Robust Sparse Bayesian Learning for Sparse Signal Recovery Under Unknown Noise Distributions
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Robust Sparse Bayesian Learning for Sparse Signal Recovery Under Unknown Noise Distributions Kaide Huang1
· Zhiyong Yang2
Received: 27 April 2020 / Revised: 14 August 2020 / Accepted: 18 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This paper considers the robust recovery problem of sparse signal with sparse Bayesian learning (SBL) in noisy environments. Most of the current SBL algorithms are constructed on the optimization problem using the square loss, which mainly deals with Gaussian noise. However, real measurements are often contaminated by an unknown distributed noise that is unlikely to be Gaussian. To prevent performance degradation of SBL in such cases, we propose a robust sparse Bayesian learning method with a simple but effective hierarchical noise model. Using this model, the resultant loss is made up of a weighted error measure and a priori-dependent constraint on the weight, and then provides the flexibility for resisting the outliers and adapting to the real noise. A type-II Bayesian estimate is performed to infer the related model parameter and the unknown sparse signal. The advantage of our method is demonstrated by extensive experiments on synthetic data and real radio tomographic imaging data. Keywords Sparse signal recovery · Compressive sensing · Robust recovery · Sparse Bayesian learning · Unknown noise distributions
1 Introduction Sparse Bayesian learning (SBL), also known as relevance vector machine (RVM), is a supervised learning method based on a parameterized prior model [21]. SBL
This work was supported by the National Natural Science Foundation of China with Grant No. 61906041 and by the Natural Science Foundation of Jiangxi Province, China with Grant No. 20181BAB202015.
B
Kaide Huang [email protected] Zhiyong Yang [email protected]
1
School of Mathematics and Big Data, Foshan University, Guangdong 528000, China
2
School of Software, Nanchang Hangkong University, Jiangxi 330000, China
Circuits, Systems, and Signal Processing
plays an important role in sparse signal recovery in the field of compressive sensing (CS) [13,29]. Compared with the widely used 1 minimization [33] and greedy CS algorithms [23], SBL has the following advantages. Firstly, prior information such as sparsity or correlation of the solutions can be exploited for improving the signal recovery performance via the prior model [3,35]. Secondly, SBL is not insensitive to the coherence of the measurement matrix [29], which brings significant benefits to practical applications. Thirdly, it has been shown that SBL is equivalent to an iterative reweighted 1 minimization algorithm, and it can potentially be advantageous to avoid local minima and find maximally sparse solutions [30]. SBL has been explored for many fields, such as direction of arrival (DOA) estimation [32], electrical impedance tomography [15] and so on. Given a measurement data vector y ∈ R M and the assumption of Gaussian noise, the SBL problem can be formulated as xSBL = arg min y − x22 + λ
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