Quaternion adaptive approximation normalization graph guided implicit low rank for robust matrix completion

We study online robust matrix completion on graphs. At each iteration a vector with some entries missing is revealed and our goal is to reconstruct it by identifying the underlying low-dimensional subspace from which the vectors are drawn. We assume there is an underlying graph structure to the data, that is, the components of each vector correspond to nodes of a certain known and fixed graph, and their values are related accordingly. We propose an algorithm that exploits the graph to reconstruct the incomplete data, in the scenario where there is outlier noise. The theoretical properties of the algorithms are studied and numerical experiments using both synthetic and real world datasets verify the improved performance of the proposed technique compared to other state of the art algorithms.

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.

As a convex relaxation of the rank minimization model, the nuclear norm minimization (NNM) problem has been attracting significant research interest in recent years. The standard NNM regularizes each singular value equally, composing an easily calculated convex norm. However, this restricts its capability and flexibility in dealing with many practical problems, where the singular values have clear physical meanings and should be treated differently. In this paper we study the weighted nuclear norm minimization (WNNM) problem, which adaptively assigns weights on different singular values. As the key step of solving general WNNM models, the theoretical properties of the weighted nuclear norm proximal (WNNP) operator are investigated. Albeit nonconvex, we prove that WNNP is equivalent to a standard quadratic programming problem with linear constrains, which facilitates solving the original problem with off-the-shelf convex optimization solvers. In particular, when the weights are sorted in a non-descending order, its optimal solution can be easily obtained in closed-form. With WNNP, the solving strategies for multiple extensions of WNNM, including robust PCA and matrix completion, can be readily constructed under the alternating direction method of multipliers paradigm. Furthermore, inspired by the reweighted sparse coding scheme, we present an automatic weight setting method, which greatly facilitates the practical implementation of WNNM. The proposed WNNM methods achieve state-of-the-art performance in typical low level vision tasks, including image denoising, background subtraction and image inpainting.

On robust matrix completion with prescribed eigenvalues

In this work, a novel technique for real-time clutter rejection in ultrasound Color Flow Imaging (CFI) is proposed. Suppressing undesired clutter signal is important because clutter prohibits an unambiguous view of the vascular network. Although conventional eigen-based filters are potentially efficient in suppressing clutter signal, their performance is highly dependent on proper selection of a clutter to blood boundary which is done manually. Herein, we resolve this limitation by formulating the clutter suppression problem as a foreground-background separation problem to extract the moving blood component. To that end, we adapt the fast Robust Matrix Completion (fRMC) algorithm, and utilize the in-face extended Frank-Wolfe method to minimize the rank of the matrix of ultrasound frames. Our method automates the clutter suppression process, which is critical for clinical use. We name the method RAPID (Robust mAtrix decomPosition for suppressIng clutter in ultrasounD) since the automation step can substantially streamline clutter suppression. The technique is validated with simulation, flow phantom and two sets of in-vivo data. RAPID code as well as most of the data in this paper can be downloaded from RAPID.sonography.ai.

Due to the insufficient measurements in the distribution system state estimation (DSSE), full observability and redundant measurements are difficult to achieve without using the pseudo measurements. The matrix completion state estimation (MCSE) combines the matrix completion and power system model to estimate voltage by exploring the low-rank characteristics of the matrix. This paper proposes a robust matrix completion state estimation (RMCSE) to estimate the voltage in a distribution system under a low-observability condition. Tradition state estimation weighted least squares (WLS) method requires full observability to calculate the states and needs redundant measurements to proceed a bad data detection. The proposed method improves the robustness of the MCSE to bad data by minimizing the rank of the matrix and measurements residual with different weights. It can estimate the system state in a low-observability system and has robust estimates without the bad data detection process in the face of multiple bad data. The method is numerically evaluated on the IEEE 33-node radial distribution system. The estimation performance and robustness of RMCSE are compared with the WLS with the largest normalized residual bad data identification (WLS-LNR), and the MCSE.

We study the problem of low-rank and sparse decomposition from possibly noisy observations, known as Robust PCA, where the sparse component can be seen as outliers. We first propose a modified objective function where the nuclear norm captures the low-rank term, $\ell _0$ -“norm” addresses the sparse outlier term, and an $\ell _1$ -norm to deal with the additive noise term. The associated algorithm, termed sparsity regularized principal component pursuit (SRPCP), is shown to converge. Under certain model and algorithm parameter settings, it is shown that SRPCP can recover the low-rank component and sparse component exactly in the noiseless case. In the noisy case, we first prove that the widely used principal component pursuit (PCP) method, which was designed for the noiseless case, is actually stable to dense noise. Then, we show that SRPCP has smaller estimation error bound, and can identify outlier entries without any false alarm. Another important byproduct of our analysis is the result that PCP with missing entries is also stable to dense noise. We further propose another objective function which replaces the above nuclear norm by the log-determinant. The proposed algorithm, termed iterative reweighted sparsity regularized principal component pursuit, is also shown to converge. In each iteration, it solves a weighted nuclear norm regularized robust matrix completion problem. We propose an alternating direction method of multipliers algorithm to solve this nonconvex subproblem, which also converges. Empirical studies demonstrate the efficacy of the proposed $\ell _{0}$ - $\ell _{1}$ regularization framework to deal with the outliers as well as its advantage over the existing state-of-the-art methods.

This paper proposes a robust matrix completion method to solve the localization problem in wireless sensor networks. A novel cost function is formulated which inherently copes with missing measures and corrupted data. In particular, the proposed algorithm robustly completes the range map between pairs of sensors by casting the problem as a low-rank and sparse matrix decomposition, while constraining the solution to be close to an Euclidean Distance Matrix. Numerical accuracy and computational efficiency are demonstrated by synthetic experiments. The empirical results also show that our method outperforms state-of-the art algorithms in several scenarios.

We propose a factorized robust matrix completion (FRMC) algorithm with global motion compensation to solve the video background subtraction problem. The algorithm decomposes a sequence of video frames into the sum of a low rank background component and a sparse motion component. The algorithm alternates between the solution of each component following a Pareto curve trajectory for each subproblem. For videos with moving background, we utilize the motion vectors extracted from the coded video bitstream to compensate for the change in the camera perspective. Performance evaluations show that our approach is faster than state-of-the-art solvers and results in highly accurate motion segmentation.

Automotive radar plays an important role in the field of advanced driver assistant systems (ADASs), in the detection of unmanned aerial vehicles (UAV), and so on. Most automotive radar products can only estimate the azimuth, and however, it is necessary for automatic driving technology to acquire height information. In this article, we propose a signal processing algorithm based on the 2-D sparse multiple-input–multiple-output (MIMO) antenna array, which can detect range, velocity, azimuth, and elevation of scattering points, and this location information can form a clear point-cloud image. First, a 2-D hybrid sparse array is designed with the advantage of reducing hardware resources, such as the number of transceiver channels. Based on the sparse antenna array, a robust matrix completion recovery algorithm, combining the low-rankness of the snapshot and the sparsity of the grid map, is further proposed to extract azimuth and elevation, and alleviate the impact of grating lobes on them. Sequentially, azimuth and elevation of targets can be shown through the correction of motion-induced phase in time-division MIMO (time-division multiplexing (TDM)-MIMO) radar and by the 2-D fast Fourier transform (2DFFT) digital beamforming (DBF). Finally, many experiments validate that a point-cloud image, such as LiDAR, can be shown, which will be hopefully applied for ADASs, UAVs, and other fields.

We consider the problem of removing and replacing clouds in satellite image sequences, which has a wide range of applications in remote sensing. Our approach first detects and removes the cloud-contaminated part of the image sequences. It then recovers the missing scenes from the clean parts using the proposed "TECROMAC" (TEmporally Contiguous RObust MAtrix Completion) objective. The objective function balances temporal smoothness with a low rank solution while staying close to the original observations. The matrix whose the rows are pixels and columns are days corresponding to the image, has low-rank because the pixels reflect land-types such as vegetation, roads and lakes and there are relatively few variations as a result. We provide efficient optimization algorithms for TECROMAC, so we can exploit images containing millions of pixels. Empirical results on real satellite image sequences, as well as simulated data, demonstrate that our approach is able to recover underlying images from heavily cloud-contaminated observations.