Robust sparse signal reconstructions against basis mismatch and their applications
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 , which has less measurements than the parameters to be estimated, namely . The recovery theory states that this system can be solved by using as less as measurements, provided that 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,
The key idea behind a sparse signal reconstruction is that the sparsity is exploited by -norm to enable one to use less measurements. The matrix is the basis at which the 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 , where 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 , where is the system additive noise. The error matrix 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 , two robust reconstruction approaches are developed in this paper. The first approach is the stochastic one by assuming that is the random matrix and its covariance is known. The second one is the worst case approach by assuming that is bounded. In the algorithm development, an extra regularization -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 , the philosophy of signal reconstruction is to estimation the parameter from the given measurements . Least Squares (LS) is the most important estimation tool used and its minimum -norm variant can be described in [3]. That is,where and is the bounded additive noise. This optimization has a closed-form solution , but unfortunately this -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 .
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
References (36)
The restricted isometry property and its implications for compressed sensing
Académie des Sci. I
(2008)- et al.
Robust bilinear factorization with missing and grossly corrupted observations
Inf. Sci.
(2015) A comparison of range-D oppler and wavenumber domain SAR focusing algorithms
IEEE Trans. Geosci. Remote Sens.
(1992)- et al.
Atomic norm denoising with applications to line spectral estimation
IEEE Trans. Signal Process.
(2013) - et al.
Convex Optimization
(2004) - M. Braunstein, J. Ralston, D. Sparrow, Signal processing approaches to radio frequency interference suppression, in:...
- et al.
The dantzig selector: statistical estimation when is much larger than
Ann. Statist.
(2007) - et al.
An introduction to compressive sampling
IEEE Signal Process. Magaz.
(2008) - D. Chae, P. Sadeghi, R. Kennedy, Effects of basis-mismatch in compressive sampling of continuous sinusoidal signals,...
- et al.
Compressive two-dimensional harmonic retrieval via atomic norm minimization
IEEE Trans. Signal Process.
(2015)
Sensitivity to basis mismatch in compressed sensing
IEEE Trans. Signal Process.
Compressed sensing
IEEE Trans. Inf. Theory
Blind compressed sesning
IEEE Trans. Inf. Theory
General deviants: an analysis of perturbations in compressed sensing
IEEE J. Select. Topics Signal Process.
Modern Spectral Estimation: Theory and Application
Fundamentals of Statistical Signal Processing: Estimation Theory
Time variant RFI suppression for SAR using iterative adaptive approach
IEEE Geosci. Remote Sens. Lett.
Cited by (14)
Representation recovery via L<inf>1</inf>-norm minimization with corrupted data
2022, Information SciencesCitation Excerpt :Recovery of the original sparse signals or representations has been one of the fast growing field in signal processing in the past decades [1–12].
An integrated optimisation algorithm for feature extraction, dictionary learning and classification
2018, NeurocomputingCitation Excerpt :For high-dimensional data, it is difficult to obtain sparse representation coefficients for classification purposes because the dictionary is usually not over-complete. To obtain sparse representation coefficients, the dimensionality of the high-dimensional data has to be pre-reduced using a random matrix or PCA [18–28]. Therefore, these SRC-based classifications [18–28] are suboptimal because such dimensionality reduction methods are independent from the classification criterion.
l<inf>p</inf>-norm regularization optimization of impulsive disturbance removal
2020, Xi'an Dianzi Keji Daxue Xuebao/Journal of Xidian UniversitySimultaneous Radio Frequency and Wideband Interference Suppression in SAR Signals via Sparsity Exploitation in Time-Frequency Domain
2018, IEEE Transactions on Geoscience and Remote Sensing