Parsimony-Enhanced Sparse Bayesian Learning for Robust Discovery of Partial Differential Equations

Compressive sensing is a sub-Nyquist sampling technique for efficient signal acquisition and reconstruction of sparse or compressible signals. In order to account for the sparsity of the underlying signal of interest, it is common to use sparsifying priors such as Bernoulli–Gaussian-inverse Gamma (BGiG) and Gaussian-inverse Gamma (GiG) priors on the components of the signal. With the introduction of variational Bayesian inference, the sparse Bayesian learning (SBL) methods for solving the inverse problem of compressive sensing have received significant interest as the SBL methods become more efficient in terms of execution time. In this paper, we consider the sparse signal recovery problem using compressive sensing and the variational Bayesian (VB) inference framework. More specifically, we consider two widely used Bayesian models of BGiG and GiG for modeling the underlying sparse signal for this problem. Although these two models have been widely used for sparse recovery problems under various signal structures, the question of which model can outperform the other for sparse signal recovery under no specific structure has yet to be fully addressed under the VB inference setting. Here, we study these two models specifically under VB inference in detail, provide some motivating examples regarding the issues in signal reconstruction that may occur under each model, perform comparisons and provide suggestions on how to improve the performance of each model.

Sparse Bayesian learning using correlated hyperparameters for recovery of block sparse signals

This paper addresses parameter estimation of superimposed signals jointly with their number within the Bayesian framework. We combine sparse Bayesian machine learning methods with the state of the art SAGE-based parameter estimation algorithm. Existing sparse Bayesian methods allow to assess model order through priors over model parameters, but do not consider models nonlinear in parameters. SAGE-based parameter estimation does allow nonlinear model structures, but lacks a mechanism for model order estimation. Here we show how Gaussian and Laplace priors can be applied to enforce sparsity and determine the model order in case of superimposed signals, as well as develop an EM-based learning algorithm that efficiently estimate parameters of the superimposed signals as well as prior parameters that control the sparsity of the learned models. Our work extends the existing approaches to complex data and models nonlinear in parameters. We also present new analytical and empirical studies of the Laplace sparsity priors applied to complex data. The performance of the proposed algorithm is analyzed using synthetic data.

This paper concerns direction of arrival (DOA) estimation based on a sparse Bayesian learning (SBL) approach. We address two inherent problems of this class of DOA estimation methods: (i) a predefined dictionary can generate off-grid problems to a SBL DOA estimator; (ii) a parametric prior generally enforces the solution to be sparse, but the existence of noise can greatly affect the sparsity of the solution. Both of these issues may have a negative impact on the estimation accuracy. In this paper, we propose an improved root SBL (IRSBL) method for off-grid DOA estimation that adopts a coarse grid to generate an initial dictionary. To reduce the bias caused by dictionary mismatch, we integrate the polynomial rooting approach into the SBL method to refine the spatial angle grid. Then, we integrate a constant false alarm rate rule in the SBL framework to enforce sparsity and improve computational efficiency. Finally, we generalize the IRSBL method to the case of non-uniform linear arrays. Numerical analysis demonstrates that the proposed IRSBL method provides improved performance in terms of both estimation accuracy and computational complexity over the most relevant existing method.

Sparse arrays are special geometrical arrangements of sensors which overcome some of the drawbacks associated with dense uniform arrays and require fewer sensors. For direction finding applications, sparse arrays with the same number of sensors can resolve more sources while providing higher resolution than a dense uniform array. This has been verified numerically and with real data for one-dimensional microphone arrays. In this study the use of nested and co-prime arrays is examined with sparse Bayesian learning (SBL), which is a compressive sensing algorithm, for estimating sparse vectors and support. SBL is an iterative parameter estimation method and can process multiple snapshots as well as multiple frequency data within its Bayesian framework. A multi-frequency variant of SBL is proposed, which accounts for non-flat frequency spectra of the sources. Experimental validation of azimuth and elevation [two-dimensional (2D)] direction-of-arrival (DOA)estimation are provided using sparse arrays and real data acquired in an anechoic chamber with a rectangular array. Both co-prime and nested arrays are obtained by sampling this rectangular array. The SBL method is compared with conventional beamforming and multiple signal classification for 2D DOA estimation of experimental data.

Sparse Bayesian learning (SBL) method for direction of arrival (DOA) estimation of near-field sources is implemented, and subspace decomposition is used to estimate range based on the estimated DOAs. Given the prior for sources and the likelihood Gaussian, the posterior distribution is derived. Hyperparameters (i.e., the variance of source power) are updated by maximizing the evidence to estimate DOAs. The number of sources is estimated simultaneously by the model-order selection criterion, which reduces the demand of a priori of the number of sources. Theoretical analysis and simulation experiments show better performance compared with two dimensional multiple signal classification (2D-MUSIC). The method reduces the dependence on high signal-to-noise ratio (SNR) and multiple snapshots.

ABSTRACTSparse Bayesian learning (SBL) is widely applied in direction‐of‐arrival (DOA) estimation. However, it is limited by complexity, grid mismatch and inappropriate initial values of noise. To address these problems, a DOA estimation algorithm using grid pruning and adaptation based on noise integral‐based sparse Bayesian learning (AGNISBL) method is proposed in this letter. To reduce complexity, grid pruning is introduced for expectation maximization (EM) framework. Furthermore, a novel adaptive‐grid method is proposed for solving grid mismatch. A noise integral‐based inference framework is used to improve the robustness of the sparse Bayesian method. Simulation results show that the performance of the proposed AGNISBL approaches CRB at high signal‐to‐noise ratios (SNR) and with lower time complexity compared to other methods.

The underwater maneuvering platform generates self-noise when sailing, which shows spatial directionality to the arrays fixed on the platform. In this paper, it is called spatially colored noise (SCN). The direction of arrival (DOA) estimation results are often influenced by this self-noise, leading to a decrease in estimation accuracy and to the appearance of spurious peaks. To resolve this problem, a sparse Bayesian learning (SBL) method adapted to underwater maneuvering platform noise is proposed in this paper. The SBL framework with unknown SCN is established first. Then, the SCN covariance matrix is estimated by projecting the received data covariance matrix into the noise subspace, and the DOA estimation results are finally obtained through multiple iterations. The simulation results show that the proposed method avoids spurious peaks, and compared to the existing methods, the proposed method achieves a higher accuracy in the case of low SNRs and small snapshot numbers. The sea trial data processing results show that the proposed method provides lower and flatter noise spectrum levels without spurious peaks.

For the traditional uniform linear array (ULA) direction of arrival (DOA) estimation method with a limited array aperture, a non-circular signal off-grid sparse Bayesian DOA estimation method based on nested arrays is proposed. Firstly, the extended matrix of the received data is constructed by taking advantage of the fact that the statistical properties of non-circular signals are not rotationally invariant. Secondly, we use the difference and sum co-arrays for the nested array technique, thus increasing the array aperture and improving the estimation accuracy. Finally, we take the noise as part of the interest signal and iteratively update the grid points using the sparse Bayesian learning (SBL) method to eliminate the modeling errors caused by off-grid gaps. The simulation results show that the proposed algorithm can improve the accuracy of DOA estimation compared with the existing algorithms.

Sparse recovery algorithms have been applied to the Space‐time adaptive processing for reducing the requirement of samples over the past 15 years. However, many Sparse recovery algorithms are not robust and need accurate user parameters. Conventional sparse Bayesian learning (SBL) algorithms are insensitivity to user parameters but converge slowly. To remedy the limitation, two iterative reweighted algorithms are proposed based on SBL. In order to minimise the SBL penalty function, we construct its upper‐bounding surrogate function via the concave conjugate function and apply iterative reweighted algorithms to minimise the surrogate function. Theoretical analysis and numerical experiments all exhibit great performance of the proposed algorithms.

The most of existing off-grid direction of arrival (DOA) estimation methods are based on the perfect array-manifold. However, in practice, it is often hard to obtain a perfect array-manifold. In this paper, to achieve the DOA estimation under mutual coupling condition with low computational complexity, we propose a robust root Sparse Bayesian Learning (SBL) method. In the proposed method, firstly, we adopt the banded complex symmetric Toeplitz structure of the mutual coupling matrix to remove the negative influence of mutual coupling on DOA estimation. Then the DOA with off-grid is estimated by formulating the root-SBL strategy. Compared with the existing SBL-based algorithms, our method can not only maintain superior DOA estimation performance under the condition of mutual coupling, especially with strong mutual coupling, but also have lower computational complexity. Simulation results demonstrate that the proposed method can still accurately estimate DOAs under strong mutual coupling conditions, while other SBL-based methods fail to work.

Aiming at DOA estimation under impulsive noise, this paper propose a real-valued sparse Bayesian learning (SBL) method. A unitary transformation is utilized to convert complex-valued direction-of-arrival (DOA) estimation into real ones. The variational Bayesian inference (VBI) technique is then adopted to perform the Bayesian inference with such real-valued prior. Consequently, the computational complexity of this Bayesian inference is significantly reduced. Simulation outcomes demonstrate the great robust performance and low computational load of the new method.

In this paper, we develop a new sparse Bayesian learning method for recovery of block-sparse signals with un- known cluster patterns. A pattern-coupled hierarchical Gaussian prior model is introduced to characterize the statistical depen- dencies among coefficients, where a set of hyperparameters are employed to control the sparsity of signal coefficients. Unlike the conventional sparse Bayesian learning framework in which each individual hyperparameter is associated independently with each coefficient, in this paper, the prior for each coefficient not only involves its own hyperparameter, but also the hyperparameters of its immediate neighbors. In doing this way, the sparsity patterns of neighboring coefficients are related to each other and the hierarchical model has the potential to encourage structured- sparse solutions. The hyperparameters, along with the sparse signal, are learned by maximizing their posterior probability via an expectation-maximization (EM) algorithm. Index Terms— Sparse Bayesian learning, pattern-coupled hi- erarchical model, block-sparse signal recovery.

Feature fusion and distillation embedded sparse Bayesian learning model for in-situ foreknowledge of robotic machining errors

This article addresses the problem of fault diagnosis in multistation assembly systems. Fault diagnosis is to identify process faults that cause excessive dimensional variation of the product using dimensional measurements. For such problems, the challenge is solving an underdetermined system caused by a common phenomenon in practice; namely, the number of measurements is less than that of the process errors. To address this challenge, this article attempts to solve the following two problems: (i) how to utilize the temporal correlation in the time series data of each process error and (ii) how to apply prior knowledge regarding which process errors are more likely to be process faults. A novel sparse Bayesian learning method is proposed to achieve the above objectives. The method consists of three hierarchical layers. The first layer has parameterized prior distribution that exploits the temporal correlation of each process error. Furthermore, the second and third layers achieve the prior distribution representing the prior knowledge of process faults. Since posterior distributions of process faults are intractable, this article derives approximate posterior distributions via Variational Bayes inference. Numerical and simulation case studies using an actual autobody assembly process are performed to demonstrate the effectiveness of the proposed method.