Rank Of Matrices Research Articles

Joint analysis of longitudinal feed intake and single recorded production traits in pigs using a novel Horizontal model1

A novel Horizontal model is presented for multitrait analysis of longitudinal traits through random regression analysis combined with single recorded traits. Weekly ADFI on test for Danish Duroc, Landrace, and Yorkshire boars were available from the national test station and were collected from 30 to 100 kg BW. Single recorded production traits of ADG from birth to 30 kg BW (ADG30), ADG from 30 to 100 kg BW (ADG100), and lean meat percentage (LMP) were available from breeding herds or the national test station. The Horizontal model combined random regression analysis of feed intake (FI) with single recorded traits of ADG100, LMP, and ADG30. In the Horizontal model, the FI data were horizontally structured with FI on each week as a "trait." The additive genetic and litter effects were modeled to be common across different FI records by reducing the rank of the covariance matrices using second- and first-order Legendre polynomials of age on test, respectively. The fixed effect and random residual variance were estimated for each weekly FI trait. Residual feed intake (RFI) was derived from the conditional distribution of FI given the breeding values of ADG100 and LMP. The heritability of FI varied by week on test in Duroc (0.12 to 0.19), Landrace (0.13 to 0.22), and Yorkshire (0.21 to 0.23). The heritability of RFI was lowest and highest in wk 6 (0.03) and 10 (0.10), respectively, in Duroc and wk 7 (0.04 and 0.02) and 1 (0.09 and 0.20), respectively, in Landrace and Yorkshire. The proportion of FI genetic variance explained by RFI ranged from 20 to 75% in Duroc, from 19 to 75% in Landrace, and from 11 to 91% in Yorkshire. Average daily gain from 30 to 100 kg BW and ADG30 heritabilities were moderate in Duroc (0.24 and 0.22, respectively), Landrace (0.34 and 0.25, respectively), and Yorkshire (0.34 and 0.22, respectively). Lean meat percentage heritability was moderate in Duroc (0.37) and large in Landrace (0.62) and Yorkshire (0.60). The genetic correlation of FI with ADG100 increased by week on test followed by a 32% decrease from wk 7 in Duroc and a 7% decrease in dam line breeds. Defining RFI as genetically independent of production traits leads to consistent and easy interpretable breeding values. The genetic parameters of traits in the feed efficiency complex and their dynamics over the test period showed breed differences that could be related to the fatness and growth potential of the breeds. The Horizontal model can be used to simultaneously analyze repeated and single recorded traits through proper modeling of the environmental variances and covariances.

An efficient quantum algorithm for spectral estimation

We develop an efficient quantum implementation of an important signal processing algorithm for line spectral estimation: the matrix pencil method, which determines the frequencies and damping factors of signals consisting of finite sums of exponentially damped sinusoids. Our algorithm provides a quantum speedup in a natural regime where the sampling rate is much higher than the number of sinusoid components. Along the way, we develop techniques that are expected to be useful for other quantum algorithms as well—consecutive phase estimations to efficiently make products of asymmetric low rank matrices classically accessible and an alternative method to efficiently exponentiate non-Hermitian matrices. Our algorithm features an efficient quantum–classical division of labor: the time-critical steps are implemented in quantum superposition, while an interjacent step, requiring much fewer parameters, can operate classically. We show that frequencies and damping factors can be obtained in time logarithmic in the number of sampling points, exponentially faster than known classical algorithms.

The minimal measurement number for low-rank matrix recovery

The paper presents several results that address a fundamental question in low-rank matrix recovery: how many measurements are needed to recover low-rank matrices? We begin by investigating the complex matrices case and show that 4 n r − 4 r 2 generic measurements are both necessary and sufficient for the recovery of rank- r matrices in C n × n . Thus, we confirm a conjecture which is raised by Eldar, Needell and Plan for the complex case. We next consider the real case and prove that the bound 4 n r − 4 r 2 is tight provided n = 2 k + r , k ∈ Z + . Motivated by Vinzant's work [19] , we construct 11 matrices in R 4 × 4 by computer random search and prove they define injective measurements on rank-1 matrices in R 4 × 4 . This disproves the conjecture raised by Eldar, Needell and Plan for the real case. Finally, we use the results in this paper to investigate the phase retrieval by projection and show fewer than 2 n − 1 orthogonal projections are possible for the recovery of x ∈ R n from the norm of them, which gives a negative answer for a question raised in [1] .

Symmetric arctic ranks of nonnegative matrices and their linear preservers

A rank one matrix can be factored as for vectors and of appropriate orders. The perimeter of this rank one matrix is the number of nonzero entries in plus the number of nonzero entries in . A matrix of rank k is the sum of k rank one matrices, a rank one decomposition. The perimeter of a matrix A of rank k is the minimum over all rank one of A of the sums of perimeters of the rank one matrices. The arctic rank of a matrix is one half the perimeter. In this article, we characterize the linear operators that preserve the symmetric arctic ranks of nonnegative symmetric matrices.

A Tensor-Train accelerated solver for integral equations in complex geometries

We present a framework using the Quantized Tensor Train (qtt) decomposition to accurately and efficiently solve volume and boundary integral equations in three dimensions. We describe how the qtt decomposition can be used as a hierarchical compression and inversion scheme for matrices arising from the discretization of integral equations. For a broad range of problems, computational and storage costs of the inversion scheme are extremely modest O(log⁡N) and once the inverse is computed, it can be applied in O(Nlog⁡N).We analyze the qtt ranks for hierarchically low rank matrices and discuss its relationship to commonly used hierarchical compression techniques such as fmm and hss. We prove that the qtt ranks are bounded for translation-invariant systems and argue that this behavior extends to non-translation invariant volume and boundary integrals.For volume integrals, the qtt decomposition provides an efficient direct solver requiring significantly less memory compared to other fast direct solvers. We present results demonstrating the remarkable performance of the qtt-based solver when applied to both translation and non-translation invariant volume integrals in 3d.For boundary integral equations, we demonstrate that using a qtt decomposition to construct preconditioners for a Krylov subspace method leads to an efficient and robust solver with a small memory footprint. We test the qtt preconditioners in the iterative solution of an exterior elliptic boundary value problem (Laplace) formulated as a boundary integral equation in complex, multiply connected geometries.

Broadcast Erasure Channel with Feedback and Message Side Information, and Related Index Coding Results

We consider the $N$ -receiver broadcast erasure channel with feedback and message side information at the receivers prior to beginning of transmission. Specifically, the transmitter must deliver different, independent messages to each of the receivers, and each receiver knows a function of these messages before transmission begins. This situation can arise in multi-hop wireless networks, where a receiver may overhear transmissions consisting of possibly encoded combinations of messages (e.g., encoded using a network coding technique) prior to beginning of transmission over a given broadcast channel. We provide an outer bound to the capacity region of this system. For the case, where each message consists of a number of symbols taking values in a finite field and each receiver knows linear combinations of these symbols, the outer bound is given in terms of ranks of matrices expressing the linear combinations. For the latter case and when $N=2$ , the outer bound is tight under mild conditions on the limiting behavior of the ranks of matrices expressing the side information. We provide a capacity achieving code for this case. The special case, where each receiver either knows the entire message of another receiver or has no information about it, constitutes a generalization of the index coding problem that incorporates channel erasures. For this instance, and when there are no channel errors, we show that the outer bound reduces to the known maximum weighted acyclic induced subgraph bound.

Adaptive Low Rank Matrix Completion

The low-rank matrix completion problem is fundamental to a number of tasks in data mining, machine learning, and signal processing. This paper considers the problem of adaptive matrix completion in time-varying scenarios. Given a sequence of incomplete and noise-corrupted matrices, the goal is to recover and track the underlying low rank matrices. Motivated from the classical least-mean square (LMS) algorithms for adaptive filtering, three LMS-like algorithms are proposed for estimating and tracking low-rank matrices. Performance of the proposed algorithms is provided in form of nonasymptotic bounds on the tracking mean-square error. Tracking performance of the algorithms is also studied via detailed simulations over real-world datasets.

A geometrical interpretation of the inverse matrix

Utilizing a new method to structure parallellotopes, a geometrical interpretation of the inverse matrix is given, which includes the generalized inverse of full column rank or a full row rank matrices. Further, some relational volume formulas of parallellotopes are established.

Matrices with high completely positive semidefinite rank

A real symmetric matrix M is completely positive semidefinite if it admits a Gram representation by (Hermitian) positive semidefinite matrices of any size d. The smallest such d is called the (complex) completely positive semidefinite rank of M, and it is an open question whether there exists an upper bound on this number as a function of the matrix size. We construct completely positive semidefinite matrices of size 4k2+2k+2 with complex completely positive semidefinite rank 2k for any positive integer k. This shows that if such an upper bound exists, it has to be at least exponential in the matrix size. For this we exploit connections to quantum information theory and we construct extremal bipartite correlation matrices of large rank. We also exhibit a class of completely positive matrices with quadratic (in terms of the matrix size) completely positive rank, but with linear completely positive semidefinite rank, and we make a connection to the existence of Hadamard matrices.

Exploring local knowledge and preferences for shea (Vitellaria paradoxa) ethnovarieties in Southwest Burkina Faso through a gender and ethnic lens

In Africa’s ‘shea belt’, the shea tree (Vitellaria paradoxa) plays a central role in rural livelihoods and ecosystems. Yet, it faces many threats. The aim of this study is to examine local people’s classification systems and preferences for shea ethnovarieties in Burkina Faso to support domestication efforts that respect local priorities. Work was carried out among the Bobo, Sambla, Mosse and FulBe ethnic groups in two villages in south-west Burkina Faso. Participatory characterization and ranking matrices were used with 10 groups segregated by gender and ethnicity to understand if knowledge and preferences for shea ethnovarieties vary between gender and ethnic groups. Results show a general agreement across groups about top-cited ethnovariety names, characteristics and key criteria defining the classification system. Participants identified a total of 25 shea ethnovarieties according to 11 primary fruit and nut variants. The number of ethnovarieties cited varied slightly across groups, with greater consistency across ethnicities than across gender groups. Each ethnic and gender group identified certain ethnovarieties not cited by their ethnic or gender counterparts. Two ethnovarieties – ‘small shea fruit’ and ‘big shea fruit’ – were preferred across groups. The study highlights the value of gender-sensitive participatory research for understanding local botanical knowledge and preferences.

Tensor decomposition in potential energy surface representations.

In order to reduce the operation count in vibration correlation methods, e.g., vibrational configuration interaction (VCI) theory, a tensor decomposition approach has been applied to the analytical representations of multidimensional potential energy surfaces (PESs). It is shown that a decomposition of the coefficients within the individual n-mode coupling terms in a multimode expansion of the PES is feasible and allows for convenient contractions of one-dimensional integrals with these newly determined factor matrices. Deviations in the final VCI frequencies of a set of small molecules were found to be negligible once the rank of the factors matrices is chosen appropriately. Recommendations for meaningful ranks are provided and different algorithms are discussed.

Rational realization of the minimum ranks of nonnegative sign pattern matrices

A sign pattern matrix (or nonnegative sign pattern matrix) is a matrix whose entries are from the set {+,−, 0} ({+, 0}, respectively). The minimum rank (or rational minimum rank) of a sign pattern matrix A is the minimum of the ranks of the matrices (rational matrices, respectively) whose entries have signs equal to the corresponding entries of A. Using a correspondence between sign patterns with minimum rank r ≥ 2 and point-hyperplane configurations in R r−1 and Steinitz’s theorem on the rational realizability of 3-polytopes, it is shown that for every nonnegative sign pattern of minimum rank at most 4, the minimum rank and the rational minimum rank are equal. But there are nonnegative sign patterns with minimum rank 5 whose rational minimum rank is greater than 5. It is established that every d-polytope determines a nonnegative sign pattern with minimum rank d + 1 that has a (d + 1) × (d + 1) triangular submatrix with all diagonal entries positive. It is also shown that there are at most min{3m, 3n} zero entries in any condensed nonnegative m × n sign pattern of minimum rank 3. Some bounds on the entries of some integer matrices achieving the minimum ranks of nonnegative sign patterns with minimum rank 3 or 4 are established.

Confluence of hypergeometric functions and integrable hydrodynamic-type systems

It is known that a large class of integrable hydrodynamic type systems can be constructed through the Lauricella function, a generalization of the classical Gauss hypergeometric function. In this paper, we construct novel class of integrable hydrodynamic type systems which govern the dynamics of critical points of confluent Lauricella type functions defined on finite dimensional Grassmannian Gr(2,n), the set of 2xn matrices of rank two. Those confluent functions satisfy certain degenerate Euler-Poisson-Darboux equations. It is also shown that in general, hydrodynamic type system associated to the confluent Lauricella function is given by an integrable and non-diagonalizable quasi-linear system of a Jordan matrix form. The cases of Grassmannian Gr(2,5) for two component systems and Gr(2,6) for three component systems are considered in details.

Fieller, N.: Basics of Matrix Algebra for Statistics with R

Matrices are becoming more important than ever before in the big-data era of the modern multidimensional world.With advances in computing and statistical packages such as R, it has become easier to manipulate matrices and analyse data efficiently. The book begins with an introduction to R, including installation, functions and basic R commands and calculations. There follows a guide to basic matrix algebra in R with applications to statistics. It covers the major topics in matrix algebra including vectors, matrices, rank of matrices, determinants, inverses, eigenanalysis, matrix calculus and their key applications to statistics. The book also includes a chapter on advanced topics such as matrix decompositions, generalised inverses, and Hadamard and Kronecker products. Somewhat unfortunately, the book’s notation for matrix calculus is different from the notation advocated by Magnus and Neudecker (1999). The book also include tricks formanipulatingmatrices, plus derivation of key results with step-by-step cross-referenced explanations and demonstrations of the techniques in R using numerical examples. The chapter on key applications to statistics covers principal component analysis, linear discriminant analysis and canonical correlation analysis, in addition to linear models. Some chapters have a summary of key results which is useful for easy reference. End of chapter exercises are also available with solutions provided at the end of the book. The book avoids advanced mathematics and is therefore a good beginner’s guide to understanding and manipulating matrices in R. It is suitable for early year undergraduate students and anyone looking for an introduction to matrix algebra in R in preparation for high-level or specialised studies in statistics. The book’s collection of

The numerical rank of Krylov matrices

In this paper we expose interesting features of Krylov matrices and Vandermonde matrices. Let S be a large symmetric matrix of order n. Let x be a real n-vector, and let Kℓ denote the related n×ℓ Krylov matrix. The question considered in this paper is how the numerical rank of Kℓ grows as ℓ increases. The key for answering this question lies in the close link between Kℓ and the n×ℓ Vandermonde matrix which is generated by the eigenvalues of S. Analysis of large Vandermonde matrices shows that the numerical rank is expected to remain much smaller than ℓ. The proof is based on partition theorems and clustering theorems. The basic tool is a new matrix equality: The Vandermonde–Pascal–Toeplitz equality. The actual numerical rank of a Vandermonde (or Krylov) matrix depends on the distribution of the eigenvalues, but often the rank is remarkable small. Numerical experiments illustrate these points. The observation that the numerical rank of Krylov matrices stays small has important practical consequences.

Two competing linear random-effects models and their connections

Assume that an observed random vector is represented in two alternative linear random-effects models (LRMs). In this case, the results of statistical inference under the two LRMs are not necessarily the same. The objective of this paper is to study the performance of the best linear unbiased predictors/best linear unbiased estimators (BLUPs/BLUEs) of all unknown parameters in two LRMs under most general assumptions on the covariance matrices of the random-effects terms and error terms. We establish necessary and sufficient conditions for the BLUPs/BLUEs to be equivalent under the two LRMs using some well-known formulas for calculating ranks of matrices and their generalized inverses, and also present various consequences and conclusions on some special special cases of LRMs.

Finding a low-rank basis in a matrix subspace

For a given matrix subspace, how can we find a basis that consists of low-rank matrices? This is a generalization of the sparse vector problem. It turns out that when the subspace is spanned by rank-1 matrices, the matrices can be obtained by the tensor CP decomposition. For the higher rank case, the situation is not as straightforward. In this work we present an algorithm based on a greedy process applicable to higher rank problems. Our algorithm first estimates the minimum rank by applying soft singular value thresholding to a nuclear norm relaxation, and then computes a matrix with that rank using the method of alternating projections. We provide local convergence results, and compare our algorithm with several alternative approaches. Applications include data compression beyond the classical truncated SVD, computing accurate eigenvectors of a near-multiple eigenvalue, image separation and graph Laplacian eigenproblems.

Algebraic boundary of matrices of nonnegative rank at most three

Understanding the boundary of the set of matrices of nonnegative rank at most r is important for applications in nonconvex optimization. The Zariski closure of the boundary of the set of matrices of nonnegative rank at most 3 is reducible. We give a minimal generating set for the ideal of each irreducible component. In fact, this generating set is a Gröbner basis with respect to the graded reverse lexicographic order. This solves a conjecture by Robeva, Sturmfels and the last author.

Low-rank updates of balanced incomplete factorization preconditioners

Let Ax = b be a large and sparse system of linear equations where A is a nonsingular matrix. An approximate solution is frequently obtained by applying preconditioned iterations. Consider the matrix B = A + PQT where P,Qźźn×k$P, Q \in \mathbb {R}^{n \times k}$ are full rank matrices. In this work, we study the problem of updating a previously computed preconditioner for A in order to solve the updated linear system Bx = b by preconditioned iterations. In particular, we propose a method for updating a Balanced Incomplete Factorization preconditioner. The strategy is based on the computation of an approximate Inverse Sherman-Morrison decomposition for an equivalent augmented linear system. Approximation properties of the preconditioned matrix and an analysis of the computational cost of the algorithm are studied. Moreover, the results of the numerical experiments with different types of problems show that the proposed method contributes to accelerate the convergence.

Efficient local search for L_1 and L_2 binary matrix factorization

Rank K Binary Matrix Factorization (BMF) approximates a binary matrix by the product of two binary matrices of lower rank, K. Several researchers have addressed this problem, focusing on either approximations of rank 1 or higher, using either the L 1 or L 2-norms for measuring the quality of the approximation. The rank 1 problem (for which the L 1 and L 2-norms are equivalent) has been shown to be related to the Integer Linear Programming (ILP) problem. We first show here that the alternating strategy with the L 2-norm, at the core of several methods used to solve BMF, can be reformulated as an Unconstrained Binary Quadratic Programming (UBQP) problem. This reformulation allows us to use local search procedures designed for UBQP in order to improve the solutions of BMF. We then introduce a new local search dedicated to the BMF problem. We show in particular that this solution is in average faster than the previously proposed ones. We then assess its behavior on several collections and methods and show that it significantly improves methods targeting the L 2-norms on all the datasets considered; for the L 1-norm, the improvement is also significant for real, structured datasets and for the BMF problem without the binary reconstruction constraint.