Rank Of Matrices Research Articles

Constructions of perfect bases for classes of 3-tensors

A well studied problem in algebraic complexity theory is the determination of the complexity of problems relying on evaluations of bilinear maps. One measure of the complexity of a bilinear map (or 3-tensor) is the optimal number of non-scalar multiplications required to evaluate it. This quantity is also described as its tensor rank, which is the smallest number of rank-1 matrices whose span contains its first slice space. In this paper we derive upper bounds on the tensor ranks of certain classes of 3-tensors and give explicit constructions of sets of rank-1 matrices containing their first slice spaces. We also show how these results can be applied in coding theory to derive upper bounds on the tensor rank of some rank-metric codes. In particular, we compute the tensor rank of some families of Fqm-linear codes and we show that they are extremal with respect to Kruskal's tensor rank bound.

Matrices of linear forms of constant rank from vector bundles on projective spaces

We consider the problem of constructing matrices of linear forms of constant rank by focusing on the associated vector bundles on projective spaces. Important examples are given by the classical Steiner bundles, as well as some special (duals of) syzygy bundles that we call Drézet bundles. Using the classification of globally generated vector bundles with small first Chern class on projective spaces, we are able to describe completely the indecomposable matrices of constant rank up to six; some of them come from rigid homogeneous vector bundles, some other from Drézet bundles related either to plane quartics or to instanton bundles on P3.

Variable selection with the knockoffs: Composite null hypotheses

The fixed-X knockoff filter is a flexible framework for variable selection with false discovery rate (FDR) control in linear models with arbitrary design matrices (of full column rank) and it allows for finite-sample selective inference via the Lasso estimates. In this paper, we extend the theory of the knockoff procedure to tests with composite null hypotheses, which are usually more relevant to real-world problems. The main technical challenge lies in handling composite nulls in tandem with dependent features from arbitrary designs. We develop two methods for composite inference with the knockoffs, namely, shifted ordinary least-squares (S-OLS) and feature-response product perturbation (FRPP), building on new structural properties of test statistics under composite nulls. We also propose two heuristic variants of S-OLS method that outperform the celebrated Benjamini–Hochberg (BH) procedure for composite nulls, which serves as a heuristic baseline under dependent test statistics. Finally, we analyze the loss in FDR when the original knockoff procedure is naively applied on composite tests.

Tangent Space Based Alternating Projections for Nonnegative Low Rank Matrix Approximation

In this paper, we develop a new alternating projection method to compute nonnegative low rank matrix approximation for nonnegative matrices. In the nonnegative low rank matrix approximation method, the projection onto the manifold of fixed rank matrices can be expensive as the singular value decomposition is required. We propose to use the tangent space of the point in the manifold to approximate the projection onto the manifold in order to reduce the computational cost. We show that the sequence generated by the alternating projections onto the tangent spaces of the fixed rank matrices manifold and the nonnegative matrix manifold, converge linearly to a point in the intersection of the two manifolds where the convergent point is sufficiently close to optimal solutions. This convergence result based inexact projection onto the manifold is new and is not studied in the literature. Numerical examples in data clustering, pattern recognition and hyperspectral data analysis are given to demonstrate that the performance of the proposed method is better than that of nonnegative matrix factorization methods in terms of computational time and accuracy.

Partial-Sum Matrix and its Rank

A partial-sum matrix is a matrix whose entries are partial sums of a seires associate with a sequence. The rank of a partial-sum matrix associate with any recurrence sequence can be related to the rank of an associate recurrence matrix, a matrix whose entries are from the same recurrence sequence. In particular, we find ranks of partial-sum matrices associated with arithmetic series and geometric series.

Bures–Wasserstein Minimizing Geodesics between Covariance Matrices of Different Ranks

Bures–Wasserstein Minimizing Geodesics between Covariance Matrices of Different Ranks

Influences of long-duration NO exposure on CO2 adsorption on coals: insights into oxy-coal burning flue gas storage in coal reservoirs

ABSTRACT Introducing oxy-coal burning flue gas into coal reservoirs has the advantages of mitigating emissions of CO2, NO x , and SO2, and producing in-situ coalbed methane (CBM). Given the characteristics of the geologic time scale for fluid sequestration, the long-duration NO exposure was conducted for various ranks of coal matrices to access the potential impacts of NO exposure on CO2 storage. Afterwards, the changes in critical physico-chemical properties and adsorbability of various ranks of coal matrices to CO2 because of long-duration NO exposure were revealed. Finally, the implications to implement oxy-coal burning flue gas sequestration in unmineable coal reservoirs with CBM production were indicated. The results demonstrate that the long-duration interaction with NO reduces the number of micro- and meso-pores of various ranks of coals, especially those with diametres below 4.00 nm. Moreover, the long-duration NO exposure reduces the oxygen-containing functional groups while significantly increasing the amine/amide-N for all the coals. The aforementioned alterations in the surface chemistry property imply that the coal matrix is capable of chemically adsorbing NO, thereby achieving its stable storage in target coal reservoirs. Given the electron donor–acceptor interactions between amine/amide-N and CO2, the NO exposure can elevate the CO2 storage capability of various ranks of coals under typical reservoir temperature and pressure. In summary, introducing oxy-coal burning flue gas into coal reservoirs is capable of stably storing critical gaseous pollutants and simultaneously enhancing the CO2 storage potential of coal reservoirs, thereby updating the existing oxy-coal burning technology and CO2 sequestration in unmineable coal reservoirs with the enhanced CBM recovery technology.

Generalized Matrix Local Low Rank Representation by Random Projection and Submatrix Propagation.

Matrix low rank approximation is an effective method to reduce or eliminate the statistical redundancy of its components. Compared with the traditional global low rank methods such as singular value decomposition (SVD), local low rank approximation methods are more advantageous to uncover interpretable data structures when clear duality exists between the rows and columns of the matrix. Local low rank approximation is equivalent to low rank submatrix detection. Unfortunately, existing local low rank approximation methods can detect only submatrices of specific mean structure, which may miss a substantial amount of true and interesting patterns. In this work, we develop a novel matrix computational framework called RPSP (Random Probing based submatrix Propagation) that provides an effective solution for the general matrix local low rank representation problem. RPSP detects local low rank patterns that grow from small submatrices of low rank property, which are determined by a random projection approach. RPSP is supported by theories of random projection. Experiments on synthetic data demonstrate that RPSP outperforms all state-of-the-art methods, with the capacity to robustly and correctly identify the low rank matrices when the pattern has a similar mean as the background, background noise is heteroscedastic and multiple patterns present in the data. On real-world datasets, RPSP also demonstrates its effectiveness in identifying interpretable local low rank matrices.

On Structural Rank and Resilience of Sparsity Patterns

A sparsity pattern in $\mathbb{R}^{n \times m}$, for $m\geq n$, is a vector subspace of matrices admitting a basis consisting of canonical basis vectors in $\mathbb{R}^{n \times m}$. We represent a sparsity pattern by a matrix with $0/\star$-entries, where $\star$-entries are arbitrary real numbers and $0$-entries are equal to $0$. We say that a sparsity pattern has full structural rank if the maximal rank of matrices contained in it is $n$. In this paper, we investigate the degree of resilience of patterns with full structural rank: We address questions such as how many $\star$-entries can be removed without decreasing the structural rank and, reciprocally, how many $\star$-entries one needs to add so as to increase the said degree of resilience to reach a target. Our approach goes by translating these questions into max-flow problems on appropriately defined bipartite graphs. Based on these translations, we provide algorithms that solve the problems in polynomial time.

Structured Matrices and Their Application in Neural Networks: A Survey

Modern neural network architectures are becoming larger and deeper, with increasing computational resources needed for training and inference. One approach toward handling this increased resource consumption is to use structured weight matrices. By exploiting structures in weight matrices, the computational complexity for propagating information through the network can be reduced. However, choosing the right structure is not trivial, especially since there are many different matrix structures and structure classes. In this paper, we give an overview over the four main matrix structure classes, namely semiseparable matrices, matrices of low displacement rank, hierarchical matrices and products of sparse matrices. We recapitulate the definitions of each structure class, present special structure subclasses, and provide references to research papers in which the structures are used in the domain of neural networks. We present two benchmarks comparing the classes. First, we benchmark the error for approximating different test matrices. Second, we compare the prediction performance of neural networks in which the weight matrix of the last layer is replaced by structured matrices. After presenting the benchmark results, we discuss open research questions related to the use of structured matrices in neural networks and highlight future research directions.

Constructive subsampling of finite frames with applications in optimal function recovery

In this paper we present new constructive methods, random and deterministic, for the efficient subsampling of finite frames in Cm. Based on a suitable random subsampling strategy, we are able to extract from any given frame with bounds 0<A≤B<∞ (and condition B/A) a similarly conditioned reweighted subframe consisting of merely O(mlog⁡m) elements. Further, utilizing a deterministic subsampling method based on principles developed by Batson, Spielman, and Srivastava to control the spectrum of sums of Hermitian rank-1 matrices, we are able to reduce the number of elements to O(m) (with a constant close to one). By controlling the weights via a preconditioning step, we can, in addition, preserve the lower frame bound in the unweighted case. This permits the derivation of new quasi-optimal unweighted (left) Marcinkiewicz-Zygmund inequalities for L2(D,ν) with constructible node sets of size O(m) for m-dimensional subspaces of bounded functions. Those can be applied e.g. for (plain) least-squares sampling reconstruction of functions, where we obtain new quasi-optimal results avoiding the Kadison-Singer theorem. Numerical experiments indicate the applicability of our results.

2D Linear CA with Mixing Boundary Conditions and Reversibility

In this paper, we consider two-dimensional cellular automata (CA) with the von Neumann neighborhood. We study the characterization of 2D linear cellular automata defined by the von Neumann neighborhood with new type of boundary conditions over the field [Formula: see text]. Furthermore, we investigate the rule matrices of 2D von Neumann CA by applying the group of permutations [Formula: see text]. Moreover, the algorithm for computing the rank of rule matrices is given. Finally, necessary and sufficient conditions for the existence of Garden of Eden configurations for considered two-dimensional cellular automata are obtained.

Constructing solutions of the Yang–Baxter-like matrix equation for singular matrices

Let A=UVH be an n×n complex singular matrix of rank r, where U, V are two n×r matrices of full column rank. Some sufficient and necessary conditions are derived for X being a nontrivial (commuting) solution of the quadratic matrix equation AXA=XAX such that we can construct infinitely many (commuting) solutions of the matrix equation. When VHU is nonsingular, all the solutions of the matrix equation are characterized and can be found and constructed by solving smaller matrix equations. In particular, all the commuting solutions can be obtained by finding the projections that commute with VHU. The method of constructing the solutions is free of computations of Jordan canonical forms and matrix inverses.

Multilinear multitask learning by transformed tensor singular value decomposition

In this paper, we study the problem of multilinear multitask learning (MLMTL), in which all tasks are stacked into a third-order tensor for consideration. In contrast to conventional multitask learning, MLMTL can explore inherent correlations among multiple tasks in a better manner by utilizing multilinear low rank structure. Existing approaches about MLMTL are mainly based on the sum of singular values for approximating low rank matrices obtained by matricizing the third-order tensor. However, these methods are suboptimal in the Tucker rank approximation. In order to elucidate intrinsic correlations among multiple tasks, we present a new approach by the use of transformed tensor nuclear norm (TTNN) constraint in the objective function. The main advantage of the proposed approach is that it can acquire a low transformed multi-rank structure in a transformed tensor by applying suitable unitary transformations which is helpful to determine principal components in grouping multiple tasks for describing their intrinsic correlations more precisely. Furthermore, we establish an excess risk bound of the minimizer of the proposed TTNN approach. Experimental results including synthetic problems and real-world images, show that the mean-square errors of the proposed method is lower than those of the existing methods for different number of tasks and training samples in MLMTL.

Characterizations of output controllability for LTI systems

The objective of this paper is to make, in a simple and rigorous way, some contributions to the notion of output controllability. We first examine, for general linear time-invariant systems, the notion of state to output controllability, introduced in the 60s by Bertram and Sarachik. More precisely, we extend the Hautus test, well-known in the case of state controllability, to state to output controllability, and propose a controllability Gramian matrix, allowing us to build a continuous control achieving a transfer with minimal energy. We also give two other notions of output controllability, namely output to output and globally output to output controllability. For each of these two other new notions, we give necessary and sufficient conditions, in terms of Kalman rank, Hautus test and Gramian matrices.

Some explorations of linear algebra

It is a fundamental problem in quantum information whether a particular quantum state of a composite system is entangled. It has enormous potential in quantum error correction, quantum cryptography, and quantum teleportation applications. This problem can be transferred in the form of a mathematical conjecture in language of linear algebra. In this paper, the authors explain the important applications, convenience, efficiency of using linear algebra in math physics, and computer science. The authors give some examples of linear algebra used in various areas, including datum coordinate system finding a location, encryption and decryption algorithm, storage of images, classical mechanics, and quantum physics. The authors list the definition of the matrix, real matrix, complex matrix, diagonal matrix, identity matrix, scalar matrix, trace, rank, and determinants of matrices. The authors explored Laplace expansion, transpose, inverse, conjugate of a matrix and addition, multiplication between matrices and between a scalar and a matrix, Kronecker product and their properties of matrices, the definitions and solving method of eigenvalue and eigenvector, and the diagonalisation and its conditions of matrices. The authors introduce the applications of matrix transformations and operations in programs. The authors explain two encryption and decryption algorithms based on linear algebra and their strengths and weaknesses.

A novel method based on shifted rank-1 reconstruction for removing EMG artifacts in ECG signals

Electrocardiogram (ECG) signal is an essential feature of human health monitoring and detection. Electromyogram (EMG) artifacts will be involved as the noise distortion and make it difficult for the doctor’s diagnosis. By comparing with the characteristics of these two signals, ECG signals have periodicity while EMG artifacts are random. However, they overlap in the frequency domain. According to the periodicity of ECG signals, several methods of extracting ECG signals from EMG artifacts are proposed. The core of ECG extraction is to form a rank-1 trajectory matrix for reconstruction. However, due to the EMG artifacts, most formed matrices are not strict rank-1 matrices, and the length of traversal segment selected for pure ECG reconstructions is not easily determined. Therefore, a new ECG extraction method based on shifted rank-1 reconstruction is proposed. Given an approximated traversal segment length, the trajectory matrix is constructed. Through the optimization of lambda in the low rank matrix, the constructed approximation matrix is approximated by shift vectors to obtain a strict rank-1 matrix. Pure ECG signals can be reconstructed via the singular values of shifted matrix. The validation of the proposed method is made by applying the algorithm to ECG records from four different databases. The quantitative and qualitative analysis are carried out and compared with other methods. The results indicate that the proposed SR1 method can remove EMG artifacts and does not require any specific traversal segment lengths to extract clean ECG signals. The QRS complexes and ST segments can be retained in reconstructed ECG signals.

On Ranks of Matrices over Noncommutative Domains

We consider matrices with entries in some domain, i.e., in a ring, not necessarily commutative, not containing non-trivial zero divisors. The concepts of the row rank and the column rank are discussed. (Coefficients of linear dependencies belong to the domain ; left coefficients are used for rows, right coefficients for columns.) Assuming that the domain satisfies the Ore conditions, i.e., the existence of non-zero left and right common multiples for arbitrary non-zero elements, it is proven that these row and column ranks are equal, which allows us to speak about the rank of a matrix without specifying which rank (row or column) is meant. In fact, the existence of non-zero left and right common multiples for arbitrary non-zero elements of is a necessary and sufficient condition for the equality of the row and column ranks of an arbitrary matrix over. An algorithm for calculating the rank of a given matrix is proposed. Our Maple implementation of this algorithm covers the domains of differential and (-)difference operators, both ordinary and with partial derivatives and differences.

On Ranks of Matrices over Noncommutative Domains

On Ranks of Matrices over Noncommutative Domains

Inference on the maximal rank of time-varying covariance matrices using high-frequency data

We study the rank of the instantaneous or spot covariance matrix ΣX(t) of a multidimensional process X(t). Given high-frequency observations X(i/n), i=0,…,n, we test the null hypothesis rank(ΣX(t))≤r for all t against local alternatives where the average (r+1)st eigenvalue is larger than some signal detection rate vn. A major problem is that the inherent averaging in local covariance statistics produces a bias that distorts the rank statistics. We show that the bias depends on the regularity and spectral gap of ΣX(t). We establish explicit matrix perturbation and concentration results that provide nonasymptotic uniform critical values and optimal signal detection rates vn. This leads to a rank estimation method via sequential testing. For a class of stochastic volatility models, we determine data-driven critical values via normed p-variations of estimated local covariance matrices. The methods are illustrated by simulations and an application to high-frequency data of U.S. government bonds.