Multiple Measurement Vector Problem Research Articles

A nonconvex sparse recovery method for DOA estimation based on the trimmed lasso

Sparse direction-of-arrival (DOA) estimation methods can be formulated as a group-sparse optimization problem. Meanwhile, sparse recovery methods based on nonconvex penalty terms have been a hot topic in recent years due to their several appealing properties. Herein, this paper studies a new nonconvex regularized approach called the trimmed lasso for DOA estimation. We define the penalty term of the trimmed lasso in the multiple measurement vector model by ℓ2,1-norm. First, we use the smooth approximation function to change the nonconvex objective function to the convex weighted problem. Next, we derive sparse recovery guarantees based on the extended Restricted Isometry Property and regularization parameter for the trimmed lasso in the multiple measurement vector problem. Our proposed method can control the desired level of sparsity of estimators exactly and give a more precise solution to the DOA estimation problem. Numerical simulations show that our proposed method overperforms traditional approaches, which is more close to the Cramer-Rao bound.

Algorithm unfolding for block-sparse and MMV problems with reduced training overhead

In this study, we consider algorithm unfolding for the multiple measurement vector (MMV) problem in the case where only few training samples are available. Algorithm unfolding has been shown to empirically speed-up in a data-driven way the convergence of various classical iterative algorithms, but for supervised learning, it is important to achieve this with minimal training data. For this, we consider learned block iterative shrinkage thresholding algorithm (LBISTA) under different training strategies. To approach almost data-free optimization at minimal training overhead, the number of trainable parameters for algorithm unfolding has to be substantially reduced. We therefore explicitly propose a reduced-size network architecture based on the Kronecker structure imposed by the MMV observation model and present the corresponding theory in this context. To ensure proper generalization, we then extend the analytic weight approach by Liu and Chen to LBISTA and the MMV setting. Rigorous theoretical guarantees and convergence results are stated for this case. We show that the network weights can be computed by solving an explicit equation at the reduced MMV dimensions which also admits a closed-form solution. Toward more practical problems, we then considered convolutional observation models and show that the proposed architecture and the analytical weight computation can be further simplified and thus open new directions for convolutional neural networks. Finally, we evaluate the unfolded algorithms in numerical experiments and discuss connections to other sparse recovering algorithms.

MMV-Net: A Multiple Measurement Vector Network for Multifrequency Electrical Impedance Tomography.

Multifrequency electrical impedance tomography (mfEIT) is an emerging biomedical imaging modality to reveal frequency-dependent conductivity distributions in biomedical applications. Conventional model-based image reconstruction methods suffer from low spatial resolution, unconstrained frequency correlation, and high computational cost. Deep learning has been extensively applied in solving the EIT inverse problem in biomedical and industrial process imaging. However, most existing learning-based approaches deal with the single-frequency setup, which is inefficient and ineffective when extended to the multifrequency setup. This article presents a multiple measurement vector (MMV) model-based learning algorithm named MMV-Net to solve the mfEIT image reconstruction problem. MMV-Net considers the correlations between mfEIT images and unfolds the update steps of the Alternating Direction Method of Multipliers for the MMV problem (MMV-ADMM). The nonlinear shrinkage operator associated with the weighted l2,1 regularization term of MMV-ADMM is generalized in MMV-Net with a cascade of a Spatial Self-Attention module and a Convolutional Long Short-Term Memory (ConvLSTM) module to better capture intrafrequency and interfrequency dependencies. The proposed MMV-Net was validated on our Edinburgh mfEIT Dataset and a series of comprehensive experiments. The results show superior image quality, convergence performance, noise robustness, and computational efficiency against the conventional MMV-ADMM and the state-of-the-art deep learning methods.

Revisiting Model Order Selection: A Sub-Nyquist Sampling Blind Spectrum Sensing Scheme

Wideband spectrum sensing based on sub-Nyquist sampling is an attractive approach to advance dynamic spectrum sharing (DSS), which can improve frequency resource utilization while overcoming sampling bottlenecks. Under the Compressive Sensing (CS) framework, finding occupied subbands can be equivalent to computing the support set for the Multiple Measurement Vectors (MMV) problem. To guarantee the performance of joint support recovery in noisy environments, different kinds of prior information are required, one of which is sparsity, a time-varying parameter. To address the dependence of recovery performance on signal sparsity, this paper proposed a two-step scheme for blind wideband spectrum sensing using a Modulated Wideband Converter (MWC) sub-Nyquist sampling front-end. The scheme first adopts the model order selection (MOS) method to estimate sparsity from the compressed covariance matrix, and then uses the estimates to dynamically adjust joint support recovery. The complete theoretical derivation innovatively applies MOS to sub-Nyquist sampling and presents a design method for MOS penalty constant. Extensive simulation results show that the proposed scheme can not only achieve blind sensing under the spectrum occupancy up to 40%, reduce the overall computational complexity of iterative SOMP, but also significantly improve the false alarm performance, meeting the requirements of the IEEE 802.22 standard.

Alternating Direction Method of Multipliers Based on $\ell_{2,0}$-Norm for Multiple Measurement Vector Problem

Alternating Direction Method of Multipliers Based on $\ell_{2,0}$-Norm for Multiple Measurement Vector Problem

On the Support Recovery of Jointly Sparse Gaussian Sources via Sparse Bayesian Learning

In this work, we provide non-asymptotic, probabilistic guarantees for successful recovery of the common nonzero support of jointly sparse Gaussian sources in the multiple measurement vector (MMV) problem. The support recovery problem is formulated as the marginalized maximum likelihood (or type-II ML) estimation of the variance hyperparameters of a joint sparsity inducing Gaussian prior on the source signals. We derive conditions under which the resulting nonconvex constrained optimization perfectly recovers the nonzero support of a joint-sparse Gaussian source ensemble with arbitrarily high probability. The support error probability decays exponentially with the number of MMVs at a rate that depends on the smallest restricted singular value and the nonnegative null space property of the self Khatri-Rao product of the sensing matrix. Our analysis confirms that nonzero supports of size as high as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(m^{2})$ </tex-math></inline-formula> are recoverable from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> measurements per sparse vector. Our derived sufficient conditions for support consistency of the proposed constrained type-II ML solution also guarantee the support consistency of any global solution of the multiple sparse Bayesian learning (M-SBL) optimization whose nonzero coefficients lie inside a bounded interval. For the case of noiseless measurements, we further show that a single MMV is sufficient for perfect recovery of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> -sparse support by M-SBL, provided all subsets of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k + 1$ </tex-math></inline-formula> columns of the sensing matrix are linearly independent.

Advanced direction of arrival estimation using step‐learnt iterative soft‐thresholding for frequency‐modulated continuous wave multiple‐input multiple‐output radar

The number of antennas in automotive frequency-modulated continuous wave (FMCW) multiple-input multiple-output (MIMO) radar systems is increasing. Existing greedy or subspace-based methods cannot quickly and accurately estimate the direction of arrival (DoA) of the target. Therefore, we propose a fast and accurate DoA estimation algorithm for the automotive FMCW MIMO radar. To achieve both fastness and accuracy, we exploit the group sparsity in DoA estimation by defining the problem as a multiple measurement vector (MMV) compressive sensing and extend the step-learnt iterative soft-thresholding algorithm (SLISTA) to the MMV problem. To apply the extended SLISTA, we train the network in an unsupervised manner and normalise the input. We conduct experiments to evaluate the performance of the proposed method. Compared to the algorithms such as ISTA/FISTA/MFOCUSS that solve the same optimisation problem, the extended SLISTA exhibits the most accurate DoA estimation results for actual targets, with less execution time than a subspace-based method. Moreover, the results show that the extended SLISTA prevents false detections, whereas greedy and subspace-based methods do not.

Bayesian Learning-Based Multiuser Detection for Grant-Free NOMA Systems

Grant-Free Non-Orthogonal Multiple Access (GF-NOMA) is considered as a promising technology to support the massive connectivity of Machine-Type Communications (MTC). The design of efficient and high-performance multi-user detection (MUD) scheme is a challenging issue of GF-NOMA, especially when the number of active users is unknown and relatively high. This paper adopts Sparse Bayesian Learning (SBL) approaches to solve the MUD problem of GF-NOMA in MTC. The MUD problem within a certain access slot is formulated as a Single Measurement Vector (SMV) model and efficiently solved via SBL-based methods. To further improve the MUD performance, we set up a Multiple Measurement Vector (MMV) model and develop block SBL-based MUD methods, by exploiting the temporal correlation of user activity over successive access slots. Then to extend the usage of the aforementioned algorithms to the scenarios with relatively high, or quasi-sparse, user activity, we propose novel SBL-based MUD algorithms via post sparse error recovery methodology, for both the SMV and MMV problem models. Simulation results show that the proposed SBL-based MUD algorithms achieve substantial performance gain over traditional ones, especially when the number of active users is unknown and relatively high.

MMV-Based Sequential AoA and AoD Estimation for Millimeter Wave MIMO Channels

The fact that the millimeter-wave (mmWave) multiple-input multiple-output (MIMO) channel has sparse support in the spatial domain has motivated recent compressed sensing (CS)-based mmWave channel estimation methods, where the angles of arrivals (AoAs) and angles of departures (AoDs) are quantized using angle dictionary matrices. However, the existing CS-based methods usually obtain the estimation result through one-stage channel sounding that have two limitations: (i) the requirement of large-dimensional dictionary and (ii) unresolvable quantization error. These two drawbacks are irreconcilable; improvement of the one implies deterioration of the other. To address these challenges, we propose, in this paper, a two-stage method to estimate the AoAs and AoDs of mmWave channels. In the proposed method, the channel estimation task is divided into two stages, Stage I and Stage II. Specifically, in Stage I, the AoAs are estimated by solving a multiple measurement vectors (MMV) problem. In Stage II, based on the estimated AoAs, the receive sounders are designed to estimate AoDs. The dimension of the angle dictionary in each stage can be reduced, which in turn reduces the computational complexity substantially. We then analyze the successful recovery probability (SRP) of the proposed method, revealing the superiority of the proposed framework over the existing one-stage CS-based methods. We further enhance the reconstruction performance by performing resource allocation between the two stages. We also overcome the unresolvable quantization error issue present in the prior techniques by applying the atomic norm minimization method to each stage of the proposed two-stage approach. The simulation results illustrate the substantially improved performance with low complexity of the proposed two-stage method.

Some results on OMP algorithm for MMV problem

In this paper, the robustness of the orthogonal matching pursuit (OMP) algorithm for multiple measurement vector (MMV) problem under the restricted isometry property (RIP) is presented. Moreover, it is shown that the OMP algorithm can exactly recover K‐row sparse matrices in K iterations under the RIP condition we have presented. Furthermore, the condition is sharp.

Extreme Compressed Sensing of Poisson Rates from Multiple Measurements.

Compressed sensing (CS) is a signal processing technique that enables the efficient recovery of a sparse high-dimensional signal from low-dimensional measurements. In the multiple measurement vector (MMV) framework, a set of signals with the same support must be recovered from their corresponding measurements. Here, we present the first exploration of the MMV problem where signals are independently drawn from a sparse, multivariate Poisson distribution. We are primarily motivated by a suite of biosensing applications of microfluidics where analytes (such as whole cells or biomarkers) are captured in small volume partitions according to a Poisson distribution. We recover the sparse parameter vector of Poisson rates through maximum likelihood estimation with our novel Sparse Poisson Recovery (SPoRe) algorithm. SPoRe uses batch stochastic gradient ascent enabled by Monte Carlo approximations of otherwise intractable gradients. By uniquely leveraging the Poisson structure, SPoRe substantially outperforms a comprehensive set of existing and custom baseline CS algorithms. Notably, SPoRe can exhibit high performance even with one-dimensional measurements and high noise levels. This resource efficiency is not only unprecedented in the field of CS but is also particularly potent for applications in microfluidics in which the number of resolvable measurements per partition is often severely limited. We prove the identifiability property of the Poisson model under such lax conditions, analytically develop insights into system performance, and confirm these insights in simulated experiments. Our findings encourage a new approach to biosensing and are generalizable to other applications featuring spatial and temporal Poisson signals.

Unitary Approximate Message Passing for Sparse Bayesian Learning

Sparse Bayesian learning (SBL) can be implemented with low complexity based on the approximate message passing (AMP) algorithm. However, it does not work well for a generic measurement matrix, which may cause AMP to diverge. Damped AMP has been used for SBL to alleviate the problem at the cost of reducing convergence speed. In this work, we propose a new SBL algorithm based on structured variational inference, leveraging AMP with a unitary transformation (UAMP). Both single measurement vector and multiple measurement vector problems are investigated. It is shown that, compared to stateof- the-art AMP-based SBL algorithms, the proposed UAMPSBL is more robust and efficient, leading to remarkably better performance.

Multiple Measurement Vector-Based Complex-Valued Multifrequency ECT

Complex-valued, multifrequency electrical capacitance tomography (CVMF-ECT) is a recently developed tomographic concept which is capable of simultaneously reconstructing spectral permittivity and conductivity properties of target objects within the region of interest. To date, this concept has been limited to simulation and another key issue restricting its wide adoption lies in its poor image quality. This letter reports a CVMF-ECT system to verify its practical feasibility and further proposes a novel image reconstruction framework to effectively and efficiently reconstruct multifrequency images using complex-valued capacitance data. The image reconstruction framework utilizes the inherent spatial correlations of the multifrequency images as a priori information and encodes it by using multiple measurement vector (MMV) model. Alternating direction method of multipliers was introduced to solve the MMV problem. Real-world experiments validate the feasibility of CVMF-ECT, and MMV-based CVMF-ECT method demonstrates superior performance compared with conventional ECT approaches.

Orthogonal AMP for Massive Access in Channels With Spatial and Temporal Correlations

We address the joint device activity detection and channel estimation (JACE) problem in a massive MIMO connectivity scenario in which a large number of mobile devices are connected to a base station (BS), while only a small portion are active at any given time. The main objective is to provide an efficient transmission and detection scheme with both spatial and temporal correlations. We formulate JACE as a multiple measurement vector (MMV) problem with correlated entries in the vectors to be estimated. We propose an MMV form of the orthogonal approximate message passing algorithm (OAMP-MMV). We derive a group Gram-Schmidt orthogonalization (GGSO) procedure for the realization of OAMP-MMV. We outline a state evolution (SE) procedure for OAMP-MMV and examine its accuracy using numerical results. We also compare OAMP-MMV with existing alternatives, including AMP-MMV and GTurbo-MMV. We show that OAMP-MMV outperforms AMP-MMV when pilot sequences are generated using Hadamard pilot matrices. Such a pilot design is attractive due to the low-cost signal processing technique using the fast Hadamard transform (FHT). We also show that OAMP-MMV outperforms GTurbo-MMV in correlated channels.

Neural Network Based AMP Method for Multi-User Detection in Massive Machine-Type Communication

In massive machine-type communications (mMTC) scenarios, grant-free non-orthogonal multiple access becomes crucial due to the small transmission latency, limited signaling overhead and the ability to support massive connectivity. In a multi-user detection (MUD) problem, the base station (BS) is unaware of the active users and needs to detect active devices. With sporadic devices transmitting signals at any moment, the MUD problem can be formulated as a multiple measurement vector (MMV) sparse recovery problem. Through the Khatri–Rao product, we prove that the MMV problem is transformed into a single measurement vector (SMV) problem. Based on the basis pursuit de-noising approximate message passing (BPDN-AMP) algorithm, a novel learning AMP network (LAMPnet) algorithm is proposed, which is designed to reduce the false alarm probability when the required detection probability is high. Simulation results show that when the required detection probablity is high, the AMP algorithm based on LAMPnet noticeably outperforms the traditional algorithms with acceptable computational complexity.

Distributed compressive sensing via LSTM-Aided sparse Bayesian learning

Model-driven algorithms on distributed compressive sensing with multiple measurement vectors (MMVs) have been generally based on the assumption that the vectors in the signal matrix are jointly sparse. However, the signal matrix in many practical scenarios violates the above assumption, since there might exist unknown dependency between vectors. It highlights the limitation of model-driven approaches and the necessity to move toward data-driven ones. In this paper, we propose a data-driven algorithm, which interprets the MMV problem as sequence modeling to infer the unknown dependency and encourage sparse signal recovery within the framework of sparse Bayesian learning (SBL). Specifically, we extend the fast-SBL algorithm suitable for MMV sparse recovery. Then the long short-term memory (LSTM) is introduced into the extended fast-SBL, serving as a strategy for selecting the basis function from the dictionary. Compared with existing data-driven algorithms, the proposed LSTM-SBL algorithm inheriting the characteristics of fast-SBL has fewer local minimums and better robustness to noise. Extensive experiments are conducted on MNIST and MSR datasets to evaluate the accuracy, robustness and speed of the LSTM-SBL algorithm. Experimental results are presented and analyzed to illustrate the potential advantages of the LSTM-SBL algorithm in contrast to state-of-the-art MMV recovery algorithms.

Remote Measurement of Human Vital Signs Based on Joint-Range Adaptive EEMD

Remote techniques for measuring human vital signs have attracted great interests due to the benefits shown in medical monitoring and military applications. Compared with continuous-wave Doppler radar, frequency-modulated continuous-wave (FMCW) radar which can discriminate vital signs from different distances, shows potential for reducing the interferences from other targets and the environment. However, in the state-of-the-art algorithms, only one chirp per frame is utilized for FMCW-based vital sign monitoring. Moreover, the vital signal is extracted from only one range bin of the fast Fourier transform. Which does not make full utilization of the long system idle time, and loses the power distributed on other range bins. By exploiting the relationship between respiration and heartbeat vibrations, an adaptive identification embedded ensemble empirical mode decomposition (EEMD) method for joint-range spectral estimation is proposed to measure the heart rate. First, A multi-chirp processing is presented for a 2-dimensional phase accumulation. Then the phase signals from a sequence of range bins are decomposed with a fast adaptive identification process. Finally, with the identified heartbeat components, we solve a multiple measurement vectors problem to estimate the heart rate. Experimental results showed that, at the detection range from 1 m~ 2.5 m, the proposed method can robustly distinguish the heartbeat from respiration and its harmonics and accurately estimate the heart rate with a root mean square error less than 6 bpm.

Guaranteed Localization of More Sources Than Sensors With Finite Snapshots in Multiple Measurement Vector Models Using Difference Co-Arrays

The Multiple Measurement Vector (MMV) problem is central to sparse signal processing, where the goal is to recover the common support of a set of unknown sparse vectors of size $N$ , from $L$ compressed measurement vectors, each of size $M . Recent advances in correlation-aware and Bayesian techniques for MMV models show promising evidence that under appropriate assumptions, it is possible to recover supports of size ( $s$ ) larger than the dimension ( $M$ ) of each measurement vector. However, these results are primarily asymptotic in $L$ , and cannot provide support recovery guarantees for finite $L$ . This paper overcomes such drawback by focusing on a broader family of correlation-aware optimization problems (which include convex problems), and establishing rigorous non-asymptotic probabilistic guarantees in the regime $s>M$ when the measurements are collected using appropriately designed sparse sensor arrays (such as nested and coprime). Assuming the source locations obey a certain “minimum separation” condition, we develop uniform upper bounds on the estimation error that is obeyed by any algorithm belonging to this family, and utilize this bound to ensure probabilistic support recovery in the regime $s>M$ . Our results crucially rely upon (i) the unique geometry of the difference co-array of the sparse arrays, and (ii) certain non-negativity constraints on the optimization variable. As a result of independent interest, we also show that this upper bound is tight with respect to the dimension $N$ . Extensive numerical simulations (including phase transition plots) are presented to validate the theoretical claims.

Evolutionary Multitasking Sparse Reconstruction: Framework and Case Study

Real-world applications typically have multiple sparse reconstruction tasks to be optimized. In order to exploit the similar sparsity pattern between different tasks, this paper establishes an evolutionary multitasking framework to simultaneously optimize multiple sparse reconstruction tasks using a single population. In the proposed method, the evolutionary algorithm aims to search the locations of nonzero components or rows instead of searching sparse vector or matrix directly. Then the within-task and between-task genetic transfer operators are employed to reinforce the exchange of genetic material belonging to the same or different tasks. The proposed method can solve multiple measurement vector problems efficiently because the length of decision vector is independent of the number of measurement vectors. Finally, a case study on hyperspectral image unmixing is investigated in an evolutionary multitasking setting. It is natural to consider a sparse unmixing problem in a homogeneous region as a task. Experiments on signal reconstruction and hyperspectral image unmixing demonstrate the effectiveness of the proposed multitasking framework for sparse reconstruction.

Generalized Channel Estimation and User Detection for Massive Connectivity With Mixed-ADC Massive MIMO

This paper aims to provide a partial discrete Fourier transform (DFT) pilot sequence assisted joint channel estimation and user activity detection scheme for massive connectivity, in which a large number of devices with sporadic transmission communicate with a base station (BS) in the uplink. The joint channel estimation and device detection problem can be formulated as a compressed sensing single measurement vector or multiple measurement vector (MMV) problem depending on whether the BS is equipped with single or large number of antennas. Due to high hardware cost and power consumption in massive multiple-input multiple-output (MIMO) systems, a mixed analog-to-digital converter (ADC) architecture is considered. In order to accommodate a large number of simultaneously transmitting devices, the joint channel estimation and active user detection are formulated as an MMV problem for the massive connectivity scenario; and the proposed GTurbo-MMV algorithm can precisely estimate the channel state information and detect active devices with relatively low overhead. Furthermore, we study the state evolution (SE) for the MMV problem to obtain achievable bounds on channel estimation and device detection performance, in which both the missing and false detection probabilities can be made tend to zero in the massive MIMO regime. The simulation results confirm the theoretical accuracy of our analysis.