Ad-hoc microphone array calibration from partial distance measurements

Localizing a cloud of points from noisy measurements of a subset of pairwise distances has applications in various areas, such as sensor network localization and reconstruction of protein conformations from NMR measurements. In [1], Drineas et al. proposed a natural two-stage approach, named SVD-MDS, for this purpose. This approach consists of a low-rank matrix completion algorithm, named SVD-Reconstruct, to estimate random missing distances, and the classic multidimensional scaling (MDS) method to estimate the positions of nodes. In this paper, we present a detailed analysis for this method. More specifically, we first establish error bounds for Euclidean distance matrix (EDM) completion in both expectation and tail forms. Utilizing these results, we then derive the error bound for the recovered positions of nodes. In order to assess the performance of SVD-Reconstruct, we present the minimax lower bound of the zero-diagonal, symmetric, low-rank matrix completion problem by Fano's method. This result reveals that when the noise level is low, the SVD-Reconstruct approach for Euclidean distance matrix completion is suboptimal in the minimax sense; when the noise level is high, SVD-Reconstruct can achieve the optimal rate up to a constant factor.

Accurate node localization is essential for many Internet of Things (IoT) applications. However, incomplete and noisy distance measurements often degrade the reliability of the Euclidean Distance Matrix (EDM), which is critical for range-based localization. To address this issue, a Low-Rank Matrix Completion approach based on nonconvex rank approximation (LRMCN) is proposed to recover the true EDM. First, the observed EDM is decomposed into a low-rank matrix representing the true distances and a sparse matrix capturing noise. Second, a nonconvex surrogate function is used to approximate the matrix rank, while the l1-norm is utilized to model the sparsity of the noise component. Third, the resulting optimization problem is solved using the Alternating Direction Method of Multipliers (ADMMs). This enables accurate recovery of a complete and denoised EDM from incomplete and corrupted measurements. Finally, relative node locations are estimated using classical multi-dimensional scaling, and absolute coordinates are determined based on a small set of anchor nodes with known locations. The experimental results show that the proposed method achieves superior performance in both matrix completion and localization accuracy, even in the presence of missing and corrupted data.

One of the main challenges faced by wireless sensor network (WSN) localization is the positioning accuracy of the WSN node. The existing algorithms are arduous to use for dealing with the pulse noise that is universal and ineluctable in practical considerations, resulting in lower positioning accuracy. Aimed at this problem and introducing Bregman divergence, we propose in this paper a novel WSN localization algorithm via matrix completion (LBDMC). Based on the natural low-rank character of the Euclidean Distance Matrix (EDM), the problem of EDM recovery is formulated as an issue of matrix completion in a noisy environment. A regularized matrix completion model is established, smoothing the pulse noise by leveraging -norm and the multivariate function Bregman divergence is defined to solve the model to obtain the EDM estimator. Furthermore, node localization is available based on the multi-dimensional scaling (MDS) method. Multi-faceted comparison experiments with existing algorithms, under a variety of noise conditions, demonstrate the superiority of LBDMC to other algorithms regarding positioning accuracy and robustness, while ensuring high efficiency. Notably, the mean localization error of LBDMC is about ten times smaller than that of other algorithms when the sampling rate reaches a certain level, such as >30%.

In this paper, we present an efficient localization algorithm for sensor networks using range information. Since each sensor can only communicate with its neighbors, the Euclidean distance matrix (EDM), composed by squared distances between each pair of sensors, is incomplete. The first step of the proposed algorithm is to fulfill the EDM completion and relative maps of sensor networks are then achieved by the multidimensional scaling technique. Besides the EDM, the centralized Gram matrix of sensors’ coordinates is also used to model the localization problem. Compared with other EDM-based localization algorithms, mathematical properties of both the EDM and the Gram matrix are appropriately exploited so as to improve the estimation accuracy. The resulting localization model is formulated as a semidefinite programming problem. An alternating direction method of multipliers is further developed to enhance the scalability of the proposed algorithm. The numerical experiments demonstrate that the proposed algorithm can effectively improve the estimation accuracy of both the EDM completion and the final localization.

Given an undirected graph G=(V,E) with node set V=[1,n], a subset $S\subseteq V$, and a rational vector $a\in {\rm\bf Q}^{S\cup E}$, the positive semidefinite matrix completion problem consists of determining whether there exists a real symmetric n × n positive semidefinite matrix X=(xij) satisfying xii=ai ($i\in S$) and xij=aij ($ij\in E$). Similarly, the Euclidean distance matrix completion problem asks for the existence of a Euclidean distance matrix completing a partially defined given matrix. It is not known whether these problems belong to NP. We show here that they can be solved in polynomial time when restricted to the graphs having a fixed minimum fill-in, the minimum fill-in of graph G being the minimum number of edges needed to be added to G in order to obtain a chordal graph. A simple combinatorial algorithm permits us to construct a completion in polynomial time in the chordal case. We also show that the completion problem is polynomially solvable for a class of graphs including wheels of fixed length (assuming all diagonal entries are specified). The running time of our algorithms is polynomially bounded in terms of n and the bitlength of the input a. We also observe that the matrix completion problem can be solved in polynomial time in the real number model for the class of graphs containing no homeomorph K4.

The 3-D seismic data reconstruction can be understood as an underdetermined inverse problem, and thus, some additional constraints need to be provided to achieve reasonable results. A prevalent scheme in 3-D seismic data reconstruction is to compute the best low-rank approximation of a formulated Hankel matrix by rank-reduction methods with a rank constraint. However, the predefined Hankel structure is easily damaged by the low-rank approximation, which leads to harming its recovery performance. In this article, we present a structured tensor completion (STC) framework to simultaneously exploit both the Hankel structure and the low-tubal-rank constraint to further enhance the performance. Unfortunately, under the assumption of elementwise sampling used by existing methods, STC is intractable to be solved since Hankel constraints cannot be expressed as linear tensor equations. Instead, tubal sampling is proposed to describe the missing trace behavior more accurately and further build a bridge between tensor and matrix completion (MC) to overcome the solving issue in two aspects: through the bridge from tensor to MC, STC can be solved efficiently using MC from random samplings of each frontal slice in the Fourier domain. Through the bridge from matrix to tensor completion, various tensor models within the framework can be developed from noise-specific MC to meet the need for data reconstruction in changeable noise environments. Moreover, alternating-minimization and alternating-direction methods of multipliers are developed to solve the proposed STC. The superior performance of STC is demonstrated in both synthetic and field seismic data.

In this study, a new semidefinite programming- based algorithm is introduced for the Euclidean distance matrix completion (EDMC) problem with noisy and incomplete distance measurements. A penalty metric that measures the data misfit is minimized over the spectrahedron with projection-free convex optimization. The optimality trade-off between the trace of a positive semidefinite matrix and a loss function is traced over a Pareto frontier using the Regula falsi method, which is simple, derivative-free, and cost-effective nonlinear equation root finding iteration. One field that the EDMC problem is often used in is wireless sensor networks (WSNs) localization. A test setup has been created by using real signal strength measurements for a 3D- localization problem in WSNs to inspect the performance of the introduced algorithm. It is shown that the proposed approach efficiently solves the optimization problem and the results are consistent with the generic semidefinite solver CVX.

Due to the surge in COVID-19 cases, hospitals have had to receive many more patients than before, which has brought unprecedented pressure to the hospital system. Therefore, the emphasis of medical decision-making has shifted from reaching the best treatment effect to prioritizing the treatment of COVID-19 patients by hospitals, which is key to relieving the pressure on the hospital system and reducing the overall mortality rate of COVID-19. There is no doubt that establishing the prioritization of COVID-19 cases is fundamental and pivotal for hospitals to achieve the shift in medical decision-making. Prioritization of COVID-19 cases in previous studies was mostly based on one patient characteristic, mainly including age, health conditions, and gender. This paper focuses on two patient characteristics at the same time. The probability that a COVID-19 patient who died had a given health condition in a given age group is calculated using the matrix completion technique based on the high-rank assumption of Bayesian matrices and the properties of Markov matrices. The calculated results show that doctors should give patients over 55 with respiratory diseases, patients over 65 with circulatory diseases, and patients over 65 with diabetes a higher prioritization in COVID-19 treatment.

Low-rank matrix completion has achieved great success in many real-world data applications. A matrix factorization model that learns latent features is usually employed and, to improve prediction performance, the similarities between latent variables can be exploited by pairwise learning using the graph regularized matrix factorization (GRMF) method. However, existing GRMF approaches often use the squared loss to measure the pairwise differences, which may be overly influenced by dissimilar pairs and lead to inferior prediction. To fully empower pairwise learning for matrix completion, we propose a general optimization framework that allows a rich class of (non-)convex pairwise penalty functions. A new and efficient algorithm is developed to solve the proposed optimization problem, with a theoretical convergence guarantee under mild assumptions. In an important situation where the latent variables form a small number of subgroups, its statistical guarantee is also developed. In particular, we theoretically characterize the performance of the complexity-regularized maximum likelihood estimator, as a special case of our framework, which is shown to have smaller errors when compared to the standard matrix completion framework without pairwise penalties. We conduct extensive experiments on both synthetic and real datasets to demonstrate the superior performance of this general framework.

Manufacturing industries collect massive amounts of multivariate measurement through automated inspection processes. Noisy measurements and high-dimensional, irrelevant features make it difficult to identify useful patterns in the data. Principal component analysis provides linear summaries of datasets with fewer latent variables. Kernel Principal Component Analysis (KPCA), however, identifies nonlinear patterns. One challenge in KPCA is to inverse map the denoised signal from a high-dimensional feature space into its preimage in input space to visualize the nonlinear variation sources. However, such an inverse map is not always defined. This article provides a new meta-method applicable to any KPCA algorithm to approximate the preimage. It improves upon previous work where a strong assumption was the availability of noise-free training data. This is problematic for applications such as manufacturing variation analysis. To attenuate noise in kernel subspace estimation the final preimage is estimated as the average from bagged samples drawn from the original dataset. The improvement is most pronounced when the parameters differ from those that minimize the error rate. Consequently, the proposed approach improves the robustness of any base KPCA algorithm. The usefulness of the proposed method is demonstrated by analyzing a classic handwritten digit dataset and a face dataset. Significant improvement over the existing methods is observed.

In this chapter, we discuss a problem of estimation of a large target matrix based 4 on a finite number of noisy measurements of linear functionals (often, random) 5 of this matrix. The underlying assumption is that the target matrix is of small 6 rank and the goal is to determine how the estimation error depends on the rank 7 as well as on other important parameters of the problem such as the number of 8 measurements and the variance of the noise. This problem can be viewed as a 9 natural noncommutative extension of sparse recovery problems discussed in the 10 previous chapters. As a matter of fact, low rank recovery is equivalent to sparse 11 recovery when all the matrices in question are diagonal. There are several important 12 instances of such problems, in particular, matrix completion [41, 45, 70, 123], 13 matrix regression [40, 90, 126] and the problem of density matrix estimation in 14 quantum state tomography [70, 71, 88]. We will study some of these problems 15 using general empirical processes techniques developed in the first several chapters. 16 Noncommutative Bernstein type inequalities established in Sect. 2.4 will play a 17 very special role in our analysis. The main results will be obtained for Hermitian 18 matrices. So called “Paulsen dilation” (see Sect. 2.4) can be then used to tackle 19 the case of rectangular matrices. Throughout the chapter, we use the notations 20 introduced in Sect. A.4.

We study the recovery of missing data from multiple smart grid datasets within a matrix completion framework. The datasets contain the electrical magnitudes required for monitoring and control of the electricity distribution system. Each dataset is described by a low rank matrix. Different datasets are correlated as a result of containing measurements of different physical magnitudes generated by the same distribution system. To assess the validity of matrix completion techniques in the recovery of missing data, we characterize the fundamental limits when two correlated datasets are jointly recovered. We then proceed to evaluate the performance of Singular Value Thresholding (SVT) and Bayesian SVT (BSVT) in this setting. We show that BSVT outperforms SVT by simulating the recovery for different correlated datasets. The performance of BSVT displays the tradeoff behaviour described by the fundamental limit, which suggests that BSVT exploits the correlation between the datasets in an efficient manner.

This paper presents a new systolic VLSI architecture for computing inverses and divisions in finite fields GF(2/sup m/) based on a variant of Euclid's algorithm. It is highly regular, modular, and thus well suited to VLSI implementation. It has O(m/sup 2/) area complexity and can produce one result per clock cycle with a latency of 8m-2 clock cycles. As compared to existing related systolic architectures with the same throughput performance, the proposed one gains a significant improvement in area complexity.

Millimeter-wave interferometric synthetic aperture radiometer (InSAR) can provide high-resolution observations for many applications by using small antennas to achieve very large synthetic aperture. However, reconstruction of a millimeter-wave InSAR image has been proven to be an ill-posed inverse problem that degrades the performance of InSAR imaging. In this paper, a novel millimeter-wave InSAR image reconstruction approach, referred to as InSAR-TVMC, by total variation (TV) regularized matrix completion (MC) in two-dimensional data space, is proposed. Based on the a priori knowledge that natural millimeter-wave images statistically hold the low-rank property, the proposed approach represents the object images as low-rank matrices and formulates the data acquisition of InSAR in two-dimensional data space directly to undersample visibility function samples. Subsequently, using the undersampled visibility function samples, the optimal solution of the InSAR image reconstruction problem is obtained by simultaneously adopting MC techniques and TV regularization. Experimental results on simulated and real millimeter-wave InSAR image data demonstrate the effectiveness and the significant improvement of the reconstruction performance of the proposed InSAR-TVMC approach over conventional and one-dimensional sparse InSAR image reconstruction approaches.

The authors develop a design technique for associative memories with learning and forgetting abilities via artificial feedback neural networks. The proposed method utilizes the theory of large-scale dynamical systems, instead of the usual energy methods. Networks synthesized by this design method are capable of learning new patterns as well as forgetting existing patterns without the necessity of recomputing the entire interconnection weights and external inputs. The method employs the properties of pseudo-inverse matrices to iteratively solve systems of linear equations, and provides significant improvements over the outer product method and the projection learning rule. Several specific examples are given to illustrate the strengths and weaknesses of the methodology. >