Rank Of Matrix Research Articles

Spectra of zero-divisor graphs of finite reduced rings

Let [Formula: see text] be a finite reduced ring with [Formula: see text] maximal ideals [Formula: see text] and let [Formula: see text] be the zero-divisor graph associated to [Formula: see text] The class of rings [Formula: see text] contains the Boolean rings as a subclass. When [Formula: see text] for all [Formula: see text] where [Formula: see text] is a finite field, we associate two [Formula: see text] sized matrices [Formula: see text] and [Formula: see text] to the graph [Formula: see text] having combinatorial entries and use these matrices to determine the spectrum of this graph. More precisely, we show that every eigenvalue of [Formula: see text] and of [Formula: see text] is an eigenvalue of [Formula: see text] To do this, we give a recursive description of the adjacency matrix of this graph and also exhibit its equitable partition. This is used in computing the determinant, rank and nullity of the adjacency matrix. Further, we propose that the eigenvalues of [Formula: see text] [Formula: see text] and the eigenvalue [Formula: see text] exhaust all the eigenvalues of [Formula: see text]

Large factor model estimation by nuclear norm plus [formula omitted] norm penalization

This paper provides a comprehensive estimation framework via nuclear norm plus ℓ1 norm penalization for high-dimensional approximate factor models with a sparse residual covariance. The underlying assumptions allow for non-pervasive latent eigenvalues and a prominent residual covariance pattern. In that context, existing approaches based on principal components may lead to misestimate the latent rank. On the contrary, the proposed optimization strategy recovers with high probability both the covariance matrix components and the latent rank and the residual sparsity pattern. Conditioning on the recovered low rank and sparse matrix varieties, we derive the finite sample covariance matrix estimators with the tightest error bound in minimax sense and we prove that the ensuing estimators of factor loadings and scores via Bartlett’s and Thomson’s methods have the same property. The asymptotic rates for those estimators of factor loadings and scores are also provided.

Structure-Aware Low-Rank Adaptation for Parameter-Efficient Fine-Tuning

With the growing scale of pre-trained language models (PLMs), full parameter fine-tuning becomes prohibitively expensive and practically infeasible. Therefore, parameter-efficient adaptation techniques for PLMs have been proposed to learn through incremental updates of pre-trained weights, such as in low-rank adaptation (LoRA). However, LoRA relies on heuristics to select the modules and layers to which it is applied, and assigns them the same rank. As a consequence, any fine-tuning that ignores the structural information between modules and layers is suboptimal. In this work, we propose structure-aware low-rank adaptation (SaLoRA), which adaptively learns the intrinsic rank of each incremental matrix by removing rank-0 components during training. We conduct comprehensive experiments using pre-trained models of different scales in both task-oriented (GLUE) and task-agnostic (Yelp and GYAFC) settings. The experimental results show that SaLoRA effectively captures the structure-aware intrinsic rank. Moreover, our method consistently outperforms LoRA without significantly compromising training efficiency.

Effective Lower Bounds on the Matrix Rank and Their Applications

Effective Lower Bounds on the Matrix Rank and Their Applications

A Novel Nested Array Structure for DOA Based on CCM of Subarrays

An innovative nested array structure that can notably increase the DOF and the linear arrays’ direction of arrival (DOA) estimated performance is proposed. For the conventional nested array, we vectorize the ACM’s output of the whole array and then remove the repeated rows to get a virtual array’s output. In contrast with classical nested arrays, the superiority of the new nested array is that it utilizes the combination of reverse ordering and conjugation of the inner uniform linear array (ULA) for one of the two subarrays and the sparse outer ULA for another to increase the array aperture. This means the outer sparse ULA’s sensor spacing is approximately the length of the entire inner ULA in the classical nested array structure. Yet, it is about twice the length in the new nested array structure. We generate one lengthy consecutive virtual uniform array excluding any redundant virtual sensors by vectorizing the two subarrays’ cross-correlation matrix (CCM), which is mentioned above. With the export signal of the virtual array and its conjugate form of it, an equivalent covariance matrix of full rank is constructed, which is called the Toeplitz matrix, to compensate for the rank lack of the virtual array’s ACM. For the sake of obtaining the DOA of input signals, the conventional DOA estimation method will be implemented on the ACM. The increase in outer sparse ULA’s sensor spacing ensures an increase in virtual array aperture in principle, which means it is capable of increasing the number of DOA estimations, which is called DOF, and optimizing the DOA estimated performance in contrast with the ULA and classical nested arrays. Results from trials certify the preponderance of the proposed structure in the aspect of DOF and the accuracy of DOA estimation.

Codes arising from directed strongly regular graphs with μ = 1

The rank of adjacency matrix plays an important role in construction of linear codes from a directed strongly regular graph using different techniques, namely, code orthogonality, adjacency matrix determinant and adjacency matrix spectrum. The problem of computing the dimensions of such codes is an intriguing one. Several conjectures to determine the rank of adjacency matrix of a DSRG [Formula: see text] over a finite field, keep researchers working in this area. To address the same to an extent, we have considered the problem of finding the rank over a finite field of the adjacency matrix of a DSRG [Formula: see text] with [Formula: see text], including some mixed Moore graphs and corresponding codes arising from them, in this paper.

Low-rank matrix recovery problem minimizing a new ratio of two norms approximating the rank function then using an ADMM-type solver with applications

Since the nuclear norm may lead to suboptimal solutions to rank minimization problems, many nonconvex surrogates have been proposed to provide better rank function approximations. In this paper, we use the ratio of the nuclear norm and the Frobenius norm, denoted as N/F, as a new nonconvex surrogate of the rank function. The N/F can provide a close approximation for the matrix rank. In particular, the N/F is the same as the rank for rank 1 matrices. Furthermore, compared to other popular nonconvex approximations, the N/F model has no extra parameters to tune. Theoretically, we establish the exact recovery condition and the stable recovery condition for the N/F minimization problem with or without noise, respectively. Computationally, we use the alternating direction method of multipliers with a specific variable-splitting scheme to solve the N/F minimization problem. What is more, we incorporate a box constraint in the N/F model, and propose the ADMM-box to solve it. The convergence analysis of ADMM and ADMM-box are also given. Extensive experiments on both synthetic data and color images verify the effectiveness of our proposed methods.

Background initialization in video data using singular value decomposition and robust principal component analysis

Background initialization is used in video processing applications to extract a scene without the foreground scene. In recent times, the issue of background initialization has gained researchers’ attention in different fields such as video surveillance, video segmentation, computational photography, and so on. The initialization of the background is affected due to different complex dissimilarities such as shadow, intermittent movement, illumination, camera jitter, and clutter. To overcome the aforementioned issues, this paper proposes a decomposition using the combination of the Singular Value Decomposition (SVD) and Robust Principal Component Analysis (RPCA) for Singular Spectrum Analysis (SSA) to perform an effective background initialization. The incorporation of RPCA in SVD is used to overcome the issues related to non-Gaussian noise and it also uses an effective structural knowledge of the video input i.e. sparse and low rank matrix which improves the Peak-Signal-to-Noise-Ratio (PSNR) of the background image. The SBI dataset was used to analyze the performances of SSA-SVDRPCA. The SSA-SVDRPCA is analyzed using MultiScale Structural Similarity Index (MSSSIM), Average gray-level error (AGE), Percentage of clustered error pixels (pCEPS), Percentage of error pixels (pEPs), and PSNR. The existing approaches such as Background Initialization Singular Spectrum Analysis (BISSA) and Quaternion-based Dynamic Mode Decomposition (Q-DMD) are used to compare with the SSA-SVDRPCA method. The PSNR of the SSA-SVDRPCA for Board class is 30.39 dB which is higher than the BISSA and Q-DMD.

Effective Lower Bounds on the Matrix Rank and Their Applications

We propose an efficiently verifiable lower bound on the rank of a sparse fully indecomposable square matrix that contains two non-zero entries in each row and each column. The rank of this matrix is equal to its order or differs from it by one. Bases of a special type are constructed in the spaces of quadratic forms in a fixed number of variables. The existence of these bases allows us to substantiate a heuristic algorithm for recognizing whether a given affine subspace passes through a vertex of a multidimensional unit cube. In the worst case, the algorithm may output a computation denial warning; however, for the general subspace of sufficiently small dimension, it correctly rejects the input. The algorithm is implemented in Python. The running time of its implementation is estimated in the process of testing.

Adaptive Rank-One Matrix Completion Using Sum of Outer Products

Matrix completion refers to recovering a matrix from a small subset of its entries. It is an important topic because numerous real-world data can be modeled as low-rank matrices. One popular approach for matrix completion is based on low-rank matrix factorization, but it requires knowing the matrix rank, which is difficult to accurately determine in many practical scenarios. We propose a novel algorithm based on rank-one approximation that a matrix can be decomposed as a sum of outer products. The key idea is to find the basis vectors of the underlying matrix according to the observed entries, and gradually increase the vector number until an appropriate rank estimate is reached. In contrast to the conventional rank-one schemes that employ unchanging rank-one basis matrices, our algorithm performs completion from the vector viewpoint and is able to generate continuously updated rank-one basis matrices. Besides, we theoretically show that the developed method has a linear convergence rate and a smaller recovery error than existing rank-one based algorithms. Experimental results using both synthetic data and real-world images demonstrate that our solution has the best recovery performance among the competing algorithms when the observations are contaminated by Gaussian noise.

Low Rank Matrix Factorization Algorithm Based on Multi-Graph Regularization for Detecting Drug-Disease Association.

Detecting potential associations between drugs and diseases plays an indispensable role in drug development, which has also become a research hotspot in recent years. Compared with traditional methods, some computational approaches have the advantages of fast speed and low cost, which greatly accelerate the progress of predicting the drug-disease association. In this study, we propose a novel similarity-based method of low-rank matrix decomposition based on multi-graph regularization. On the basis of low-rank matrix factorization with L2 regularization, the multi-graph regularization constraint is constructed by combining a variety of similarity matrices from drugs and diseases respectively. In the experiments, we analyze the difference in the combination of different similarities, resulting that combining all the similarity information on drug space is unnecessary, and only a part of the similarity information can achieve the desired performance. Then our method is compared with other existing models on three data sets (Fdataset, Cdataset and LRSSLdataset) and have a good advantage in the evaluation measurement of AUPR. Besides, a case study experiment is conducted and showing that the superior ability for predicting the potential disease-related drugs of our model. Finally, we compare our model with some methods on six real world datasets, and our model has a good performance in detecting real world data.

Localization of mixed‐field sources based on fourth‐order cumulant matrix rank reduction

AbstractIn this letter, a localization algorithm for mixed‐field sources based on fourth‐order cumulant matrix rank reduction is proposed. First, a cumulant matrix is constructed, and a spectral peak search function associated with the direction‐of‐arrival (DOA) of the sources only is constructed using the rank reduction principle to achieve DOA estimation. Then the range estimation is also obtained by a spectral peak search. Compared to existing algorithms, the algorithm is able to utilize all the elements of the cumulant matrix and has a higher accuracy and degrees‐of‐freedom. The simulation results verify the excellence of the algorithm.

M-IHS: An accelerated randomized preconditioning method avoiding costly matrix decompositions

Momentum Iterative Hessian Sketch (▪) techniques, a group of solvers for large scale regularized linear Least Squares (LS) problems, are proposed and analyzed in detail. The proposed ▪ techniques are obtained by incorporating Polyak's heavy ball acceleration into the Iterative Hessian Sketch algorithm and they provide significant improvements over the randomized preconditioning techniques. By solving the linear systems arising in the sub-problems during the iterations approximately, the proposed techniques are capable of avoiding all matrix decompositions and inversions, which is one of the main advantages over the alternative solvers such as the Blendenpik and the LSRN. Similar to the Chebyshev semi-iterations, the ▪ variants do not use any inner products and eliminate the corresponding synchronization steps in hierarchical or distributed memory systems, yet the ▪ converges faster than the Chebyshev semi-iteration based solvers. Lower bounds on the required sketch size for various randomized distributions are established through the error analyses. Unlike the previously proposed approaches to produce a solution approximation, the proposed ▪ techniques can use sketch sizes that are proportional to the statistical dimension which is always smaller than the rank of the coefficient matrix.

Through the Wall Radar Imaging via Kronecker-structured Huber-type RPCA

The detection of multiple targets in an enclosed scene, from its outside, is a challenging topic of research addressed by Through-the-Wall Radar Imaging (TWRI). Traditionally, TWRI methods operate in two steps: first the removal of wall clutter then followed by the recovery of targets positions. Recent approaches manage in parallel the processing of the wall and targets via low rank plus sparse matrix decomposition and obtain better performances. In this paper, we reformulate this precisely via a RPCA-type problem, where the sparse vector appears in a Kronecker product. We extend this approach by adding a robust distance with flexible structure to handle heterogeneous noise and outliers, which may appear in TWRI measurements. The resolution is achieved via the Alternating Direction Method of Multipliers (ADMM) and variable splitting to decouple the constraints. The removal of the front wall is achieved via a closed-form proximal evaluation and the recovery of targets is possible via a tailored Majorization–Minimization (MM) step. The analysis and validation of our method is carried out using Finite-Difference Time-Domain (FDTD) simulated data, which show the advantage of our method in detection performance over complex scenarios.

Using low-rank approximations of gridded data for spline surface fitting

The paper is devoted to the problem of finding bivariate tensor-product spline (or similar) functions that approximate gridded data in the least-squares sense. We propose to apply a low rank approximation of matrices to the data, to find the result by solving a sequence of univariate fitting problems that can be handled efficiently. This can be seen as a generalization of the method proposed by Georgieva and Hofreither (2017) that combines cross approximation (which is a particular method for low rank matrix approximation) with spline interpolation.While the algorithm yields the best least-squares approximation after r steps, where r denotes the rank of the data matrix, terminating it earlier yields a low-rank approximation, which often provides a sufficient level of accuracy. We also present a stopping criterion (based on a lower error estimate) that allows to use the method efficiently in the situation when the required number of degrees of freedom is not known in advance.

Discrete fixed-time observers over sensor networks with unknown noise

This paper investigates fixed-time observers for a class of sensor networks with unknown noise. Specifically, the strongly observable condition and observer matching condition are provided for the convenience of analysis. Two kinds of fixed-time observers for sensor networks are developed based on the system’s observability and the properties of nilpotent matrices. The convergence time of the first kind of observer depends on the target state’s dimension and the rank of the noise input matrix. Besides, to further reduce the convergence time, another kind of observer is designed with the thought of the reduced-order observer. Its convergence time is related to the dimension of the target state and the rank of the augmented measurement matrix. Finally, the effectiveness of the proposed fixed-time observers is demonstrated by simulation examples.

An efficient randomized QLP algorithm for approximating the singular value decomposition

The rank-revealing pivoted QLP decomposition approximates the computationally prohibitive singular value decomposition (SVD) via two consecutive column-pivoted QR (CPQR) decomposition. It furnishes information on all four fundamental subspaces of the matrix. In this paper, we introduce a randomized version of the QLP decomposition called Rand-QLP. Operating on a matrix A, Rand-QLP gives A=QLPT, where Q and P are orthonormal, and L is lower-triangular. Under the assumption that the rank of the input matrix is k, we derive several error bounds for Rand-QLP: bounds for the first k approximate singular values and for the trailing block of the middle factor L, which show that the decomposition is rank-revealing; bounds for the distance between approximate subspaces and the exact ones for all four fundamental subspaces of a given matrix; and bounds for the errors of low-rank approximations constructed by the columns of Q and P. Rand-QLP is able to effectively leverage modern computational architectures, due to the utilization of random sampling and the unpivoted QR decomposition, thus addressing a serious bottleneck associated with classical algorithms such as the SVD, CPQR and most recent matrix decomposition algorithms. We utilize a modern computing architecture to assess the performance behavior of different algorithms. In comparison to CPQR and the SVD, Rand-QLP respectively achieves a speedup of up to 5 times and 6.6 times on the CPU and up to 3.8 times and 4.4 times with the hybrid GPU architecture. In terms of quality of approximation, our results on synthetic and real data show that the approximations by Rand-QLP are comparable to those of pivoted QLP and the optimal SVD, and in most cases are considerably better than those of CPQR.

The Smith normal form of the walk matrix of the extended Dynkin graph [formula omitted

In this paper, we provide an explicit formula for the rank of the walk matrix of the extended Dynkin graph D˜n. Moreover, we prove that the Smith normal form of the walk matrix of the extended Dynkin graph D˜n is{diag(1,…,1︸r−1,n−12,0,…,0),ifnis odd,diag(1,…,1︸r−1,n−1,0,…,0),ifnis even, where r=⌊n2⌋.

Deterministic sampling method using simplex ensemble and scaling method for efficient and robust uncertainty quantification

ABSTRACT Uncertainty quantification (UQ) of the neutron multiplication factor is important to investigate the appropriate safety margin for a target system. Although the random sampling method is a practical and useful UQ method, a large computational cost is required to reduce the statistical error of the estimated uncertainty. Furthermore, if an input variable follows a normal distribution with a large standard deviation, the perturbed input variable by the random sampling method may become a physically inappropriate or negative value. To address these issues for the efficient and robust UQ, a modified deterministic sampling method using the simplex ensemble and the scaling method is proposed. The features of the proposed method are summarized as follows: The sample size is r + 2 , where r corresponds to the effective rank of the covariance matrix between the input variables; depending on a situation of target UQ, the amounts of perturbations for the input parameters can be arbitrarily given by the scaling factor method; the scaling factor can be updated to avoid physically inappropriate in the perturbed input variables. The effectiveness of the proposed method is demonstrated through the UQ of the neutron multiplication factor due to fuel manufacturing uncertainties for a typical PWR pin-cell burnup calculation.

A hesitant fuzzy multiple-attribute decision-making method considering rank relation

A decision matrix is typically used to express hesitant information when solving multi-attribute decision problems in an uncertain environment. To further investigate the decision problem, this paper takes the property of matrix rank as the starting point, introduces it into the hesitant fuzzy theory, presents the concept of the rank of the hesitant fuzzy decision matrix and discusses the related properties, and then studies the hesitant fuzzy linear relation, obtaining the attribute reduction method based on the hesitant fuzzy linear relation and applying it to the multi-attribute decision making. It adds to the theoretical understanding of the hesitant fuzzy decision matrix. The aggregation operator first transforms the hesitant fuzzy information into a comprehensive decision matrix, and the row echelon transformation determines the rank of the matrix. Second, the hesitant fuzzy linear relationship is obtained using the rank property. A new hesitant fuzzy matrix is obtained after attribute reduction based on the hesitant fuzzy linear relationship, and the alternatives are sorted using the TOPSIS method. Finally, the effectiveness and superiority of the proposed method are demonstrated through a comparison of actual case analysis and existing methods, and the expected research purpose is met.