Matrix Rank Minimization Research Articles

A novel ship trajectory reconstruction approach based on low-rank tensor completion

The Automatic Identification System (AIS) enhances maritime safety and environmental monitoring by providing crucial ship trajectory data. However, this data is often compromised by missing or abnormal values due to signal blockage, transmission errors, and equipment failures, jeopardizing safety and the accuracy of environmental analyses. Addressing these challenges, we propose a novel Customized Tensor-Based Maritime Trajectory Reconstruction (CTMTR) framework that leverages the self-similarity of ship trajectories to reformulate trajectory reconstruction as a matrix rank minimization issue. The CTMTR framework consists of three steps: data preprocessing, trajectory matrix construction, and trajectory reconstruction. We conducted simulation experiments using a dataset comprising vessel trajectories from the east coast of Dover in the United States in 2022. To validate its effectiveness, the CTMTR is compared with four advanced methods (LRMC, LRTC-TNN, TRPCA, and TNN-DCT) under diverse missing and anomalous scenarios. The results substantiate that the performance of CTMTR outperforms other approaches, especially in anomalous cases. Our method outperforms existing methods by an order of magnitude under random missing combined with anomaly scenarios. The CTMTR framework thus has the potential to advance maritime trajectory reconstruction methodologies, providing a solid foundation for future innovations in maritime data analysis and navigation safety technologies.

Nonlocal Matrix Rank Minimization Method for Multiplicative Noise Removal

Nonlocal Matrix Rank Minimization Method for Multiplicative Noise Removal

Image cartoon-texture decomposition by a generalized non-convex low-rank minimization method

Image cartoon-texture decomposition is an important problem in image processing. In recent years, by exploiting low-rank priors of images, low-rank minimization methods have been widely adopted for image cartoon-texture decomposition. Since matrix rank minimization is an NP-hard problem, the convex nuclear norm is often used as a substitute for the matrix’s rank to realize the low-rank minimization methods. In this paper, we utilize a generalized non-convex surrogate of the matrix rank function to develop a novel low-rank minimization model for image cartoon-texture decomposition. We design a proximal alternating algorithm to solve the non-convex model and further demonstrate the global convergence of the algorithm. Numerical experiments illustrate that the proposed method can show much better performances than the existing state-of-the-art methods for image cartoon-texture decomposition.

Low Tucker rank tensor completion using a symmetric block coordinate descent method

AbstractLow Tucker rank tensor completion has wide applications in science and engineering. Many existing approaches dealt with the Tucker rank by unfolding matrix rank. However, unfolding a tensor to a matrix would destroy the data's original multi‐way structure, resulting in vital information loss and degraded performance. In this article, we establish a relationship between the Tucker ranks and the ranks of the factor matrices in Tucker decomposition. Then, we reformulate the low Tucker rank tensor completion problem as a multilinear low rank matrix completion problem. For the reformulated problem, a symmetric block coordinate descent method is customized. For each matrix rank minimization subproblem, the classical truncated nuclear norm minimization is adopted. Furthermore, temporal characteristics in image and video data are introduced to such a model, which benefits the performance of the method. Numerical simulations illustrate the efficiency of our proposed models and methods.

Multivariable Robust Proportional-Integral-Derivative Control for Linear Quadratic Compensation of a Class of Norm-Bounded Uncertain Systems

AbstractThis paper is concerned with designing multi-input multi-output (MIMO) proportional-integral-derivative (PID) control having filtered derivative terms to achieve optimal linear quadratic performance for a class of linear uncertain MIMO systems with norm-bounded time-varying uncertainties. A necessary and sufficient condition for existence of the PID controller is obtained in terms of rank constrained linear matrix inequalities (LMIs). Using an existing penalty-function-based approximation for matrix rank minimization, an algorithm is proposed to solve these rank constrained LMIs and find PID gains. A suitable numerical example is considered to show the efficacy of the proposed approach in ensuring robust performance as compared to the existing ones. The proposed algorithm is also tested on several benchmark examples and found to be computationally much more efficient than some well-known methods.

A smoothing proximal gradient algorithm for matrix rank minimization problem

In this paper, we study the low-rank matrix minimization problem, where the loss function is convex but nonsmooth and the penalty term is defined by the cardinality function. We first introduce an exact continuous relaxation, that is, both problems have the same minimizers and the same optimal value. In particular, we introduce a class of lifted stationary points of the relaxed problem and show that any local minimizer of the relaxed problem must be a lifted stationary point. In addition, we derive lower bound property for the nonzero singular values of the lifted stationary point and hence also of the local minimizers of the relaxed problem. Then the smoothing proximal gradient (SPG) algorithm is proposed to find a lifted stationary point of the continuous relaxation model. Moreover, it is shown that any accumulating point of the sequence generated by SPG algorithm is a lifted stationary point. At last, numerical examples show the efficiency of the SPG algorithm.

Local low-rank approach to nonlinear matrix completion

This paper deals with a problem of matrix completion in which each column vector of the matrix belongs to a low-dimensional differentiable manifold (LDDM), with the target matrix being high or full rank. To solve this problem, algorithms based on polynomial mapping and matrix-rank minimization (MRM) have been proposed; such methods assume that each column vector of the target matrix is generated as a vector in a low-dimensional linear subspace (LDLS) and mapped to a pth order polynomial and that the rank of a matrix whose column vectors are dth monomial features of target column vectors is deficient. However, a large number of columns and observed values are needed to strictly solve the MRM problem using this method when p is large; therefore, this paper proposes a new method for obtaining the solution by minimizing the rank of the submatrix without transforming the target matrix, so as to obtain high estimation accuracy even when the number of columns is small. This method is based on the assumption that an LDDM can be approximated locally as an LDLS to achieve high completion accuracy without transforming the target matrix. Numerical examples show that the proposed method has a higher accuracy than other low-rank approaches.

A Note on a Lower Bound on the Minimum Rank of a Positive Semidefinite Hankel Matrix Rank Minimization Problem

This paper investigates the problem of approximating the global minimum of a positive semidefinite Hankel matrix minimization problem with linear constraints. We provide a lower bound on the objective of minimizing the rank of the Hankel matrix in the problem based on conclusions from nonnegative polynomials, semi-infinite programming, and the dual theorem. We prove that the lower bound is almost half of the number of linear constraints of the optimization problem.

Multiple Subspace Model and Image-Inpainting Algorithm Based on Multiple Matrix Rank Minimization

Multiple Subspace Model and Image-Inpainting Algorithm Based on Multiple Matrix Rank Minimization

Form-finding of tensegrity structures via rank minimization of force density matrix

This study proposes a general computational framework for the form-finding of tensegrity structures. The procedure is divided into two stages in which the member force densities and nodal coordinates are obtained respectively. In the first stage, the determination of force densities is transformed into a rank minimization problem regarding the force density matrix and then formulated into a semi-definite programming. The unilaterality condition of member forces and the positive semi-definiteness condition of force density matrix are incorporated as constraints. In the second stage, the determination of nodal coordinates is formulated into a constrained nonlinear programming model. The nodal positions and member lengths are assigned as constraints and auxiliary variables are introduced to homogenize the member lengths. The proposed formulation bypasses the number of self-stress states in a tensegrity structure thus applies to both tensegrity structures with single and multiple self-stress states. Different from existing studies that are based on the minimum required rank deficiency condition of force density matrix, the proposed method can handle tensegrity structures that have a force density matrix with rank deficiency greater than the required minimum number. Several examples are presented to verify the effectiveness of the proposed method on different types of tensegrity structures.

Frontal face modeling using morphing-based averaging and Low-rank decomposition

Head pose variations are a major problem in face recognition. Many advanced methods exist for synthesizing frontal face images with head pose variations. These methods are generally categorized into three groups: 3D based methods which use 3D face models, path based methods which exploit linear transformations of small segments of the face, and deep learning methods. Frontalization methods usually use a single head pose image for frontal face synthesizing and heuristics for filling the hidden parts of the face in the reference model. This causes some artifact in the image, and reduces similarity between the synthesized and actual frontal face image. In this research, frontalization is done by a weighted averaging algorithm, as an extended version of the path based frontalization method powered by matrix rank minimization, which exploits from classical image domain transform. This preprocessing technique grants matrix decomposition methods the ability to eliminate head pose variations. Representing the actual face models and robustness to facial landmark detection errors are the advantages of this method compared to other existing methods. Quantitative and qualitative comparison results indicate superiority of the proposed method in face frontalization.

Guarantees of Fast Band Restricted Thresholding Algorithm for Low-Rank Matrix Recovery Problem

Affine matrix rank minimization problem is a famous problem with a wide range of application backgrounds. This problem is a combinatorial problem and deemed to be NP-hard. In this paper, we propose a family of fast band restricted thresholding (FBRT) algorithms for low rank matrix recovery from a small number of linear measurements. Characterized via restricted isometry constant, we elaborate the theoretical guarantees in both noise-free and noisy cases. Two thresholding operators are discussed and numerical demonstrations show that FBRT algorithms have better performances than some state-of-the-art methods. Particularly, the running time of FBRT algorithms is much faster than the commonly singular value thresholding algorithms.

Design strategy of thresholding operator for low-rank matrix recovery problem

This paper proposes the band restricted thresholding (BRT) operator and studies theoretically the performance of the corresponding singular value thresholding algorithms for the affine matrix rank minimization problem. We first introduce a new notion named matrix-type generalized restricted isometry constant (MgRIC) as a generalization of restricted isometry constant (RIC). Then we show the stability results characterized via gRIC and RIC. Finally, we design a new thresholding operator and numerical demonstration on random low-rank matrix completion problems and image inpainting problems have been shown that the design strategy of thresholding operator is effective.

Low-Rank Matrix Recovery via Modified Schatten-p Norm Minimization with Convergence Guarantees.

In recent years, low-rank matrix recovery problems have attracted much attention in computer vision and machine learning. The corresponding rank minimization problems are both combinational and NP-hard in general, which are mainly solved by both nuclear norm and Schatten-p (0<p<1) norm based optimization algorithms. However, inspired by weighted nuclear norm and Schatten-p norm as the relaxations of rank function, the main merits of this work firstly provide a modified Schatten-p norm in the affine matrix rank minimization problem, denoted as the modified Schatten-p norm minimization (MSpNM). Secondly, its surrogate function is constructed and the equivalence relationship with the MSpNM is further achieved. Thirdly, the iterative singular value thresholding algorithm (ISVTA) is devised to optimize it, and its accelerated version, i.e., AISVTA, is also obtained to reduce the number of iterations through the well-known Nesterov's acceleration strategy. Most importantly, the convergence guarantees and their relationship with objective function, stationary point and variable sequence generated by the proposed algorithms are established under some specific assumptions, e.g., Kurdyka-Łojasiewicz (KŁ) property. Finally, numerical experiments demonstrate the effectiveness of the proposed algorithms in the matrix completion problem for image inpainting and recommender systems. It should be noted that the accelerated algorithm has a much faster convergence speed and a very close recovery precision when comparing with the proposed non-accelerated one.

Homotopy method for matrix rank minimization based on the matrix hard thresholding method

Based on the matrix hard thresholding method, a homotopy method is proposed for solving the matrix rank minimization problem. This method iteratively solves a series of regularization subproblems, whose solutions are given in closed form by the matrix hard thresholding operator. Under some mild assumptions, convergence of the proposed method is proved. The proposed method does not depend on a prior knowledge of exact rank value. Numerical experiments demonstrate that the proposed homotopy method weakens the affection of the choice of the regularization parameter, and is more efficient and effective than the existing sate-of-the-art methods.

Exact recovery low-rank matrix via transformed affine matrix rank minimization

Abstract The goal of affine matrix rank minimization problem is to reconstruct a low-rank or approximately low-rank matrix under linear constraints. In general, this problem is combinatorial and NP-hard. In this paper, a nonconvex fraction function is studied to approximate the rank of a matrix and translate this NP-hard problem into a transformed affine matrix rank minimization problem. The equivalence between these two problems is established, and we proved that the uniqueness of the global minimizer of transformed affine matrix rank minimization problem also solves affine matrix rank minimization problem if some conditions are satisfied. Moreover, we also proved that the optimal solution to the transformed affine matrix rank minimization problem can be approximately obtained by solving its regularization problem for some proper smaller λ > 0. Lastly, the DC algorithm is utilized to solve the regularization transformed affine matrix rank minimization problem and the numerical experiments on image inpainting problems show that our method performs effectively in recovering low-rank images compared with some state-of-art algorithms.

Affine matrix rank minimization problem via non-convex fraction function penalty

Affine matrix rank minimization problem is a fundamental problem in many important applications. It is well known that this problem is combinatorial and NP-hard in general. In this paper, a continuous promoting low rank non-convex fraction function is studied to replace the rank function in this NP-hard problem. An iterative singular value thresholding algorithm is proposed to solve the regularization transformed affine matrix rank minimization problem. With the change of the parameter in non-convex fraction function, we could get some much better results, which is one of the advantages for the iterative singular value thresholding algorithm compared with some state-of-art methods. Some convergence results are established. Moreover, we proved that the value of the regularization parameter λ > 0 cannot be chosen too large. Indeed, there exists λ ̄ > 0 such that the optimal solution of the regularization transformed affine matrix rank minimization problem is equal to zero for any λ > λ ̄ . Numerical experiments on matrix completion problems and image inpainting problems show that our method performs effective in finding a low-rank matrix compared with some state-of-art methods.

Restoration of Graph Signals Based on AR Models and Structured Matrix Rank Minimization

Restoration of Graph Signals Based on AR Models and Structured Matrix Rank Minimization

Multiple Matrix Rank Minimization Approach to Audio Declipping

Multiple Matrix Rank Minimization Approach to Audio Declipping

A Rank Minimization Formulation for Identification of Linear Parameter Varying Models

Abstract We explore the problem of identification of LPV models when the scheduling variables are not known in advance and the model parameters exhibit a dynamic dependence on them. We consider an affine ARX model structure whose parameters vary with time. We solve for the model’s parameters and scheduling variables in two steps. In the first step, we use the measured input-output data to realize a parameter trajectory by solving a regularized Hankel matrix rank minimization problem. The regularization penalty is guided by the prior knowledge regarding the nature of system’s time variation. In the second step, the scheduling variables are estimated as parameters of a sparse ARX structure relating the model’s parameters to the measured input-output variables. The effectiveness of the proposed approach is illustrated with two practical examples.