Location is one of the basic information required for underwater optical wireless sensor networks (UOWSNs) for different purposes, such as relating the sensing measurements with precise sensor positions, enabling efficient geographic routing techniques, and sustaining link connectivity between the nodes. Even though various 2-D UOWSNs’ localization methods have been proposed in the past, the directive nature of optical wireless communications and 3-D deployment of sensors require to develop 3-D underwater localization methods. Additionally, the localization accuracy of the network strongly depends on the placement of the anchors. Therefore, we propose a robust 3-D localization method for partially connected UOWSNs, which can accommodate the outliers and optimize the placement of the anchors to improve the localization accuracy. The proposed method formulates the problem of missing pairwise distances and outliers as an optimization problem, which is solved through half quadratic minimization. Furthermore, analysis is provided to optimally place the anchors in the network, which improves the localization accuracy. The problem of optimal anchor placement is formulated as a combination of Fisher information matrices for the sensor nodes where the condition of D-optimality is satisfied. The numerical results indicate that the proposed method outperforms the literature substantially in the presence of outliers.
As the centralized localization methods are based on dimensionality reduction techniques which require all of the pairwise distance estimations between the nodes, robust matrix completion methods are needed to estimate the missing pairwise distances and remove the outliers. A number of researchers have provided theoretical constraints about the matrix rank, sampling scheme, and missing rate to effectively complete a matrix in low rank. In general, matrix completion schemes can be classified into three classes. The first class consists of matrix factorization based schemes where the missing entries of a partially filled matrix of size @math and rank @math are estimated from the product of two matrices @math and @math @cite_56 @cite_15 . Matrix completion schemes based on matrix factorization are non-convex and they require prior information about the rank of the matrix. The authors in @cite_17 have dynamically adjusted the rank of the matrix to estimate the missing elements via matrix factorization. The matrix completion problem in @cite_17 is formulated as a non-linear successive over relaxation problem which is solved by using linear least square method.
@cite_15 @cite_17 @cite_56
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Recovering low-rank and sparse matrices from incomplete or corrupted observations is an important problem in statistics, machine learning, computer vision, as well as signal and image processing. In theory, this problem can be solved by the natural convex joint mixed relaxations (i.e., l 1 -norm and trace norm) under certain conditions. However, all current provable algorithms suffer from superlinear per-iteration cost, which severely limits their applicability to large-scale problems. In this paper, we propose a scalable, provable and structured robust bilinear factorization (RBF) method to recover low-rank and sparse matrices from missing and grossly corrupted data, i.e., robust matrix completion (RMC), or incomplete and grossly corrupted measurements, i.e., compressive principal component pursuit (CPCP). Specifically, we first present two small-scale matrix trace norm regularized bilinear factorization models for RMC and CPCP problems, in which repetitively calculating SVD of a large-scale matrix is replaced by updating two much smaller factor matrices. Then, we apply the alternating direction method of multipliers (ADMM) to efficiently solve the RMC problems. Finally, we provide the convergence analysis of our algorithm, and extend it to address general CPCP problems. Experimental results verified both the efficiency and effectiveness of our method compared with the state-of-the-art methods. We use convex relaxation techniques to provide a sequence of regularized low-rank solutions for large-scale matrix completion problems. Using the nuclear norm as a regularizer, we provide a simple and very efficient convex algorithm for minimizing the reconstruction error subject to a bound on the nuclear norm. Our algorithm SOFT-IMPUTE iteratively replaces the missing elements with those obtained from a soft-thresholded SVD. With warm starts this allows us to efficiently compute an entire regularization path of solutions on a grid of values of the regularization parameter. The computationally intensive part of our algorithm is in computing a low-rank SVD of a dense matrix. Exploiting the problem structure, we show that the task can be performed with a complexity of order linear in the matrix dimensions. Our semidefinite-programming algorithm is readily scalable to large matrices; for example SOFT-IMPUTE takes a few hours to compute low-rank approximations of a 106 X 106 incomplete matrix with 107 observed entries, and fits a rank-95 approximation to the full Netflix training set in 3.3 hours. Our methods achieve good training and test errors and exhibit superior timings when compared to other competitive state-of-the-art techniques. Matrix completion concerns the recovery of a low-rank matrix from a subset of its revealed entries, and nuclear norm minimization has emerged as an effective surrogate for this combinatorial problem. Here, we show that nuclear norm minimization can recover an arbitrary n×n matrix of rank r from O(nr log2(n)) revealed entries, provided that revealed entries are drawn proportionally to the local row and column coherences (closely related to leverage scores) of the underlying matrix. Our results are order-optimal up to logarithmic factors, and extend existing results for nuclear norm minimization which require strong incoherence conditions on the types of matrices that can be recovered, due to assumed uniformly distributed revealed entries. We further provide extensive numerical evidence that a proposed two-phase sampling algorithm can perform nearly as well as local-coherence sampling and without requiring a priori knowledge of the matrix coherence structure. Finally, we apply our results to quantify how weighted nuclear norm minimization can improve on unweighted minimization given an arbitrary set of sampled entries. Matrix completion is a rank minimization problem to recover a low-rank data matrix from a small subset of its entries. Since the matrix rank is nonconvex and discrete, many existing approaches approximate the matrix rank as the nuclear norm. However, the truncated nuclear norm is known to be a better approximation to the matrix rank than the nuclear norm, exploiting a priori target rank information about the problem in rank minimization. In this paper, we propose a computationally efficient truncated nuclear norm minimization algorithm for matrix completion, which we call TNNM-ALM. We reformulate the original optimization problem by introducing slack variables and considering noise in the observation. The central contribution of this paper is to solve it efficiently via the augmented Lagrange multiplier (ALM) method, where the optimization variables are updated by closed-form solutions. We apply the proposed TNNM-ALM algorithm to ghost-free high dynamic range imaging by exploiting the low-rank structure of irradiance maps from low dynamic range images. Experimental results on both synthetic and real visual data show that the proposed algorithm achieves significantly lower reconstruction errors and superior robustness against noise than the conventional approaches, while providing substantial improvement in speed, thereby applicable to a wide range of imaging applications.