Denoised Wigner distribution deconvolution via low-rank matrix completion.

Low-rank matrix completion, which aims to recover a matrix with many missing values, has attracted much attention in many fields of computer science. A low-rank matrix fitting (LMaFit) method has been proposed for fast matrix completion recently. However, this method cannot converge accurately on matrices of real-world images. For improving the accuracy of LMaFit method, an improved low-rank matrix fitting (ILMF) method based on the weighted [Formula: see text] norm minimization is proposed in this paper, where the [Formula: see text] norm is the summation of the [Formula: see text]-power [Formula: see text] of [Formula: see text] norms of rows in a matrix. In the proposed method, i.e. the ILMF method, the incomplete matrix that may be corrupted by noises is decomposed into the summation of a low-rank matrix and a noise matrix at first. Then, a weighted [Formula: see text] norm minimization problem is solved by using an alternating direction method for improving the accuracy of matrix completion. Experimental results on real-world images show that the ILMF method has much better performances in terms of both the convergence accuracy and convergence speed than the compared methods.

The matrix completion problem consists of finding or approximating a low-rank matrix based on a few samples of this matrix. We propose a new algorithm for matrix completion that minimizes the least-square distance on the sampling set over the Riemannian manifold of fixed-rank matrices. The algorithm is an adaptation of classical nonlinear conjugate gradients, developed within the framework of retraction-based optimization on manifolds. We describe all the necessary objects from differential geometry necessary to perform optimization over this low-rank matrix manifold, seen as a submanifold embedded in the space of matrices. In particular, we describe how metric projection can be used as retraction and how vector transport lets us obtain the conjugate search directions. Finally, we prove convergence of a regularized version of our algorithm under the assumption that the restricted isometry property holds for incoherent matrices throughout the iterations. The numerical experiments indicate that our approach scales very well for large-scale problems and compares favorably with the state-of-the-art, while outperforming most existing solvers.

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.

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.

The problem of recovering a low-rank matrix from partial entries, known as low-rank matrix completion, has been extensively investigated in recent years. It can be viewed as a special case of the affine constrained rank minimization problem which is NP-hard in general and is computationally hard to solve in practice. One widely studied approach is to replace the matrix rank function by its nuclear-norm, which leads to the convex nuclear-norm minimization problem solved efficiently by many popular convex optimization algorithms. In this paper, we propose a new nonconvex approach to better approximate the rank function. The new approximation function is actually the Moreau envelope of the rank function (MER) which has an explicit expression. The new approximation problem of low-rank matrix completion based on MER can be converted to an optimization problem with two variables. We then adapt the proximal alternating minimization algorithm to solve it. The convergence (rate) of the proposed algorithm is proved and its accelerated version is also developed. Numerical experiments on completion of low-rank random matrices and standard image inpainting problems have shown that our algorithms have better performance than some state-of-art methods.

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.

Robust low-rank matrix completion (RMC), or robust principal component analysis with partially observed data, has been studied extensively for computer vision, signal processing and machine learning applications. This problem aims at decomposing a partially observed matrix into the superposition of a low-rank matrix and a sparse matrix, where the sparse matrix captures the grossly corrupted entries of the matrix. A widely used approach to tackle RMC is to consider a convex formulation, which minimizes the nuclear norm of the low-rank matrix (to promote low-rankness) plus the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _1$</tex-math></inline-formula> norm of the sparse matrix (to promote sparsity). In this paper, motivated by some recent works on low-rank matrix completion and Riemannian optimization, we formulate this problem as a nonsmooth Riemannian optimization problem over Grassmann manifold. This new formulation is scalable as the low-rank matrix is factorized to the multiplication of two much smaller matrices. We then propose an alternating manifold proximal gradient continuation method to solve the proposed new formulation. Convergence rate of the proposed algorithm is rigorously analyzed. Numerical results on both synthetic data and real data on background extraction from surveillance videos are reported to demonstrate the advantages of the proposed new formulation and algorithm over several popular existing approaches.

DOA estimation via shift-invariant matrix completion

Low Rank Matrix Completion (LRMC) is widely used in the analysis of incomplete datasets. In this paper, we propose a novel FPGA-based accelerator to speedup a matrix-factorization-based LRMC algorithm that uses stochastic gradient descent. The accelerator is a multi-pipelined architecture with parallel pipelines processing distinct data from a shared on-chip buffer. We propose two distinct on-chip buffer architectures based on a design-space exploration of the performance tradeoffs offered by two competing design methodologies: memory-efficiency versus concurrent conflict-free accesses. Our first design (i.e., memory-efficient design) organizes the buffer into banks and maximally utilizes available on-chip memory for matrix chunk processing without requiring complex address translation tables for on-chip addressing; however, it could incur bank conflicts when concurrent accesses to the same bank occur. The second design (i.e., bank-conflict-free design) exploits parallel multiport memory access and completely eliminates bank conflicts by duplicating the stored data; however, it has much higher on-chip RAM consumption. Intuitively, design one enables (slower) acceleration of (larger) chunks of the input matrix whereas design two enables (faster) processing of (smaller) matrix chunks but requires more iterations for processing the complete matrix. We propose a simple but efficient partitioning approach for supporting large input matrices that do not fit in the on-chip memory of FPGA. We also develop algorithmic optimizations based on matching to reduce data dependencies for parallel pipeline execution. We implement our designs on a state-of-the-art UltraScale+ FPGA device. We use real-life datasets for the evaluation and compare these two designs by varying the number of pipelines. The data dependency optimization results in at least 21.6 x data dependency reduction and improves the execution time by up to 66.3 x compared with non-optimized baseline designs. The memory-efficient design is also shown to be more scalable than the bank-conflict-free design. Compared with the state-of-the-art multi-core implementation and GPU implementation, the bank-conflict-free design achieves 5.4 x and 5.2 x speedup, respectively; the memory-efficient design achieves 16.7 x and 16.2 x speedup, respectively.

In this paper, a novel video super-resolution image reconstruction algorithm is proposed. We design a patch-based low rank matrix completion algorithm. The proposed algorithm addresses the problem of generating a high-resolution (HR) image from several low-resolution (LR) images, based on sparse representation and low-rank matrix completion. The approach represents observed LR frames in the form of sparse matrices and rearranges those frames into low dimensional constructions. Experimental results demonstrate that, high-frequency details in the super resolved images are recovered from the LR frames. The gains in terms of PSNR and SSIM are significant.

This article studies the channel estimation for wideband multiple-input multiple-output (MIMO) systems equipped with hybrid analog/digital transceivers operating in the millimeter-wave (mmWave) or terahertz (THz) bands. By exploiting the low-rank property of the concatenated channel matrix of the delay taps, we formulate the channel estimation problem as a low-rank matrix sensing (LRMS) problem and solve it using a low-complexity generalized conditional gradient-alternating minimization (GCG-ALTMIN) algorithm. This LRMS-based solution can accommodate different precoder/combiner and training structures. In addition, it does not require knowledge about the array responses at the transceivers, in contrast to most existing solutions allowing low training overhead. Furthermore, a preconditioned conjugate gradient (PCG) algorithm-based implementation and a low-rank matrix completion (LRMC) formulation are proposed to further reduce the computational complexity. In order to enhance the channel estimation performance for fat and tall channel matrices, we introduce a matrix reshaping approach that can preserve the channel rank by exploiting the shift-invariance property of uniform arrays. We also introduce a spectrum denoising (SD) approach for further improving the performance when the array responses are known and the number of paths is small. These approaches can effectively enhance the performance at a given training overhead. Simulation results suggest that the proposed solutions can achieve higher channel estimation accuracy and reduce the computational complexity as compared to several representative channel estimation schemes.

A novel algorithm to remove rain or snow streaks from a video sequence using temporal correlation and low-rank matrix completion is proposed in this paper. Based on the observation that rain streaks are too small and move too fast to affect the optical flow estimation between consecutive frames, we obtain an initial rain map by subtracting temporally warped frames from a current frame. Then, we decompose the initial rain map into basis vectors based on the sparse representation, and classify those basis vectors into rain streak ones and outliers with a support vector machine. We then refine the rain map by excluding the outliers. Finally, we remove the detected rain streaks by employing a low-rank matrix completion technique. Furthermore, we extend the proposed algorithm to stereo video deraining. Experimental results demonstrate that the proposed algorithm detects and removes rain or snow streaks efficiently, outperforming conventional algorithms.

Low-rank matrix completion is a hot topic in the field of machine learning. It is widely used in image processing, recommendation systems and subspace clustering. However, the traditional method uses the nuclear norm to approximate the rank function, which leads to only the suboptimal solution. Inspired by the closed-form formulation of L_{2/3} regularization, we propose a new truncated schatten 2/3-norm to approximate the rank function. Our proposed regularizer takes full account of the prior rank information and achieves a more accurate approximation of the rank function. Based on this regularizer, we propose a new low-rank matrix completion model. Meanwhile, a fast and efficient algorithm are designed to solve the proposed model. In addition, a rigorous mathematical analysis of the convergence of the proposed algorithm is provided. Finally, the superiority of our proposed model and method is investigated on synthetic data and recommender system datasets. All results show that our proposed algorithm is able to achieve comparable recovery performance while being faster and more efficient than state-of-the-art methods.

This paper studies the low-rank matrix completion problem from an information theoretic perspective. The completion problem is rephrased as a communication problem of an (uncoded) low-rank matrix source over an erasure channel. The paper then uses achievability and converse arguments to present order-wise optimal bounds for the completion problem.

Low-rank matrix completion (LRMC) problems arise in a wide variety of applications. Previous theory mainly provides conditions for completion under missing-at-random samplings. This paper studies deterministic conditions for completion. An incomplete d × N matrix is finitely rank-r completable if there are at most finitely many rank-r matrices that agree with all its observed entries. Finite completability is the tipping point in LRMC, as a few additional samples of a finitely completable matrix guarantee its unique completability. The main contribution of this paper is a characterization of finitely completable observation sets. We use this characterization to derive sufficient deterministic sampling conditions for unique completability. We also show that under uniform random sampling schemes, these conditions are satisfied with high probability if O(max{r; log d}) entries per column are observed.