Completions Of Matrices Research Articles

On Fredholm completions of partial operator matrices

The aim of this article is to study the Fredholm completion problem of two-by-two partial operator matrices in which the lower-left entry is unspecified and others are specified. By using the methods of operator matrix representation and operator equation, we obtain necessity and sufficiency conditions for the partial operator matrices to have a Fredholm completion with the property that the lower-right entry of its Fredholm inverses is specified.

On the maximum rank of completions of entry pattern matrices

In an entry pattern matrix A, all entries are indeterminates but the same indeterminate can appear in numerous positions. For a field F, an F-completion of A results from assigning a value from F to each indeterminate entry. We define the genericF-rank of an entry pattern matrix to be its rank when considered over the function field generated over F by its indeterminate entries. We investigate the situation where the generic F-rank of A is not attained by any F-completion of A, which can occur only if the generic F-rank exceeds the field order.

An efficient method to transcription factor binding sites imputation via simultaneous completion of multiple matrices with positional consistency.

Transcription factors (TFs) are DNA-binding proteins that have a central role in regulating gene expression. Identification of DNA-binding sites of TFs is a key task in understanding transcriptional regulation, cellular processes and disease. Chromatin immunoprecipitation followed by high-throughput sequencing (ChIP-seq) enables genome-wide identification of in vivo TF binding sites. However, it is still difficult to map every TF in every cell line owing to cost and biological material availability, which poses an enormous obstacle for integrated analysis of gene regulation. To address this problem, we propose a novel computational approach, TFBSImpute, for predicting additional TF binding profiles by leveraging information from available ChIP-seq TF binding data. TFBSImpute fuses the dataset to a 3-mode tensor and imputes missing TF binding signals via simultaneous completion of multiple TF binding matrices with positional consistency. We show that signals predicted by our method achieve overall similarity with experimental data and that TFBSImpute significantly outperforms baseline approaches, by assessing the performance of imputation methods against observed ChIP-seq TF binding profiles. Besides, motif analysis shows that TFBSImpute preforms better in capturing binding motifs enriched in observed data compared with baselines, indicating that the higher performance of TFBSImpute is not simply due to averaging related samples. We anticipate that our approach will constitute a useful complement to experimental mapping of TF binding, which is beneficial for further study of regulation mechanisms and disease.

Applications of Completions of Operator Matrices to Some Properties of Operator Products on Hilbert Spaces

In this paper, we will investigate certain properties of some operator products on Hilbert spaces, by applications of completions of operator matrices. It is shown that, quite surprisingly, the invariance properties of the operator product $$T_1T_2T_2^{(1,\ldots )}T_1^{(1,\ldots )}T_1T_2$$ have a neat relationship with the properties of the reverse order laws for generalized inverses of the operator product $$T_1T_2$$ . That is, the mixed-type reverse order laws $$\begin{aligned} T_2\{1,\ldots \}T_1\{1,\ldots \}\subseteq (T_1T_2)\{1\} \end{aligned}$$ hold if and only if the operator product $$T_1T_2T_2^{(1,\ldots )}T_1^{(1,\ldots )}T_1T_2$$ is invariant, where $$(1,\ldots )$$ is taken respectively as (1), (1, 2), (1, 3), (1, 4), (1, 2, 3) as well as (1, 2, 4).

Completions of upper-triangular matrices to left-Fredholm operators with non-positive index

Completions of upper-triangular matrices to left-Fredholm operators with non-positive index

Hankel Low-Rank Matrix Completion: Performance of the Nuclear Norm Relaxation

The completion of matrices with missing values under the rank constraint is a nonconvex optimization problem. A popular convex relaxation is based on minimization of the nuclear norm (sum of singular values) of the matrix. For this relaxation, an important question is whether the two optimization problems lead to the same solution. This question was addressed in the literature mostly in the case of random positions of missing elements and random known elements. In this contribution, we analyze the case of structured matrices with a fixed pattern of missing values, namely, the case of Hankel matrix completion. We extend existing results on completion of rank-one real Hankel matrices to completion of rank-r complex Hankel matrices.

Decentralized Control of Linear Switched Nested Systems With $\ell_{2}$-Induced Norm Performance

This paper considers a decentralized switched control problem where exact conditions for controller synthesis are obtained in the form of semidefinite programming (SDP). The formulation involves a discrete-time switched linear plant that has a nested structure, and whose system matrices switch between a finite number of values according to finite-state automation. The goal of this paper is to synthesize a commensurately nested switched controller to achieve a desired level of $\ell_{2}$ -induced norm performance. The nested structures of both plant and controller are characterized by block lower-triangular system matrices. For this setup, exact conditions are provided for the existence of a finite path-dependent synthesis. These include conditions for the completion of scaling matrices obtained through an extended matrix completion lemma. When individual controller dimensions are chosen at least as large as the plant, these conditions reduce to a set of linear matrix inequalities. The completion lemma also provides an algorithm to complete closed-loop scaling matrices, leading to inequalities for controller synthesis that are solvable either algebraically or numerically through SDP.

Music Therapy Using Singing Training Improves Psychomotor Speed in Patients with Alzheimer's Disease: A Neuropsychological and fMRI Study

Background/Aims: To investigate the effect of singing training on the cognitive function in Alzheimer's disease (AD) patients. Methods: Ten AD patients (mean age 78.1 years) participated in music therapy using singing training once a week for 6 months (music therapy group). Each session was performed with professional musicians using karaoke and a unique voice training method (the YUBA Method). Before and after the intervention period, each patient was assessed by neuropsychological batteries, and functional magnetic resonance imaging (fMRI) was performed while the patients sang familiar songs with a karaoke device. As the control group, another 10 AD patients were recruited (mean age 77.0 years), and neuropsychological assessments were performed twice with an interval of 6 months. Results: In the music therapy group, the time for completion of the Japanese Raven's Colored Progressive Matrices was significantly reduced (p = 0.026), and the results obtained from interviewing the patients' caregivers revealed a significant decrease in the Neuropsychiatric Inventory score (p = 0.042) and a prolongation of the patients' sleep time (p = 0.039). The fMRI study revealed increased activity in the right angular gyrus and the left lingual gyrus in the before-minus-after subtraction analysis of the music therapy intervention. Conclusion: Music therapy intervention using singing training may be useful for dementia patients by improving the neural efficacy of cognitive processing.

Completions of upper-triangular matrices to Kato nonsingularity

In this paper we consider the problem of completion of the upper-triangular operator matrix [A?0B] where A∈B(H) and B∈B(K), to Kato nonsingular operators and completely solve it in each of the following cases: one of the operators A or B is Kato nonsingular; B is injective; A is with dense range; B is with finite ascent; A is with finite descent; 0∉int(σp(B)); 0∉int(σcp(A)). In particular, the results generalize and complete those of [3] and [7] that concern the same problem.

Applications of completions of operator matrices to reverse order law for {1}-inverses of operators on Hilbert spaces

Although the reverse order law for {1}-generalized inverses of matrices has been completely resolved, in this paper we do it for the first time for operators acting on separable Hilbert spaces. The approach taken here involves some of the previous research on completions of operator matrices to left and right invertibility. We thus demonstrate usability of some of the results of that type by showing how they can be applied to one of the topics in generalized inverses of operators that has seen a great interest over the years.

Left invertible completions of upper triangular operator matrices with unbounded entries

Given two closed, in general unbounded, operators A and C, we investigate the left invertible completion of the partial operator matrix $$\left( {\begin{array}{*{20}{c}} A&? \\ 0&C \end{array}} \right)$$ . Based on the space decomposition technique, the alternative sufficient and necessary conditions are given according to whether the dimension of $$\mathcal{R}(A)^ \bot$$ is finite or infinite. As a direct consequence, the perturbation of left spectra is further presented.

Invertible and regular completions of operator matrices

In this paper, for given operators A ∈ B(X) and B ∈ B(Y), the set of all C ∈ B(Y,X) such that the operator matrix M_C = \left[ \begin{array}{cc} A & C \\ O & B \end{array} \right] is injective, invertible, left invertible and right invertible, is described. Answers to some open questions are given. Also, in the case when A and B are relatively regular operators, the set of all C ∈ B(Y,X) such that M_C is regular is described. In addition, a necessary and a sufficient conditions are given for MC to be regular with the inner inverse of a certain given form.

Completions of partial matrices

Completions of partial matrices

Characterization of Consistent Completion of Reciprocal Comparison Matrices

Analytic hierarchy process (AHP) is a leading multi-attribute decision-aiding model that is designed to help make better choices when faced with complex decisions involving several dimensions. AHP, which enables qualitative analysis using a combination of subjective and objective information, is a multiple criteria decision analysis approach that uses hierarchical structured pairwise comparisons. One of the drawbacks of AHP is that a pairwise comparison cannot be completed by an actor or stakeholder not fully familiar with all the aspects of the problem. The authors have developed a completion based on a process of linearization that minimizes the matrix distance defined in terms of the Frobenius norm (a strictly convex minimization problem). In this paper, we characterize when an incomplete, positive, and reciprocal matrix can be completed to become a consistent matrix. We show that this characterization reduces the problem to the solution of a linear system of equations—a straightforward procedure. Various properties of such a completion are also developed using graph theory, including explicit calculation formulas. In real decision-making processes, facilitators conducting the study could use these characterizations to accept an incomplete comparison body given by an actor or to encourage the actor to further develop the comparison for the sake of consistency.

TP_k completions of partial matrices with one unspecified entry

Every partial TP2 (TP1) matrix with one unspecified entry has a TP2 (TP1) com- pletion. For a given m-by-n pattern with one unspecified entry, the minimum set of conditions characterizing TP3 completability is given. These conditions are at most eight polynomial inequali- ties on the specified entries of the pattern. For k � 3, patterns with one unspecified entry that are TPk completable are also characterized, and conditions are described for completability otherwise.

On completions of symmetric and antisymmetric block diagonal partial matrices

A partial matrix is a matrix where only some of the entries are given. We determine the maximum rank of the symmetric completions of a symmetric partial matrix where only the diagonal blocks are given and the minimum rank and the maximum rank of the antisymmetric completions of an antisymmetric partial matrix where only the diagonal blocks are given.

Estimation of missing RTTs in computer networks: Matrix completion vs compressed sensing

We estimate the missing round trip time (RTT) measurements in computer networks using doubly non-negative (DN) matrix completion and compressed sensing. The major contributions of this paper are the following: (i) an iterative DN matrix completion that minimizes the mean square estimation error; (ii) mathematical conditions for the convergence of the algorithm; (iii) systematic and detailed experimental comparison of DN matrix completion and compressed sensing for estimating missing RTT estimation in computer networks. To our knowledge, this is the first work that compares the pros and cons of compressed sensing and DN matrix completion for RTT estimation using actual Internet measurement data. Results indicate that compressed sensing provides better estimation in networks with sporadic missing values while DN completion of matrices is more suitable for estimation in networks which miss blocks of measurements. Our proposed DN matrix completion method is one of the first approaches to matrix completion, that minimizes the estimation error.

The elliptic matrix completion problem

Completions of partial elliptic matrices are studied. Given an undirected graph G , it is shown that every partial elliptic matrix with graph G can be completed to an elliptic matrix if and only if the maximal cliques of G are pairwise disjoint. Further, given a partial elliptic matrix A with undirected graph G, it is proved that if G is chordal and each specified principal submatrix defined by a pair of intersecting maximal cliques is nonsingular, then A can be completed to an elliptic matrix. Conversely, if G is nonchordal or if the regularity condition is relaxed, it is shown that there exist partial elliptic matrices which are not completable to an elliptic matrix. In the process we obtain several results concerning chordal graphs that may be of independent interest.

The positive definite completion problem revisited

In [R. Grone, C.R. Johnson, E. Sa, H. Wolkowicz, Positive definite completions of partial Hermitian matrices, Linear Algebra Appl. 58 (1984) 109–124] the positive definite (semi-) completion problem in which the underlying graph is chordal was solved. For the positive definite case, the process was constructive and the completion was obtained by completing the partial matrix an entry at a time. For the positive semidefinite case, they obtained completions of a particular sequence of partial positive definite matrices with the same underlying graph and noted that there is a convergent subsequence of these completions that converges to the desired completion. Here, in the chordal case, we provide a constructive solution, based entirely on matrix/graph theoretic methods, to the positive (semi-)definite completion problem. Our solution associates a specific tree (called the “clique tree” [C.R. Johnson, M. Lundquist, Matrices with chordal inverse zero-patterns, Linear and Multilinear Algebra 36 (1993) 1–17]) with the (chordal) graph of the given partial positive (semi-)definite matrix. This tree structure allows us to complete the matrix a “block at a time” as opposed to an “entry at a time” (as in Grone et al. (1984) for the positive definite case). In Grone et al. (1984), using complex analytic techniques, the completion for the positive definite case was shown to be the unique determinant maximizing completion and was shown to be the unique completion that has zeros in its inverse in the positions corresponding to the unspecified entries of the partial matrix. Here, we show the same using only matrix/graph theoretic tools.

A spectral completion of 2x2 operator matrices

A spectral completion of 2x2 operator matrices