Eigenvalue Condition Research Articles

Analysis of eigenvalue condition numbers for a class of randomized numerical methods for singular matrix pencils

The numerical solution of the generalized eigenvalue problem for a singular matrix pencil is challenging due to the discontinuity of its eigenvalues. Classically, such problems are addressed by first extracting the regular part through the staircase form and then applying a standard solver, such as the QZ algorithm, to that regular part. Recently, several novel approaches have been proposed to transform the singular pencil into a regular pencil by relatively simple randomized modifications. In this work, we analyze three such methods by Hochstenbach, Mehl, and Plestenjak that modify, project, or augment the pencil using random matrices. All three methods rely on the normal rank and do not alter the finite eigenvalues of the original pencil. We show that the eigenvalue condition numbers of the transformed pencils are unlikely to be much larger than the δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document}-weak eigenvalue condition numbers, introduced by Lotz and Noferini, of the original pencil. This not only indicates favorable numerical stability but also reconfirms that these condition numbers are a reliable criterion for detecting simple finite eigenvalues. We also provide evidence that, from a numerical stability perspective, the use of complex instead of real random matrices is preferable even for real singular matrix pencils and real eigenvalues. As a side result, we provide sharp left tail bounds for a product of two independent random variables distributed with the generalized beta distribution of the first kind or Kumaraswamy distribution.

Condition numbers for real eigenvalues of real elliptic ensemble: weak non-normality at the edge

Sensitivity of an eigenvalue λ i to the perturbation of matrix elements is controlled by the eigenvalue condition number defined as κi=⟨Li|Li⟩⟨Ri|Ri⟩ , where ⟨Li| and |Ri⟩ are left and right eigenvectors to the eigenvalue λ i . In random matrix theory the squared eigenvalue condition number is also known as the eigenvector self-overlap. In this work we calculate the asymptotics of the joint probability density function of the real eigenvalue and the square of the corresponding eigenvalue condition number for the real elliptic ensemble in the double scaling regime of almost Hermiticity and close to the edge of the spectrum. As a byproduct, we also calculate the one-parameter deformation of the Scorer’s function.

Optimal lower bound on eigenvector overlaps for non-Hermitian random matrices

We consider large non-Hermitian N×N matrices with an additive independent, identically distributed (i.i.d.) noise for each matrix elements. We show that already a small noise of variance 1/N completely thermalises the bulk singular vectors, in particular they satisfy the strong form of Quantum Unique Ergodicity (QUE) with an optimal speed of convergence. In physics terms, we thus extend the Eigenstate Thermalisation Hypothesis, formulated originally by Deutsch [34] and proven for Wigner matrices in [23], to arbitrary non-Hermitian matrices with an i.i.d. noise. As a consequence we obtain an optimal lower bound on the diagonal overlaps of the corresponding non-Hermitian eigenvectors. This quantity, also known as the (square of the) eigenvalue condition number measuring the sensitivity of the eigenvalue to small perturbations, has notoriously escaped rigorous treatment beyond the explicitly computable Ginibre ensemble apart from the very recent upper bounds given in [7] and [45]. As a key tool, we develop a new systematic decomposition of general observables in random matrix theory that governs the size of products of resolvents with deterministic matrices in between.

Eigenvalue sensitivity from eigenstate geometry near and beyond arbitrary-order exceptional points

Systems with an effectively non-Hermitian Hamiltonian display an enhanced sensitivity to parametric and dynamic perturbations, which arises from the nonorthogonality of their eigenstates. This enhanced sensitivity can be quantified by the phase rigidity, which mathematically corresponds to the eigenvalue condition number and physically also determines the Petermann factor of quantum noise theory. Here, we derive an exact nonperturbative expression for this sensitivity measure that applies to arbitrary eigenvalue configurations. The expression separates spectral correlations from additional geometric data and retains a simple asymptotic behavior close to exceptional points (EPs) of any order, while capturing the role of additional states in the system. This reveals that such states can have a sizable effect even if they are spectrally well separated and identifies the specific matrix whose elements determine this nonperturbative effect. The employed algebraic approach, which follows the eigenvectors-from-eigenvalues school of thought, also provides direct insights into the geometry of the states near an EP. For instance, it can be used to show that the phase rigidity follows a striking equipartition principle in the quasidegenerate subspace of a system. Published by the American Physical Society 2024

A note on the solution to one-dimensional advection-dispersion-reaction problems

The series representation of the reactant concentration in one-dimensional advection-dispersion-reaction problems within a container of finite length is derived in a compact and computationally less complex form than other representations found in the literature. The initial concentration of the reactant is assumed to be uniform (C 0), while the boundary conditions are assumed to be a constant reactant flux ( U C e ) at the inlet and a zero concentration gradient at the outlet of the container, where U is the average fluid velocity. The concentration is expressed in terms of the introduced constants a and b for the steady-state part, and In , Jn and Kn for the transient part of the response. The derived solution to the advection-dispersion-reaction problem can be readily generalized to include a uniform source/sink term (zero-order production rate σ) by making the replacements of C 0 and C e with C 0 − σ / k and C e − σ / k , where k is the reaction rate constant, and by adding the particular solution σ / k . A simple form of the solution to the advection-dispersion equation with a source term is also derived; its transient part is expressed in terms of the integrals In and Kn only. The formulation of the analysis is cast in such a way that the same eigenfunctions and the same eigenvalue condition apply to all three considered cases, advection-dispersion-reaction with and without a source-term, and advection-dispersion with a source term, independently of the values of k and σ.

Positivity certificates for P-recursive sequences

We extend the decidability of the positivity problem for P-recursive sequences of Poincaré type having one simple dominant eigenvalue and generic initial conditions. We give an algorithm that produces a certificate of positivity that is a data-structure for a proof by induction. The main ingredient of our approach is the existence of a positive cone which is stable by the iteration of the recurrence.

Low-rank approximation of Hankel matrices in denoising applications for statistical damage diagnosis of wind turbine blades

Model order selection is a fundamental task in subspace identification for estimation of modal parameters, uncertainty propagation and damage diagnosis. However, the true model order and the related low-rank structure of the dynamic system are generally unknown. In this paper, a statistical methodology to actively select the dynamic signal subspace in covariance-driven subspace identification is developed on the basis of statistical analysis of the eigenvalue condition numbers of the output covariance Hankel matrix. It is shown that the condition numbers highly sensitive to random perturbations characterize the noise subspace. The signal subspace is separated from the noise subspace by analyzing two statistical parameters associated with the condition number sensitivity, whose thresholds are user-defined. A practical algorithm to retrieve the system dynamics is designed and demonstrated on a running example of a simulated wind turbine blade benchmark. The resultant framework is then applied in the context of damage detection on a medium-size wind turbine blade. It is demonstrated that the detectability of small damage is enhanced compared to the classic approaches and robustness of damage diagnosis is increased by reducing the number of false alarms.

The fundamental solution to \Box_{𝑏} on quadric manifolds with nonzero eigenvalues

This paper is part of a continuing examination into the geometric and analytic properties of the Kohn Laplacian and its inverse on general quadric submanifolds of C n × C m \mathbb {C}^n\times \mathbb {C}^m . The goal of this article is explore the complex Green operator in the case that the eigenvalues of the directional Levi forms are nonvanishing. We (1) investigate the geometric conditions on M M which the eigenvalue condition forces, (2) establish optimal pointwise upper bounds on complex Green operator and its derivatives, (3) explore the L p L^p and L p L^p -Sobolev mapping properties of the associated kernels, and (4) provide examples.

The largest eigenvalue conditions for Hamiltonian and traceable graphs

The largest eigenvalue conditions for Hamiltonian and traceable graphs

On the proofs of orthogonality of eigenfunctions for heat conduction, wave propagation, and advection-diffusion problems

The orthogonality of eigenfunctions in problems of unsteady heat conduction in an infinite slab with symmetric and nonsymmetric convective boundary conditions are demonstrated by performing actual integration of the products of the derived forms of eigenfunctions and by implementing the corresponding eigenvalue conditions. The analysis also applies to longitudinal vibrations of an elastic rod attached at its ends to linear elastic springs, and to advection-diffusion problems under appropriate boundary conditions. The same form of eigenfunctions and the same type of eigenvalue condition apply in all three considered cases. The proofs are compared with the well-known general proof of orthogonality of eigenfunctions, which circumvents the actual integration. The presented analysis is pedagogically appealing for use in engineering and applied physics education.

Singular quadratic eigenvalue problems: linearization and weak condition numbers

The numerical solution of singular eigenvalue problems is complicated by the fact that small perturbations of the coefficients may have an arbitrarily bad effect on eigenvalue accuracy. However, it has been known for a long time that such perturbations are exceptional and standard eigenvalue solvers, such as the QZ algorithm, tend to yield good accuracy despite the inevitable presence of roundoff error. Recently, Lotz and Noferini quantified this phenomenon by introducing the concept of $$\delta $$ -weak eigenvalue condition numbers. In this work, we consider singular quadratic eigenvalue problems and two popular linearizations. Our results show that a correctly chosen linearization increases $$\delta $$ -weak eigenvalue condition numbers only marginally, justifying the use of these linearizations in numerical solvers also in the singular case. We propose a very simple but often effective algorithm for computing well-conditioned eigenvalues of a singular quadratic eigenvalue problems by adding small random perturbations to the coefficients. We prove that the eigenvalue condition number is, with high probability, a reliable criterion for detecting and excluding spurious eigenvalues created from the singular part.

Stability Conditions for Cluster Synchronization in Directed Networks of Diffusively Coupled Nonlinear Systems

This paper investigates the stability issue of cluster synchronization manifold in networks of diffusively-coupled nonlinear system with directed topology. It is assumed that each cluster subdigraph is strongly connected and contains only cooperative interactions, but there may exist competitions among nodes of different clusters. In the case that the clusters meet the cluster-input-equivalence condition and each nonlinear system has a forward invariant set over which the system possesses a bounded Jacobian, some local stability conditions of cluster synchronization are derived. These conditions are expressed as an inequality of the matrix measure of the system’s Jacobian matrix, the matrix measure of a reduced-order matrix about the digraph among the clusters, the coupling strength, and some eigenvalue conditions of the subdigraphs. Finally, the theoretical findings are validated by two numerical examples about coupled Lorenz-like system and coupled Hopfield neural network, respectively.

Explicit formulas concerning eigenvectors of weakly non-unitary matrices

We investigate eigenvector statistics of the Truncated Unitary ensemble $\mathrm{TUE}(N,M)$ in the weakly non-unitary case $M=1$, that is when only one row and column are removed. We provide an explicit description of generalized overlaps as deterministic functions of the eigenvalues, as well as a method to derive an exact formula for the expectation of diagonal overlaps (or squared eigenvalue condition numbers), conditionally on one eigenvalue. This complements recent results obtained in the opposite regime when $M \geq N$, suggesting possible extensions to $\mathrm{TUE}(N,M)$ for all values of $M$.

A Simplified Algorithm for Setting the Observer Parameters for Second-Order Systems with Persistent Disturbances Using a Robust Observer.

The properties of the convergence region of the estimation error of a robust observer for second-order systems are determined, and a new algorithm is proposed for setting the observer parameters, considering persistent but bounded disturbances in the two observation error dynamics. The main contributions over closely related studies of the stability of state observers are: (i) the width of the convergence region of the observer error for the unknown state is expressed in terms of the interaction between the observer parameters and the disturbance terms of the observer error dynamics; (ii) it was found that this width has a minimum point and a vertical asymptote with respect to one of the observer parameters, and their coordinates were determined. In addition, the main advantages of the proposed algorithm over closely related algorithms are: (i) the definition of observer parameters is significantly simpler, as the fulfillment of Riccati equation conditions, solution of LMI constraints, and fulfillment of eigenvalue conditions are not required; (ii) unknown bounded terms are considered in the dynamics of the observer error for the known state. Finally, the algorithm is applied to a model of microalgae culture in a photobioreactor for the estimation of biomass growth rate and substrate uptake rate based on known concentrations of biomass and substrate.

On the Condition Number of the Shifted Real Ginibre Ensemble

We derive an accurate lower tail estimate on the lowest singular value $\sigma_1(X-z)$ of a real Gaussian (Ginibre) random matrix $X$ shifted by a complex parameter $z$. Such shift effectively changes the upper tail behaviour of the condition number $\kappa(X-z)$ from the slower $\mathbf{P}(\kappa(X-z)\ge t)\lesssim 1/t$ decay typical for real Ginibre matrices to the faster $1/t^2$ decay seen for complex Ginibre matrices as long as $z$ is away from the real axis. This sharpens and resolves a recent conjecture in [arXiv:2005.08930] on the regularizing effect of the real Ginibre ensemble with a genuinely complex shift. As a consequence we obtain an improved upper bound on the eigenvalue condition numbers (known also as the eigenvector overlaps) for real Ginibre matrices. The main technical tool is a rigorous supersymmetric analysis from our earlier work [arXiv:1908.01653].

Thermal impact induced by the environment in the transport of heavy oils in offshore insulated pipelines: Evaluation of heat transfer

In this work, the non-isothermal transport of heavy oils in offshore insulated pipelines is theoretically analyzed. Here, because heavy oils show poor thermostability in terms of the dynamic viscosity, the evaluation of the two physical phenomena involved in the mentioned process (flow hydrodynamics and heat transfer) must be carried out in a coupled manner. In addition, if the performance of this type of transport system is considered, and a novel numerical approach is employed, the present mathematical formulation allows defining the relationship between flow assurance and the thermal impact induced by certain environmental conditions. The above is the main contribution of this work, which is achieved by obtaining the results considering an eigenvalue condition, and making the numerical scheme a useful tool to determine an adequate thermal shielding. First, the heat transfer in the transport system is estimated for given thermal–hydrodynamic conditions, and then, the numerical algorithm corrects the thermal shielding (insulation thickness) in such a way that the flow rate is preserved. In summary, the main results indicate that, when a section of the pipeline has a deficient thermal shielding, the flow hydrodynamics receives a considerable thermal impact, causing a significant reduction in the flow rate. Instead, when the thermal shielding is improved, the flow rate increases as the heat flux that is emitted by the pipeline to the environment is confined. Unfortunately, introducing a strong thermal shielding along the entire length of the pipeline implies an excessive use of thermal insulation (cost–benefit); therefore, rational and adequate use of thermal insulation barriers is the most appropriate condition. Such a conclusion is demonstrated by comparing the two study cases using the same amount of thermal insulation (value of the Graetz–Nusselt relationship); here, the flow rate increases more than double for the case as the thermal shielding is numerically corrected, in comparison to the case when the thermal insulation thickness is constant.

Group Logistic Regression Models with lp,q Regularization

In this paper, we proposed a logistic regression model with lp,q regularization that could give a group sparse solution. The model could be applied to variable-selection problems with sparse group structures. In the context of big data, the solutions for practical problems are often group sparse, so it is necessary to study this kind of model. We defined the model from three perspectives: theoretical, algorithmic and numeric. From the theoretical perspective, by introducing the notion of the group restricted eigenvalue condition, we gave the oracle inequality, which was an important property for the variable-selection problems. The global recovery bound was also established for the logistic regression model with lp,q regularization. From the algorithmic perspective, we applied the well-known alternating direction method of multipliers (ADMM) algorithm to solve the model. The subproblems for the ADMM algorithm were solved effectively. From the numerical perspective, we performed experiments for simulated data and real data in the factor stock selection. We employed the ADMM algorithm that we presented in the paper to solve the model. The numerical results were also presented. We found that the model was effective in terms of variable selection and prediction.

Moving beyond sub-Gaussianity in high-dimensional statistics: applications in covariance estimation and linear regression

Abstract Concentration inequalities form an essential toolkit in the study of high-dimensional statistical methods. Most of the relevant statistics literature in this regard is, however, based on the assumptions of sub-Gaussian or sub-exponential random variables/vectors. In this paper, we first bring together, through a unified exposition, various probabilistic inequalities for sums of independent random variables under much more general exponential type (namely sub-Weibull) tail assumptions. These results extract a part sub-Gaussian tail behavior of the sum in finite samples, matching the asymptotics governed by the central limit theorem, and are compactly represented in terms of a new Orlicz quasi-norm—the Generalized Bernstein–Orlicz norm—that typifies such kind of tail behaviors. We illustrate the usefulness of these inequalities through the analysis of four fundamental problems in high-dimensional statistics. In the first two problems, we study the rate of convergence of the sample covariance matrix in terms of the maximum elementwise norm and the maximum $k$-sub-matrix operator norm that are key quantities of interest in bootstrap procedures and high-dimensional structured covariance matrix estimation, as well as in high-dimensional and post-selection inference. The third example concerns the restricted eigenvalue condition, required in high-dimensional linear regression, which we verify for all sub-Weibull random vectors through a unified analysis, and also prove a more general result related to restricted strong convexity in the process. In the final example, we consider the Lasso estimator for linear regression and establish its rate of convergence to be generally $\sqrt{k\log p/n}$, for $k$-sparse signals, under much weaker than usual tail assumptions (on the errors as well as the covariates), while also allowing for misspecified models and both fixed and random design. To our knowledge, these are the first such results for Lasso obtained in this generality. The common feature in all our results over all the examples is that the convergence rates under most exponential tails match the usual (optimal) ones obtained under sub-Gaussian assumptions. Finally, we also establish some complementary results on analogous tail bounds for the suprema of empirical processes indexed by sub-Weibull variables. All our results are finite samples.

Frequency Synchronization of a High-Order Multiconverter System

We investigate the stability properties of a multi-converter power system model, defined on a high-order manifold. For this, we identify its symmetry (i.e., rotational invariance) generated by a static angle shift and rotation of AC signals. We characterize the steady state set, primarily determined by the steady state angles and DC power input. Based on eigenvalue conditions of its Jacobian matrix, we show asymptotic stability of the multi-converter system in a neighborhood of the synchronous steady state set by applying the center manifold theory. We guarantee the eigenvalue conditions via an explicit approach. Finally, we demonstrate our results based on a numerical example involving a network of identical DC/AC converter systems.

Heterogeneous Overdispersed Count Data Regressions via Double-Penalized Estimations

Recently, the high-dimensional negative binomial regression (NBR) for count data has been widely used in many scientific fields. However, most studies assumed the dispersion parameter as a constant, which may not be satisfied in practice. This paper studies the variable selection and dispersion estimation for the heterogeneous NBR models, which model the dispersion parameter as a function. Specifically, we proposed a double regression and applied a double ℓ1-penalty to both regressions. Under the restricted eigenvalue conditions, we prove the oracle inequalities for the lasso estimators of two partial regression coefficients for the first time, using concentration inequalities of empirical processes. Furthermore, derived from the oracle inequalities, the consistency and convergence rate for the estimators are the theoretical guarantees for further statistical inference. Finally, both simulations and a real data analysis demonstrate that the new methods are effective.