Symmetric Matrix Research Articles

A note on switching eigenvalues under small perturbations

Sensitivity of eigenvectors and eigenvalues of symmetric matrix estimates to the removal of a single observation have been well documented in the literature. However, a complicating factor can exist in that the rank of the eigenvalues may change due to the removal of an observation, and with that so too does the perceived importance of the corresponding eigenvector. We refer to this problem as “switching of eigenvalues”. Since there is not enough information in the new eigenvalues, post observation removal, to indicate that this has happened, how do we know that this switching has occurred? In this article, we show that approximations to the eigenvalues can be used to help determine when switching may have occurred. We then discuss possible actions researchers can take based on this knowledge, for example making better choices when it comes to deciding how many principal components should be retained and adjustments to approximate influence diagnostics that perform poorly when switching has occurred. Our results are easily applied to any eigenvalue problem involving symmetric matrix estimators. We highlight our approach with application to two real data examples.

Multiview Clustering via Hypergraph Induced Semi-Supervised Symmetric Nonnegative Matrix Factorization

Nonnegative matrix factorization (NMF) based multiview technique has been commonly used in multiview data clustering tasks. However, previous NMF based multiview clustering approaches fail to take advantage of a small amount of supervisory information to effectively improve the clustering performance, and are easily affected by the additional post-processing method in clustering tasks. To cope with these issues, a novel framework named multiview clustering via hypergraph induced semi-supervised symmetric NMF (MVCHSS) is proposed in this paper for multiview data clustering applications. Specifically, the proposed method has the following features: 1) a new multiview based hypergraph pairwise constraints propagation (MHPCP) algorithm is developed in MVCHSS to construct a set of informative similarity matrices, revealing the high-order relationships effectively and fully utilizing the limited pairwise constraint supervisory information among samples of each view data; 2) the obtained similarity matrices with much supervisory information are not only enforced into the symmetric NMF (SNMF) model, but also incorporated into the graph regularization for each view data; 3) the optimization problem of MVCHSS is formulated for multiview data clustering tasks to acquire a more discriminative clustering indicator matrix (or called consensus assignment matrix) without additional post-processing method. Moreover, the proof of convergence and the computational complexity for MVCHSS are presented. Extensive experiments on five multiview datasets demonstrate that the proposed MVCHSS framework outperforms several state-of-the-art multiview clustering methods.

On eigenvalues of the Brownian sheet matrix

We derive a system of stochastic partial differential equations satisfied by the eigenvalues of the symmetric matrix whose entries are the Brownian sheets. We prove that the sequence Ld(s,t),(s,t)∈[0,S]×[0,T]d∈N of empirical spectral measures of the rescaled matrices is tight on C([0,S]×[0,T],P(R)) and hence is convergent as d goes to infinity by Wigner’s semicircle law. We also obtain PDEs which are satisfied by the high-dimensional limiting measure.

Riemannian SPD learning to represent and characterize fixational oculomotor Parkinsonian abnormalities

Parkinson’s disease (PD) is the second most common neurodegenerative disorder, mainly characterized by motor alterations. Despite multiple efforts, there is no definitive biomarker to diagnose, quantify, and characterize the disease early. Recently, abnormal fixational oculomotor patterns have emerged as a promising disease biomarker with high sensitivity, even at early stages. Nonetheless, the complex patterns and potential correlations with the disease remain largely unexplored, among others, because of the limitations of standard setups that only analyze coarse measures and poorly exploit the associated PD alterations. This work introduces a new strategy to represent, analyze and characterize fixational patterns from non-invasive video analysis, adjusting a geometric learning strategy. A deep Riemannian framework is proposed to discover potential oculomotor patterns aimed at withstanding data scarcity and geometrically interpreting the latent space. A convolutional representation is first built, then aggregated onto a symmetric positive definite matrix (SPD). The latter encodes second-order statistics of deep convolutional features and feeds a non-linear hierarchical architecture that processes SPD data by maintaining them into their Riemannian manifold. The complete representation discriminates between Parkinson and Healthy (Control) fixational observations, even at PD stages 2.5 and 3. Besides, the proposed geometrical representation exhibit capabilities to statistically differentiate observations among Parkinson’s stages. The developed tool demonstrates coherent results from explainability maps back-propagated from output probabilities.

Computation of the von Neumann entropy of large matrices via trace estimators and rational Krylov methods

We consider the problem of approximating the von Neumann entropy of a large, sparse, symmetric positive semidefinite matrix A, defined as {{,textrm{tr},}}(f(A)) where f(x)=-xlog x. After establishing some useful properties of this matrix function, we consider the use of both polynomial and rational Krylov subspace algorithms within two types of approximations methods, namely, randomized trace estimators and probing techniques based on graph colorings. We develop error bounds and heuristics which are employed in the implementation of the algorithms. Numerical experiments on density matrices of different types of networks illustrate the performance of the methods.

The inverse of a symmetric nonnegative matrix can be copositive

Let $A$ be an $n\times n$ symmetric matrix. We first show that if $A$ and its pseudoinverse are strictly copositive, then $A$ is positive semidefinite, which extends a similar result of Han and Mangasarian. Suppose $A$ is invertible, as well as being symmetric. We showed in an earlier paper that if $A^{-1}$ is nonnegative with $n$ zero diagonal entries, then $A$ can be copositive (for instance, this happens with the Horn matrix), and when $A$ is copositive, it cannot be of form $P+N$, where $P$ is positive semidefinite and $N$ is nonnegative and symmetric. Here, we show that if $A^{-1}$ is nonnegative with $n-1$ zero diagonal entries and one positive diagonal entry, then $A$ can be of the form $P+N$, and we show how to construct $A$. We also show that if $A^{-1}$ is nonnegative with one zero diagonal entry and $n-1$ positive diagonal entries, then $A$ cannot be copositive.

On the Forsythe conjecture

Forsythe formulated a conjecture about the asymptotic behavior of the restarted conjugate gradient method in 1968. We translate several of his results into modern terms, and pose an analogous version of the conjecture (originally formulated only for symmetric positive definite matrices) for symmetric and nonsymmetric matrices. Our version of the conjecture uses a two-sided or cross iteration with the given matrix and its transpose, which is based on the projection process used in the Arnoldi (or for symmetric matrices the Lanczos) algorithm. We prove several new results about the limiting behavior of this iteration, but the conjecture still remains largely open. We hope that our paper motivates further research that eventually leads to a proof of the conjecture.

Decompositions of optimal averaged Gauss quadrature rules

Optimal averaged Gauss quadrature rules provide estimates for the quadrature error in Gauss rules, as well as estimates for the error incurred when approximating matrix functionals of the form uTf(A)v with a large matrix A∈RN×N by low-rank approximations that are obtained by applying a few steps of the symmetric or nonsymmetric Lanczos processes to A; here u,v∈RN are vectors. The latter process is used when the measure associated with the Gauss quadrature rule has support in the complex plane. The symmetric Lanczos process yields a real tridiagonal matrix, whose entries determine the recursion coefficients of the monic orthogonal polynomials associated with the measure, while the nonsymmetric Lanczos process determines a nonsymmetric tridiagonal matrix, whose entries are recursion coefficients for a pair of sets of bi-orthogonal polynomials. Recently, it has been shown, by applying the results of Peherstorfer, that optimal averaged Gauss quadrature rules, which are associated with a nonnegative measure with support on the real axis, can be expressed as a weighted sum of two quadrature rules. This decomposition allows faster evaluation of optimal averaged Gauss quadrature rules than the previously available representation. The present paper provides a new self-contained proof of this decomposition that is based on linear algebra techniques. Moreover, these techniques are generalized to determine a decomposition of the optimal averaged quadrature rules that are associated with the tridiagonal matrices determined by the nonsymmetric Lanczos process. Also, the splitting of complex symmetric tridiagonal matrices is discussed. The new splittings allow faster evaluation of optimal averaged Gauss quadrature rules than the previously available representations. Computational aspects are discussed.

An intrinsic anisotropic feature of DTI images derived by geometric properties on the Riemannian manifold

Diffusion tensor imaging (DTI) is a promising technique for characterizing organic microstructural information by mapping a spatial point as a 3 × 3 symmetric matrix (called diffusion tensor). Analyzing DTI images directly is challenging since each element of the DTI image is a 3-order matrix rather than a scalar. It is meaningful to extract the scalar-valued feature for DTI image processing and analysis. In this paper, we propose an intrinsic anisotropic feature of DTI images using global geometric attributes of diffusion tensors on the Riemannian manifold. To measure the intrinsic similarity between diffusion tensors as symmetric matrices, we convert them into symmetric positive definite (SPD) matrices and use the normalized geodesic distance in the tensor manifold to define the visible intrinsic feature. Moreover, we extend the geodesic distance to make sure our feature can measure the intrinsic properties of diffusion tensors, and the neighborhood attributes and even the global information of a DTI image are utilized to define the proposed feature. To measure the anisotropy of DTI images, we define the average diffusion sphere at each point to measure the isotropic attributes and use the difference between the diffusion tensor and its average diffusion tensor to describe the anisotropic properties. It is experimentally demonstrated that the proposed feature performs adequately in characterizing various attributes of DTI images, including but not limited anisotropy, intrinsic geometric property, and average diffusion degree, etc. Compared with the most popular features, our anisotropy feature can represent DTI image properties more robustly and effectively since the contrast of various details in the feature is improved. Moreover, the results of fiber tracking defined by the anisotropy feature indicate that our feature performs better on representing substantial anisotropic attributes of DTI images and is potentially used in DTI image processing and analysis.

Distribution of Eigenvalues and Upper Bounds of the Spread of Interval Matrices

The distribution of eigenvalues and the upper bounds for the spread of interval matrices are significant in various fields of mathematics and applied sciences, including linear algebra, numerical analysis, control theory, and combinatorial optimization. We present the distribution of eigenvalues within interval matrices and determine upper bounds for their spread using Geršgorin’s theorem. Specifically, through an equality for the variance of a discrete random variable, we derive upper bounds for the spread of symmetric interval matrices. Finally, we give three numerical examples to illustrate the effectiveness of our results.

A new bounded real lemma for discrete‐time linear descriptor systems

AbstractThis paper presents a new bounded real lemma for discrete‐time linear descriptor systems. We first present a necessary and sufficient condition for the admissibility of discrete‐time descriptor systems expressed in terms of strict linear matrix inequalities (LMIs). We then prove that a discrete‐time descriptor system is ‐stable if and only if a symmetric matrix satisfying two strict LMIs can be found. Two illustrative examples are provided for verification.

Bordering of symmetric matrices and an application to the minimum number of distinct eigenvalues for the join of graphs

An important facet of the inverse eigenvalue problem for graphs is to determine the minimum number of distinct eigenvalues of a particular graph. We resolve this question for the join of a connected graph with a path. We then focus on bordering a matrix and attempt to control the change in the number of distinct eigenvalues induced by this operation. By applying bordering techniques to the join of graphs, we obtain numerous results on the nature of the minimum number of distinct eigenvalues as vertices are joined to a fixed graph.

The Problem of Moments: A Bunch of Classical Results with Some Novelties

We summarize significant classical results on (in)determinacy of measures in terms of their finite positive integer order moments. Well known is the role of the smallest eigenvalues of Hankel matrices, starting from Hamburger’s results a century ago and ending with the great progress made only in recent times by C. Berg and collaborators. We describe here known results containing necessary and sufficient conditions for moment (in)determinacy in both Hamburger and Stieltjes moment problems. In our exposition, we follow an approach different from that commonly used. There are novelties well complementing the existing theory. Among them are: (a) to emphasize on the geometric interpretation of the indeterminacy conditions; (b) to exploit fine properties of the eigenvalues of perturbed symmetric matrices allowing to derive new lower bounds for the smallest eigenvalues of Hankel matrices (these bounds are used for concluding indeterminacy); (c) to provide new arguments to confirm classical results; (d) to give new numerical illustrations involving commonly used probability distributions.

Binary Heterogroup-Templated Scaffolds of Polyoxopalladates as Precatalysts for Plasma-Assisted Ammonia Synthesis.

In addition to improving the synthetic efficiency, the template method can do a lot more in the chemistry of polyoxopalladates (POPs), such as the establishment of novel metal-oxo scaffolds. In this endeavor, a binary system comprising heterogroups of nonmetallic {As/SiO4} and metallic {VO4/5} successfully fulfills the templated growth of two POPs with unprecedented seesaw- and spindle-like prototypes. Of these, self-aggregation of heterogroups beacons an effective route to break the highly symmetrical PdII-oxo matrix and to force the arrangement of addenda in a nonconventional manner. Aside from the interest in their structural features, the as-made POPs are available for immobilization on the mesoporous SBA-15 as precatalysts for ammonia synthesis. The outer cover of heterogroups in the POP precursors contributes to the ultrafine size and uniform distribution of derived Pd0 nanoparticles (PdNPs). With the help of plasma activation on H2 and N2, such PdNPs-SBA15 catalysts significantly improve the production performance of NH3, showcasing the maximum synthesis rate of 64.42 μmol/(min·gcat) with the corresponding energy yield as high as 4.38 g-NH3/kWh.

Eigenvalue Tests for the Number of Latent Factors in Short Panels

Abstract This article studies new tests for the number of latent factors in a large cross-sectional factor model with small time dimension. These tests are based on the eigenvalues of variance–covariance matrices of (possibly weighted) asset returns and rely on either an assumption of spherical errors, or instrumental variables for factor betas. We establish the asymptotic distributional results using expansion theorems based on perturbation theory for symmetric matrices. Our framework accommodates semi-strong factors in the systematic components. We propose a novel statistical test for weak factors against strong or semi-strong factors. We provide an empirical application to U.S. equity data. Evidence for a different number of latent factors according to market downturns and market upturns is statistically ambiguous in the considered subperiods. In particular, our results contradict the common wisdom of a single-factor model in bear markets.

A unified model for the fairness mechanism‐based coordinated vehicle route guidance

AbstractCoordinated vehicle route guidance is recognized as an effective way to alleviate the Braess' paradox that new congestion is generated since numerous vehicles obey the same guidance from Google Maps. In conventional route games, decision‐makers are assumed to be completely rational (maximized utility is taken as an optimization objective). However, behavioral research finds that not only the expected profit is pursued, but also the fairness psychology is concerned by decision‐makers in game theory. In existing studies, the fairness concern of players is widely accepted in supply chains. Inspired by the fairness psychology of supply chain members, the vehicle fairness coefficient is originally proposed in this paper. Moreover, a unified fairness mechanism of the routing game is built to perfect the existing theoretical framework of route games. Specifically, the formulated vehicle fairness coefficient is incorporated into the payoff functions of a perfectly rational‐based routing game, which makes the single‐objective problem to be multiple. Compared with the current completely rational‐based route games, this improvement allows vehicles to focus not only on travel time but also on route schemes' fairness. The consideration of vehicle fairness concern leads the original symmetric payoff matrix of the route game to be asymmetric. Since the existing solving theorems are not necessarily valid for the asymmetrical multi‐player game, a matched reinforcement learning algorithm is adopted to solve this new game. An experimental study shows that the averaged fairness coefficient is increased from 0.798 to 1.00 on the basis of the equal travel time due to the working of the designed fairness mechanism.

Analytic relationship of relative synchronizability to network structure and motifs

Synchronization phenomena on networks have attracted much attention in studies of neural, social, economic, and biological systems, yet we still lack a systematic understanding of how relative synchronizability relates to underlying network structure. Indeed, this question is of central importance to the key theme of how dynamics on networks relate to their structure more generally. We present an analytic technique to directly measure the relative synchronizability of noise-driven time-series processes on networks, in terms of the directed network structure. We consider both discrete-time autoregressive processes and continuous-time Ornstein-Uhlenbeck dynamics on networks, which can represent linearizations of nonlinear systems. Our technique builds on computation of the network covariance matrix in the space orthogonal to the synchronized state, enabling it to be more general than previous work in not requiring either symmetric (undirected) or diagonalizable connectivity matrices and allowing arbitrary self-link weights. More importantly, our approach quantifies the relative synchronization specifically in terms of the contribution of process motif (walk) structures. We demonstrate that in general the relative abundance of process motifs with convergent directed walks (including feedback and feedforward loops) hinders synchronizability. We also reveal subtle differences between the motifs involved for discrete or continuous-time dynamics. Our insights analytically explain several known general results regarding synchronizability of networks, including that small-world and regular networks are less synchronizable than random networks.

Vigilance estimation using truncated l1 distance kernel-based sparse representation regression with physiological signals

BackgroundWith a large number of accidents caused by the decline in the vigilance of operators, finding effective automatic vigilance monitoring methods is a work of great significance in recent years. Based on physiological signals and machine learning algorithms, researchers have opened up a path for objective vigilance estimation. MethodsSparse representation (SR)-based recognition algorithms with excellent performance and simple models are very promising approaches in this field. This paper aims to study the adaptability and performance improvement of truncated l1 distance (TL1) kernel on SR-based algorithm in the context of physiological signal vigilance estimation. Compared with the traditional radial basis function (RBF), the TL1 kernel has good adaptiveness to nonlinearity and is suitable for the discrimination of complex physiological signals. A recognition framework based on TL1 and SR theory is proposed. Firstly, the inseparable physiological features are mapped to the reproducing kernel Kreĭn space through the infinite-dimensional projection of the TL1 kernel. Then the obtained kernel matrix is converted into the symmetric positive definite matrix according to the eigenspectrum approaches. Finally, the final prediction result is obtained through the sparse representation regression process. ResultsWe verified the performance of the proposed framework on the popular SEED-VIG dataset containing physiological signals (electroencephalogram and electrooculogram) associated with vigilance. In the experimental results, the TL1 kernel is superior to the RBF kernel in both performance and kernel parameter stability. ConclusionsThis demonstrates the effectiveness of the TL1 kernel in distinguishing physiological signals and the excellent vigilance estimation capability of the proposed framework. Moreover, the contribution of our research motivates the development of physiological signal recognition based on kernel methods.

Conjugate variables approach tomixedq-Araki–Woods algebras: Factoriality and non-injectivity

We establish factoriality and non-injectivity in full generality for the mixed q-Araki–Woods von Neumann algebra associated to a separable real Hilbert space HR with dimHR≥2, a strongly continuous one parameter group of orthogonal transformations on HR, a direct sum decomposition HR=⊕iHR(i), and a real symmetric matrix (qij) with supi,j|qij| < 1. This is achieved by first proving the existence of conjugate variables for a finite number of generators of the algebras (when dimHR<∞), following the lines of Miyagawa–Speicher and Kumar–Skalski–Wasilewski. The conjugate variables belong to the factors in question and satisfy certain Lipschitz conditions.

An Improved Non-Negative Latent Factor Model for Missing Data Estimation via Extragradient-Based Alternating Direction Method.

In this article, an improved double factorization-based symmetric and non-negative latent factor (Im-DF-SNLF) model is proposed to make the estimation for missing data in symmetric, high-dimensional, and sparse (SHiDS) matrices. The main idea of the Im-DF-SNLF model is fourfold: 1) considering the data variety in the practical engineering, non-negative latent factors (NLFs) in different cases are considered to better reflect the latent relationships between entries; 2) the l₂-norm regularization and the Lagrangian multiplier technique are simultaneously adopted to handle the overfitting and satisfy the non-negative constraint for latent factors (LFs); 3) the extragradient-based alternating direction (EGAD) method is utilized to accelerate the model training and rigidly guarantee the non-negativity of LFS; and 4) a rigorous proof is provided to show that, under the given assumption that the objective function is smooth and has a Lipschitz continuous gradient, the designed algorithm can find an ε-optimal solution within O(1/ε), and the upper bound of the learning rate is given by 1/2. Finally, experimental results on public datasets are given to demonstrate the effectiveness of our proposed Im-DF-SNLF model with EGAD.