Rank-one Matrix Research Articles

IRREDUCIBLE FAMILIES OF COMPLEX MATRICES CONTAINING A RANK-ONE MATRIX

We show that an irreducible family ${\mathcal{S}}$ of complex $n\times n$ matrices satisfies Paz’s conjecture if it contains a rank-one matrix. We next investigate properties of families of rank-one matrices. If ${\mathcal{R}}$ is a linearly independent, irreducible family of rank-one matrices then (i) ${\mathcal{R}}$ has length at most $n$, (ii) if all pairwise products are nonzero, ${\mathcal{R}}$ has length 1 or 2, (iii) if ${\mathcal{R}}$ consists of elementary matrices, its minimum spanning length $M$ is the smallest integer $M$ such that every elementary matrix belongs to the set of words in ${\mathcal{R}}$ of length at most $M$. Finally, for any integer $k$ dividing $n-1$, there is an irreducible family of elementary matrices with length $k+1$.

Rank-One Matrix Approximation With ℓ p -Norm for Image Inpainting

In the problem of image inpainting, one popular approach is based on low-rank matrix completion. Compared with other methods which need to convert the image into vectors or dividing the image into patches, matrix completion operates on the whole image directly. Therefore, it can preserve latent information of the two-dimensional image. An efficient method for low-rank matrix completion is to employ the matrix factorization technique. However, conventional low-rank matrix factorization-based methods often require a prespecified rank, which is challenging to determine in practice. The proposed method factorizes an image matrix as a sum of rank-one matrices so that it does not require rank information in advance as it can be automatically estimated by the algorithm itself when the algorithm has satisfactorily converged. In our study, matching pursuit is applied to search for the best rank-one matrix at each iteration. To be robust against impulsive noise, the residual error between the observed and estimated matrices is minimized by $\ell _p$ -norm with $0 . Then the resultant $\ell _p$ -norm minimization is solved by the iteratively reweighted least squares method. The proposed model is beneficial for the robustness against outliers, and does not require rank information. Experimental results verify the effectiveness and higher accuracy of the proposed method with comparison to several state-of-the-art matrix completion-based image inpainting approaches.

Chasing after flavor symmetries of quarks from the bottom up

AbstractWe explore a flavor structure of quarks in the standard model under the assumption that flavor symmetries exist in a theory beyond the standard model, and seek their properties, using a bottom-up approach. We reacknowledge that a flavor-symmetric part of the Yukawa coupling matrix can be realized by a rank-one matrix, and a democratic-type matrix occupies a special position, based on Dirac’s naturalness.

Persistent Flows in Deterministic Chains

This paper studies, the role of persistent flows in the convergence of infinite backward products of stochastic matrices of deterministic chains over networks with nonreciprocal interactions between agents. An arc describing the interaction strength between two agents is said to be persistent if its weight function has an infinite $l_1$ norm; convergence of the infinite backward products to a rank-one matrix of a deterministic chain of stochastic matrices is equivalent to achieving consensus at the node states. We discuss two balance conditions on the interactions between agents, which generalize the arc-balance and cut-balance conditions in the literature, respectively. The proposed conditions require that such a balance should be satisfied over each time window of a fixed length instead of at each time instant. We prove that in both cases global consensus is reached if and only if the persistent graph, which consists of all the persistent arcs, contains a directed spanning tree. The convergence rates of the system to consensus are also provided in terms of the interactions between agents having taken place. The results are obtained under a weak condition without assuming the existence of a positive lower bound of all the nonzero weights of arcs and are compared with the existing results. Illustrative examples are provided to validate the results and show the critical importance of the nontrivial lower boundedness of the self-confidence of the agents.

Robust generalized sidelobe canceller based on eigenanalysis and a MaxSINR beamformer

A robust generalized sidelobe canceller is proposed to combat direction of arrival (DOA) mismatches. To estimate the interference-plus-noise (IPN) statistics characteristics, conventional signal of interest (SOI) extraction methods usually collect a large number of segments where only the IPN signal is active. To avoid that collection procedure, we redesign the blocking matrix structure using an eigenanalysis method to reconstruct the IPN covariance matrix from the samples. Additionally, a modified eigenanalysis reconstruction method based on the rank-one matrix assumption is proposed to achieve a higher reconstruction accuracy. The blocking matrix is obtained by incorporating the effective reconstruction into the maximum signal-to-interferenceplus- noise ratio (MaxSINR) beamformer. It can minimize the influence of signal leakage and maximize the IPN power for further noise and interference suppression. Numerical results show that the two proposed methods achieve considerable improvements in terms of the output waveform SINR and correlation coefficients with the desired signal in the presence of a DOA mismatch and a limited number of snapshots. Compared to the first proposed method, the modified one can reduce the signal distortion even further.

The adaptive interpolation method for proving replica formulas. Applications to the Curie–Weiss and Wigner spike models

In this contribution we give a pedagogic introduction to the newly introduced adaptive interpolation method to prove in a simple and unified way replica formulas for Bayesian optimal inference problems. Many aspects of this method can already be explained at the level of the simple Curie–Weiss spin system. This provides a new method of solution for this model which does not appear to be known. We then generalize this analysis to a paradigmatic inference problem, namely rank-one matrix estimation, also refered to as the Wigner spike model in statistics. We give many pointers to the recent literature where the method has been succesfully applied.

A method for recovering Jacobi matrices with mixed spectral data

In this paper we employ the Euclidean division for polynomials to recover uniquely a Jacobi matrix in terms of the mixed spectral data consisting of its partial entries and the information given on its full spectrum together with a subset of eigenvalues of its truncated matrix obtained by deleting the last row and last column, or its rank-one modification matrix modified by adding a constant to the last element. A necessary and sufficient condition is provided for the existence of the inverse problem. A numerical algorithm and a numerical example are given.

An improved total variation regularized RPCA for moving object detection with dynamic background

Dynamic background extraction has been a fundamental research topic in video analysis. In this paper, a novel robust principal component analysis (RPCA) approach for foreground extraction is proposed by decomposing video frames into three parts: rank-one static background, dynamic background and sparse foreground. First, the static background is represented by a rank-one matrix, which can avoid the computation of singular value decomposition because usually the dimensionality of a surveillance video is very large. Secondly, the dynamic background is characterized by \begin{document}$ \ell_{2,1} $\end{document} -norm, which can exploit the shared information. Thirdly, the sparse foreground is enhanced by total variation, which can preserve sharp edges that are usually the most important for clear object extraction. An efficient symmetric Gauss-Seidel based alternating direction method of multipliers (sGS-ADMM) is established with convergence analysis. Extensive experiments on real-world datasets show that our proposed approach outperforms the existing state-of-the-art approaches. In fact, to the best of our knowledge, this is the first time to integrate the joint sparsity and total variation into a RPCA framework, which has demonstrated the superiority of performance.

Generalized Sarymsakov Matrices

Within the set of stochastic, indecomposable, aperiodic (SIA) matrices, the class of Sarymsakov matrices is the largest known subset that is closed under matrix multiplication, and more critically whose compact subsets are all consensus sets. This paper shows that a larger subset with these two properties can be obtained by generalizing the standard definition for Sarymsakov matrices. The generalization is achieved by introducing the notion of the SIA index of a stochastic matrix, whose value is 1 for Sarymsakov matrices, and then exploring those stochastic matrices with larger SIA indices. In addition to constructing the larger set, this paper introduces another class of generalized Sarymsakov matrices, which contains matrices that are not SIA, and studies their products. Sufficient conditions are provided for an infinite product of matrices from this class, converging to a rank-one matrix. Finally, as an application of the results just described and to confirm their usefulness, a necessary and sufficient combinatorial condition, the “avoiding set condition,” for deciding whether or not a compact set of stochastic matrices is a consensus set is revisited. In addition, a necessary and sufficient combinatorial condition is established for deciding whether or not a compact set of doubly stochastic matrices is a consensus set.

Alternating Minimization Algorithm for Polynomial Optimal Control Problems

Many optimal control problems can be cast into polynomial optimization problems through the discretization and conversion of expressions. The polynomial programming problem can further be transformed into a general quadratically constrained quadratic programing (QCQP) problem by introducing new variables and equality constraints. This paper develops an alternating minimization algorithm (AMA) to search for the optimal solution to a QCQP that is formulated as a rank-one constrained optimization problem. Based on the fact that a rank-one matrix is formed by two equivalent vectors, AMA alternatively solves each vector in sequence. Each sequential problem is a convex quadratic programming problem with linear constraints. A convergence analysis of AMA is provided. The efficacy of the proposed AMA is demonstrated by numerically solving two constrained optimal control problems where the conventional approach based on nonlinear programming experiences difficulty.

Line survey joint denoising via low-rank minimization

Prestack seismic data denoising is an important step in seismic processing due to the development of prestack time migration. Reduced-rank filtering is a state-of-the-art method for prestack seismic denoising that uses predictability between neighbor traces for each single frequency. Different from the original way of embedding low-rank matrix based on the Hankel or Toeplitz transform, we have developed a new multishot gathers joint denoising method in a line survey, which used a new way of rearranging data to a matrix with low rank. Inspired by video denoising, each single-shot record in the line survey can be viewed as a frame in the video sequence. Due to high redundancy and similar event structure among the shot gathers, similar patches can be selected from different shot gathers in the line survey to rearrange a low-rank matrix. Then, seismic denoising is formulated into a low-rank minimization problem that can be further relaxed into a nuclear-norm minimization problem. A fast algorithm, called the orthogonal rank-one matrix pursuit, is used to solve the nuclear-norm minimization. Using this method avoids the computation of a full singular value decomposition. Our method is validated using synthetic and field data, in comparison with [Formula: see text] deconvolution and singular spectrum analysis methods.

A new method based on the manifold-alternative approximating for low-rank matrix completion

In this paper, a new method is proposed for low-rank matrix completion which is based on the least squares approximating to the known elements in the manifold formed by the singular vectors of the partial singular value decomposition alternatively. The method can achieve a reduction of the rank of the manifold by gradually reducing the number of the singular value of the thresholding and get the optimal low-rank matrix. It is proven that the manifold-alternative approximating method is convergent under some conditions. Furthermore, compared with the augmented Lagrange multiplier and the orthogonal rank-one matrix pursuit algorithms by random experiments, it is more effective as regards the CPU time and the low-rank property.

Complex-Domain Super MDS: A New Framework for Wireless Localization With Hybrid Information

We revisit the super multidimensional scaling (SMDS) wireless localization algorithm first proposed a decade ago, recasting it onto the complex-domain. Under this new formulation, the edge kernel, which carries both angle and distance information simultaneously and plays a central role in the SMDS algorithm, becomes a complex-valued rank-one matrix, resulting in a new complex-domain SMDS framework, which yields several advantages over the original, including the elimination of redundancy, the enhancement of conditions to handle information erasure, and the possibility of designing several algorithmic variations that offer different complexity/performance improvements. To cite some concrete results, it is shown, for instance, that a distance-based localization system with 20 targets employing one of the new algorithms dubbed complex-domain SMDS (CD-SDMDS) outperforms the original SMDS at a tenth of the computational cost, and that the handling of missing information via matrix completion is superior in CD-SMDS compared to SMDS. If the same network collects also anchor-to-target angle information, it is furthermore shown that localization is achieved with nearly $20\times $ lower complexity and still higher accuracy using another new algorithm dubbed Turbo MRC-SMDS, and over $25\times $ faster using yet another method dubbed Iterative MRC-SMDS, with only a slight degradation.

The adaptive interpolation method: a simple scheme to prove replica formulas in Bayesian inference

In recent years important progress has been achieved towards proving the validity of the replica predictions for the (asymptotic) mutual information (or “free energy”) in Bayesian inference problems. The proof techniques that have emerged appear to be quite general, despite they have been worked out on a case-by-case basis. Unfortunately, a common point between all these schemes is their relatively high level of technicality. We present a new proof scheme that is quite straightforward with respect to the previous ones. We call it the adaptive interpolation method because it can be seen as an extension of the interpolation method developped by Guerra and Toninelli in the context of spin glasses, with an interpolation path that is adaptive. In order to illustrate our method we show how to prove the replica formula for three non-trivial inference problems. The first one is symmetric rank-one matrix estimation (or factorisation), which is the simplest problem considered here and the one for which the method is presented in full details. Then we generalize to symmetric tensor estimation and random linear estimation. We believe that the present method has a much wider range of applicability and also sheds new insights on the reasons for the validity of replica formulas in Bayesian inference.

Identifiability of Multichannel Blind Deconvolution and Nonconvex Regularization Algorithm

In this paper, we consider the multichannel blind deconvolution problem, where we observe the output of channels $\mathbf {h}_{i}\in \mathbb {R}^n$ ( $i=1,...,N$ ) that all convolve with the same unknown input signal $\mathbf {x}\in \mathbb {R}^n$ . We wish to estimate the input signal and blur kernels simultaneously. Existing theoretical results showed that the original inputs are identifiable under subspace assumptions. However, the subspaces discussed before were randomly or generically chosen. Here, we propose deterministic subspace assumption, which is widely used in practice, and give some theoretical results. First of all, we derive tight sufficient condition for identifiability of signal and convolution kernels, which is only violated on a set of Lebesgue measure zero. Then, we present a nonconvex regularization algorithm by a lifting method and approximate the rank-one constraint via the difference of nuclear norm and Frobenius norm. The global minimizer of the proposed nonconvex algorithm is rank-one matrix under mild conditions on parameters and noise level. The stability result is also shown under the assumption that the inputs lie in a compact set. Besides, the computation of our regularization model is carried out and any limit point of iterations converges to a stationary point of our model. Finally, we provide numerical experiments to show that our nonconvex regularization model outperforms convex relaxation models, such as nuclear norm minimization and some nonconvex methods, such as alternating minimization method and spectral method.

A subspace-approximating algorithm for matrix completion

In this paper, we propose a new subspace-approximating algorithm for matrix completion based on the subspace-approximating of the singular value decomposition.Then we use quadratic programming to produce the closest and the best feasible matrix in the subspace. This algorithm can achieve the reduction of the rank of the subspace by gradually reducing the number of the singular value of the thresholding and get the optimal low-rank matrix. It is proved that the subspace-approximating algorithm is convergent under some conditions. Besides, compared with the augmented Lagrange multiplier algorithm and orthogonal rank-one matrix pursuit algorithm by random experiments,the proposed algorithm is more effective in the CPU time and the low-rank property.

Near-Optimal Compressed Sensing of a Class of Sparse Low-Rank Matrices Via Sparse Power Factorization

Compressed sensing of simultaneously sparse and low-rank matrices enables recovery of sparse signals from a few linear measurements of their bilinear form. One important question is how many measurements are needed for a stable reconstruction in the presence of measurement noise. Unlike conventional compressed sensing for sparse vectors, where convex relaxation via the $\ell _{1}$ -norm achieves near-optimal performance, for compressed sensing of sparse low-rank matrices, it has been shown recently that convex programmings using the nuclear norm and the mixed norm are highly suboptimal even in the noise-free scenario. We propose an alternating minimization algorithm called sparse power factorization (SPF) for compressed sensing of sparse rank-one matrices. For a class of signals whose sparse representation coefficients are fast-decaying, SPF achieves stable recovery of the rank-one matrix formed by their outer product and requires number of measurements within a logarithmic factor of the information-theoretic fundamental limit. For the recovery of general sparse low-rank matrices, we propose subspace-concatenated SPF (SCSPF), which has analogous near-optimal performance guarantees to SPF in the rank-one case. Numerical results show that SPF and SCSPF empirically outperform convex programmings using the best known combinations of mixed norm and nuclear norm.

Blind Image Deblurring Using Row–Column Sparse Representations

Blind image deblurring is a particularly challenging inverse problem where the blur kernel is unknown and must be estimated en route to recover the deblurred image. The problem is of strong practical relevance since many imaging devices such as cellphone cameras, must rely on deblurring algorithms to yield satisfactory image quality. Despite significant research effort, handling large motions remains an open problem. In this paper, we develop a new method called Blind Image Deblurring using Row-Column Sparsity (BD-RCS) to address this issue. Specifically, we model the outer product of kernel and image coefficients in certain transformation domains as a rank-one matrix, and recover it by solving a rank minimization problem. Our central contribution then includes solving {\em two new} optimization problems involving row and column sparsity to automatically determine blur kernel and image support sequentially. The kernel and image can then be recovered through a singular value decomposition (SVD). Experimental results on linear motion deblurring demonstrate that BD-RCS can yield better results than state of the art, particularly when the blur is caused by large motion. This is confirmed both visually and through quantitative measures.

Iterative reconstruction of rank-one matrices in noise

Abstract We consider the problem of estimating a rank-one matrix in Gaussian noise under a probabilistic model for the left and right factors of the matrix. The probabilistic model can impose constraints on the factors including sparsity and positivity that arise commonly in learning problems. We propose a family of algorithms that reduce the problem to a sequence of scalar estimation computations. These algorithms are similar to approximate message-passing techniques based on Gaussian approximations of loopy belief propagation that have been used recently in compressed sensing. Leveraging analysis methods by Bayati and Montanari, we show that the asymptotic behavior of the algorithm is described by a simple scalar equivalent model, where the distribution of the estimates at each iteration is identical to certain scalar estimates of the variables in Gaussian noise. Moreover, the effective Gaussian noise level is described by a set of state evolution equations. The proposed approach to deriving algorithms thus provides a computationally simple and general method for rank-one estimation problems with a precise analysis in certain high-dimensional settings.

A Novel Image Registration Method Based on Phase Correlation Using Low-Rank Matrix Factorization With Mixture of Gaussian

Image registration is a critical process for the various applications in the remote sensing community, and its accuracy greatly affects the results of the subsequent applications. Image registration based on phase correlation has been widely concerned due to its robustness to gray differences and efficiency. After calculating the normalized cross-relation matrix $Q$ , the most commonly used approach is fitting the 2-D phase plane that passes through the origin, but it needs to remove contaminated spectrum carefully and the corresponding parameters are empirical. In fact, the phase correlation matrix is rank one for a noise-free translation model. This property simplifies the matching problem to finding the best rank-one approximation of the normalized cross-relation matrix. We develop a novel algorithm that performs the rank-one matrix factorization on the phase correlation matrix by assuming its noise as mixture of Gaussian (MoG) distributions. The MoG model is a general approximator for any continuous distribution, and hence is able to model a wide range of noise distribution. The parameters of the MoG model can be evaluated under the framework of maximum likelihood estimation by using an expectation-maximization method, and the subspace is calculated with standard methods. The advantages of the algorithm, high accuracy, and robustness to aliasing, noise, gray difference, and occlusions are illustrated by a series of simulated and real-image experiments.