [30] utilized a certain structure matrix perturbation to formulate the off-grid issue and performed direction of arrival (DOA) estimation. Gleichman and Eldar [12] developed a blind compressed sensing approach, where the sensing matrix is assumed to be unknown. This means that to uniquely determine the solution, additional constraints on the sensing matrix are needed. The sparse learning has found wide applications as well. In [32], a novel spare learning method in manifold learning was proposed to perform image clustering. The multimodal sparse coding was adopted in [31] to predict clicks based image reranking. The multigraph learning approach [28] and multimodal graph reranking [29] were successfully applied in video retrieval and web image search, respectively. To alleviate the influence of noisy data, robust nonnegative learning frameworks were utilized to promote robustness against noisy data, unreliable graphs, and noisy labels in graph embedding [33] and robust bilinear factorization approach was also adopted to improve robustness against data missing and noisy data in image processing [21]. Instead of using discrete dictionary, the so-called continuous dictionary was created and then atomic norm was utilized to enforce sparsity and to alleviate the impact of the off-grid issue [24], [2], [18], [9]. However, this approach is only applicable to one special kind of signal, i.e., signal having Vandermonde structures.

In this paper, we tackle the mismatch problem by developing robust sparse signal reconstruction techniques. The mismatch is modeled by using perturbed basis matrix, namely Hˆ=H+U, where U is the error matrix that models the mismatch problem. One way to view this error matrix is that it can be seen as the multiplicative noise to the unknown parameter since the system model can be rewritten as y=Hx+(Ux+n), where n is the system additive noise. The error matrix U can also be viewed as quantization error since it is introduced by griding the basis domain as shown in Section 3 later. Based on the information available on U, two robust reconstruction approaches are developed in this paper. The first approach is the stochastic one by assuming that U is the random matrix and its covariance is known. The second one is the worst case approach by assuming that U is bounded. In the algorithm development, an extra regularization 2-norm on parameters of interest is introduced in both cases to increase the robustness. Those two robust reconstruction approaches really offer superior performance than the original one both in theory and applications. The main contributions of this work are threefold. First, to overcome the off-grid problem, the basis mismatch is modeled by perturbed measurement system. Second, with the different information available on the error matrix, two robust algorithms are designed to eliminate its negative effect. Third, based on the robust algorithms developed, three important applications, such as frequency estimation, DOA estimation and imaging are studied.

The rest of paper is organized as follows: In Section 2, the two robust sparse signal recovery approaches are developed and their theoretical bound is derived as well. Three important applications by utilizing robust sparse signal recovery in this study are presented in considerate details in Section 3. In Section 4, numerical studies are presented to demonstrate the superior performance. Finally, this paper concludes with a brief summary in Section 5.

Section snippets

Robust sparse signal reconstruction

Given a linear system z=Hx+n, the philosophy of signal reconstruction is to estimation the parameter x from the given measurements z. Least Squares (LS) is the most important estimation tool used and its minimum 2-norm variant can be described in [3]. That is,minimizex2subject toz-Hx2<,where n=z-Hx and n< is the bounded additive noise. This optimization has a closed-form solution xˆ=HH(HHH+I)-1z, but unfortunately this 2-norm minimization cannot recover sparse solutions.

To construct

Applications

Spare signal reconstruction has been widely used for the applications of signal acquisition, compression, reconstruction, estimation, and medical imaging [27], [34], [10]. In this section, three important applications, such as frequency estimation, DOA estimation, and imaging, are presented to demonstrate the performance of the proposed robust sparse signal reconstruction algorithms.

Numerical results

In this section, simulation and experimental results are provided to evaluate the performance for frequency estimation, DOA estimation, and imaging. The uncertainty strength is defined by η=U/Ψ.

Conclusion

In this paper, robust sparse signal reconstruction approaches are investigated to overcome basis mismatch issue. The robust optimization techniques, such as stochastic and worst case optimizations are explored to enhance robustness of the recovery algorithms. The proposed methods are applied in three important applications of frequency estimation, DOA estimation, and imaging to verify their performances. Aside from theoretical analysis, simulation results demonstrate their effectiveness to

Acknowledgments

This work was jointly supported by Program for Changjiang Scholars and Innovative Research Team in University under Grant IRT1299, by the special fund of Chongqing Key Laboratory, by Foundation and Advanced research projects of Chongqing Municipal Science and Technology Commission under Grants cstc2014jcyjA40017 and cstc2014jcyjA40027, by Science and Technology project of Chongqing Municipal Education Commission under Grants KJ1400425 and KJ130504, by China NSF under Grant No. 61401050, by

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