Eigenvalue Condition Research Articles

Infinite time blow-up for critical heat equation with drift terms

We construct infinite time blow-up solution to the following heat equation with Sobolev critical exponent and drift terms $$\begin{aligned} {\left\{ \begin{array}{ll} u_t \,=\, \Delta u\,+\,\nabla b (x) \cdot \nabla u\,+\, u^{\frac{n+2}{n-2}} ~ \text{ in } ~ \mathbb {R}^n\times (0,+\infty ),\\ u(\cdot ,0)=u_0 ~ \text{ in } ~ \mathbb {R}^n, \end{array}\right. } \end{aligned}$$where b(x) is a smooth bounded function in $$\mathbb {R}^{n}$$ with $$n\ge 5$$ and the initial datum $$u_0$$ is positive and smooth. Let $$q_j \in \mathbb {R}^n,j=1,\ldots ,k$$, be distinct nondegenerate local minimum points of b(x). Assume that an eigenvalue condition (1.6) is satisfied. We prove the existence of a positive smooth solution u(x, t) which blows up at infinite time near those points with the form $$\begin{aligned} u(x,t) \approx \sum _{j=1}^k \alpha _n \left( \frac{ \mu _j(t)}{ \mu _j(t)^2 \,+\, |x-\xi _j(t)|^2 } \right) ^{\frac{n-2}{2}}, \quad \text{ as } t\rightarrow +\infty . \end{aligned}$$Here $$\xi _j(t) \rightarrow q_j$$ and $$0<\mu _j(t)\rightarrow 0$$ exponentially as $$t\rightarrow +\infty $$.

Matching Extendability and Connectivity of Regular Graphs from Eigenvalues

Let G be a graph on even number of vertices. A perfect matching of G is a set of independent edges which cover each vertex of G. For an integer $$t\ge 1$$, G is t-extendable if G has a perfect matching, and for any t independent edges, G has a perfect matching which contains these given t edges. The graph G is bi-critical if for any two vertices u and v, the graph $$G-\left\{ u,v\right\} $$ contains a perfect matching. Let G be a connected k-regular graph. In this paper, we obtain two sharp sufficient eigenvalue conditions for a connected k-regular graph to be 1-extendable and bi-critical, respectively. Our results are in term of the second largest eigenvalue of the adjacency matrix of G. We also give examples that show that there is no good spectral characterization (in term of the second largest eigenvalue of the adjacency matrix) for 2-extendability of regular graphs in general. Also, for any integer $$1\le \ell <\frac{k+2}{2}$$, we obtain an eigenvalue condition for G to be $$(\ell +1)$$-connected.

High-Dimensional Classification by Sparse Logistic Regression

We consider high-dimensional binary classification by sparse logistic regression. We propose a model/feature selection procedure based on penalized maximum likelihood with a complexity penalty on the model size and derive the non-asymptotic bounds for its misclassification excess risk. To assess its tightness, we establish the corresponding minimax lower bounds. The bounds can be reduced under the additional low-noise condition. The proposed complexity penalty is remarkably related to the Vapnik-Chervonenkis-dimension of a set of sparse linear classifiers. Implementation of any complexity penalty-based criterion, however, requires a combinatorial search over all possible models. To find a model selection procedure computationally feasible for high-dimensional data, we extend the Slope estimator for logistic regression and show that under an additional weighted restricted eigenvalue condition it is rate-optimal in the minimax sense.

Structured generalized eigenvalue condition numbers for parameterized quasiseparable matrices

In this paper, when A and B are {1;1}-quasiseparable matrices, we consider the structured generalized relative eigenvalue condition numbers of the pair $$(A, \, B)$$ with respect to relative perturbations of the parameters defining A and B in the quasiseparable and the Givens-vector representations of these matrices. A general expression is derived for the condition number of the generalized eigenvalue problems of the pair $$(A,\, B)$$ , where A and B are any differentiable function of a vector of parameters with respect to perturbations of such parameters. Moreover, the explicit expressions of the corresponding structured condition numbers with respect to the quasiseparable and Givens-vector representation via tangents for $$\{1; 1\}$$ -quasiseparable matrices are derived. Our proposed condition numbers can be computed efficiently by utilizing the recursive structure of quasiseparable matrices. We investigate relationships between various condition numbers of structured generalized eigenvalue problem when A and B are {1;1}-quasiseparable matrices. Numerical results show that there are situations in which the unstructured condition number can be much larger than the structured ones.

Analysis of Eigen Frequency Through a Simple Architectural Acoustic Modal.

In this paper, we have displayed an answer for the portrayal of the room acoustics field with the mix of uniform impedance limit conditions forced on a few dividers utilizing Fourier strategy. Acoustic limit conditions are communicated in the terms of retention coefficient esteems. In this issue, the Fourier technique is determined zas the blend of three one-dimensional Sturm-Liouville (S-L) issues with Robin-Robin limit conditions at the first and second measurement and Robin-Neumann ones at the third measurement. The Fourier strategy requires an assessment of eigenvalues and eigenfunctions of the Helmholtz condition, by means of the arrangement of the eigenvalue condition, every which way. The arrangements are examined utilizing chart. As a curiosity, it is exhibited that the Fourier technique gives a helpful and productive approach for a room acoustics with various estimations of divider impedances. Hypothetical contemplations are shown for rectangular cross-area of the live with specific proportion. Results got in the paper will be a perspective to the numerical figuring’s.

Bayesian sparse linear regression with unknown symmetric error

Abstract We study Bayesian procedures for sparse linear regression when the unknown error distribution is endowed with a non-parametric prior. Specifically, we put a symmetrized Dirichlet process mixture of Gaussian prior on the error density, where the mixing distributions are compactly supported. For the prior on regression coefficients, a mixture of point masses at zero and continuous distributions is considered. Under the assumption that the model is well specified, we study behavior of the posterior with diverging number of predictors. The compatibility and restricted eigenvalue conditions yield the minimax convergence rate of the regression coefficients in $\ell _1$- and $\ell _2$-norms, respectively. In addition, strong model selection consistency and a semi-parametric Bernstein–von Mises theorem are proven under slightly stronger conditions.

Local Fourier Analysis of Balancing Domain Decomposition By Constraints Algorithms

Local Fourier analysis is a commonly used tool for the analysis of multigrid and other multilevel algorithms, providing both insight into observed convergence rates and predictive analysis of the performance of many algorithms. In this paper, for the first time, we adapt local Fourier analysis to examine variants of two- and three-level balancing domain decomposition by constraints (BDDC) algorithms, to better understand the eigenvalue distributions and condition number bounds on these preconditioned operators. This adaptation is based on a modified choice of basis for the space of Fourier harmonics that greatly simplifies the application of local Fourier analysis in this setting. The local Fourier analysis is validated by considering the two-dimensional Laplacian and predicting the condition numbers of the preconditioned operators with different sizes of subdomains. Several variants are analyzed, showing that the two- and three-level performance of the “lumped” variant can be greatly improved when used in multiplicative combination with a weighted diagonal scaling preconditioner, with weight optimized through the use of local Fourier analysis.

A comparison of eigenvalue condition numbers for matrix polynomials

In this paper, we consider the different condition numbers for simple eigenvalues of matrix polynomials used in the literature and we compare them. One of these condition numbers is a generalization of the Wilkinson condition number for the standard eigenvalue problem. This number has the disadvantage of only being defined for finite eigenvalues. In order to give a unified approach to all the eigenvalues of a matrix polynomial, both finite and infinite, two (homogeneous) condition numbers have been defined in the literature. In their definition, very different approaches are used. One of the main goals of this note is to show that, when the matrix polynomial has a moderate degree, both homogeneous condition numbers are essentially the same and one of them provides a geometric interpretation of the other. We also show how the homogeneous condition numbers compare with the “Wilkinson-like” eigenvalue condition number and how they extend this condition number to zero and infinite eigenvalues.

Slope meets Lasso: Improved oracle bounds and optimality

We show that two polynomial time methods, a Lasso estimator with adaptively chosen tuning parameter and a Slope estimator, adaptively achieve the minimax prediction and $\ell_{2}$ estimation rate $(s/n)\log(p/s)$ in high-dimensional linear regression on the class of $s$-sparse vectors in $\mathbb{R}^{p}$. This is done under the Restricted Eigenvalue (RE) condition for the Lasso and under a slightly more constraining assumption on the design for the Slope. The main results have the form of sharp oracle inequalities accounting for the model misspecification error. The minimax optimal bounds are also obtained for the $\ell_{q}$ estimation errors with $1\le q\le2$ when the model is well specified. The results are nonasymptotic, and hold both in probability and in expectation. The assumptions that we impose on the design are satisfied with high probability for a large class of random matrices with independent and possibly anisotropically distributed rows. We give a comparative analysis of conditions, under which oracle bounds for the Lasso and Slope estimators can be obtained. In particular, we show that several known conditions, such as the RE condition and the sparse eigenvalue condition are equivalent if the $\ell_{2}$-norms of regressors are uniformly bounded.

Integrative multi-view regression: Bridging group-sparse and low-rank models.

Multi-view data have been routinely collected in various fields of science and engineering. A general problem is to study the predictive association between multivariate responses and multi-view predictor sets, all of which can be of high dimensionality. It is likely that only a few views are relevant to prediction, and the predictors within each relevant view contribute to the prediction collectively rather than sparsely. We cast this new problem under the familiar multivariate regression framework and propose an integrative reduced-rank regression (iRRR), where each view has its own low-rank coefficient matrix. As such, latent features are extracted from each view in a supervised fashion. For model estimation, we develop a convex composite nuclear norm penalization approach, which admits an efficient algorithm via alternating direction method of multipliers. Extensions to non-Gaussian and incomplete data are discussed. Theoretically, we derive non-asymptotic oracle bounds of iRRR under a restricted eigenvalue condition. Our results recover oracle bounds of several special cases of iRRR including Lasso, group Lasso, and nuclear norm penalized regression. Therefore, iRRR seamlessly bridges group-sparse and low-rank methods and can achieve substantially faster convergence rate under realistic settings of multi-view learning. Simulation studies and an application in the Longitudinal Studies of Aging further showcase the efficacy of the proposed methods.

Horseshoes in 4-dimensional piecewise affine systems with bifocal heteroclinic cycles.

By studying the Poincaré map in a neighborhood of the bifocal heteroclinic cycle (the corresponding subsystems only have conjugate complex eigenvalues), this paper provides a result on the existence of chaotic invariant sets for the two-zone 4-dimensional piecewise affine systems with bifocal heteroclinic cycles that cross the switching manifold transversally at two points. Different from Shil'nikov type theorems, the existence of chaotic invariant sets near the heteroclinic cycles depends not only on the eigenvalue conditions but also on the way of intersections of the stable manifolds and unstable manifolds of the subsystems.

Bifurcation solutions of Gross-Pitaevskii equations for spin-1 Bose-Einstein condensates

<p style='text-indent:20px;'>The main aim of this paper is to study the bifurcation solutions associated with the spinor Bose-Einstein condensates. Based on the Principle of Hamilton Dynamics and the Principle of Lagrangian Dynamics, a general pattern formation equation for the spinor Bose-Einstein condensates is established. Moreover, three kinds of critical conditions for eigenvalues are obtained under spectrum analysis and the different external confining potentials. With the change of different external potentials, the different topological structures of bifurcation solutions for the spinor Bose-Einstein condensates system are derived from steady state bifurcation theory.

Macroeconomic Variables and Retained Earnings of Quoted Manufacturing Firms in Nigeria: A Time Variant Study

This study examined external factors that determine retained earnings of quoted manufacturing firms in Nigeria. Annual time series data were sourced from Central Bank of Nigerian Statistical Bulletin, and Annual Reports of the selected manufacturing firms, the study modeled retained earnings the function of money supply, exchange rate, oil price, inflation rate and interest rate. The ordinary Least Square method was employed with multiple regression model based on Statistical Package for Social Sciences version (22.0). The Durbin-Watson statistics show the presence of multiple serial autocorrelation.The result shows collinearity that corresponds with the Eigen value condition index and variance constants are less than the required number, while the variance inflation factors indicate the absence of auto-correlation.It was found that Oil price have positive impact on retention rate of the selected manufacturing firms while exchange rate and interest rate have negative impact on the dependent variable. It was also found that money supply have negative effect on dividend payout rate while inflation rate have positive impact on retention rate. From the findings we conclude that oil price, interest rate, exchange rate and money supply have no significant relationship with dividend policy while inflation rate have significant relationship with dividend policy of the selected quoted manufacturing firms. We recommend the need for the manufacturing firms to formulate policies that leverage the negative effect of macroeconomic variables on retained earnings of the manufacturing firms and interest rate should properly be defined in the Nigerian financial market that is either full deregulated or regulated to determine the market rate of return, investment and the profitability of manufacturing firms. The operational efficiency of Nigerian capital market and the financial environment should be deepened, existing laws that does not encourage profitable investment should be changed and new laws enacted to enhance investment that will affect the profitability of manufacturing firms positively.

Chaotic Dynamics in Four-Dimensional Piecewise Affine Systems with Bifocal Heteroclinic Cycles

The well-known Shil’nikov type theory provides an approach to proving the existence of chaotic invariant sets for some classes of smooth dynamical systems with homoclinic orbits or heteroclinic cycles. However, it cannot be applied to nonsmooth systems directly. Based on the similar ideas, this paper studies the existence of chaotic invariant sets for a class of two-zone four-dimensional piecewise affine systems with bifocal heteroclinic cycles that cross the switching manifold transversally at two points. It turns out that there exist countable infinite chaotic invariant sets in a neighborhood of the bifocal heteroclinic cycle under some eigenvalue conditions. Moreover, the horseshoes of the corresponding Poincaré map are topologically semi-conjugated to a full shift on four symbols.

Minimax Optimal Convex Methods for Poisson Inverse Problems Under &lt;inline-formula&gt; &lt;tex-math notation="LaTeX"&gt;$\ell_{q}$ &lt;/tex-math&gt; &lt;/inline-formula&gt;-Ball Sparsity

In this paper, we study the minimax rates and provide an implementable convex algorithm for Poisson inverse problems under weak sparsity and physical constraints. In particular, we assume the model $y_{i} \sim \mathrm {Poisson}(Ta_{i}^{\top }f^{*})$ for $1 \leq i \leq n$ , where $T \in \mathbb {R}_{+}$ is the intensity, and we impose weak sparsity on $f^{*} \in \mathbb {R}^{p}$ by assuming $f^{*}$ lies in an $\ell _{q}$ -ball when rotated according to an orthonormal basis $D \in \mathbb {R}^{p \times p}$ . In addition, since we are modeling real-physical systems, we also impose positivity and flux-preserving constraints on the matrix $A = [a_{1}, a_{2},\ldots, a_{n}]^{\top }$ and the function $f^{*}$ . We prove minimax lower bounds for this model, which scale as $R_{q} ({\log p}/{T})^{1 - ({q}/{2})}$ where it is noticeable that the rate depends on the intensity $T$ and not the sample size $n$ . We also show that an $\ell _{1}$ -based regularized least-squares estimator achieves this minimax lower bound, provided a suitable restricted eigenvalue condition is satisfied. Finally, we prove that provided $n \geq \tilde {K} \log p$ where $\tilde {K} = \Theta (R_{q} ({\log p}/{T})^{- ({q}/{2})})$ represents an approximate sparsity level, and our restricted eigenvalue condition and physical constraints are satisfied for random bounded ensembles. We also provide numerical experiments that validate our mean-squared error bounds. Our results address a number of open issues from prior work on Poisson inverse problems that focuses on strictly sparse models and does not provide guarantees for convex implementable algorithms.

A block-symmetric linearization of odd degree matrix polynomials with optimal eigenvalue condition number and backward error

The standard way of solving numerically a polynomial eigenvalue problem (PEP) is to use a linearization and solve the corresponding generalized eigenvalue problem (GEP). In addition, if the PEP possesses one of the structures arising very often in applications, then the use of a linearization that preserves such structure combined with a structured algorithm for the GEP presents considerable numerical advantages. Block-symmetric linearizations have proven to be very useful for constructing structured linearizations of structured matrix polynomials. In this scenario, we analyze the eigenvalue condition numbers and backward errors of approximated eigenpairs of a block symmetric linearization that was introduced by Fiedler (Linear Algebra Appl 372:325–331, 2003) for scalar polynomials and generalized to matrix polynomials by Antoniou and Vologiannidis (Electron J Linear Algebra 11:78–87, 2004). This analysis reveals that such linearization has much better numerical properties than any other block-symmetric linearization analyzed so far in the literature, including those in the well known vector space $$\mathbb {DL}(P)$$ of block-symmetric linearizations. The main drawback of the analyzed linearization is that it can be constructed only for matrix polynomials of odd degree, but we believe that it will be possible to extend its use to even degree polynomials via some strategies in the near future.

Signal recovery under cumulative coherence

This paper considers signal recovery in the framework of cumulative coherence. First, we show that the Lasso estimator and the Dantzig selector exhibit similar behavior under the cumulative coherence. Then we estimate the approximation equivalence between the Lasso and the Dantzig selector by calculating prediction loss difference under the condition of cumulative coherence. And we also prove that the cumulative coherence implies the restricted eigenvalue condition. Last, we illustrate the advantages of cumulative coherence condition for three class matrices, in terms of the recovery performance of sparse signals via extensive numerical experiments.

Dynamical transition for S‐K‐T biological competing model with cross‐diffusion

The main objective of this paper is to study the dynamical transition for the S‐K‐T biological competition system with cross‐diffusion. Based on the spectral analysis, the principle of exchange of stabilities conditions for eigenvalues are obtained. By using the dynamical transition theory, 2 different types of dynamical transition for the S‐K‐T model are also derived. In addition, an example is given to illustrate our main results.

I-LAMM FOR SPARSE LEARNING: SIMULTANEOUS CONTROL OF ALGORITHMIC COMPLEXITY AND STATISTICAL ERROR.

We propose a computational framework named iterative local adaptive majorize-minimization (I-LAMM) to simultaneously control algorithmic complexity and statistical error when fitting high dimensional models. I-LAMM is a two-stage algorithmic implementation of the local linear approximation to a family of folded concave penalized quasi-likelihood. The first stage solves a convex program with a crude precision tolerance to obtain a coarse initial estimator, which is further refined in the second stage by iteratively solving a sequence of convex programs with smaller precision tolerances. Theoretically, we establish a phase transition: the first stage has a sublinear iteration complexity, while the second stage achieves an improved linear rate of convergence. Though this framework is completely algorithmic, it provides solutions with optimal statistical performances and controlled algorithmic complexity for a large family of nonconvex optimization problems. The iteration effects on statistical errors are clearly demonstrated via a contraction property. Our theory relies on a localized version of the sparse/restricted eigenvalue condition, which allows us to analyze a large family of loss and penalty functions and provide optimality guarantees under very weak assumptions (For example, I-LAMM requires much weaker minimal signal strength than other procedures). Thorough numerical results are provided to support the obtained theory.

Oracle inequalities for sparse additive quantile regression in reproducing kernel Hilbert space

This paper considers the estimation of the sparse additive quantile regression (SAQR) in high-dimensional settings. Given the nonsmooth nature of the quantile loss function and the nonparametric complexities of the component function estimation, it is challenging to analyze the theoretical properties of ultrahigh-dimensional SAQR. We propose a regularized learning approach with a two-fold Lasso-type regularization in a reproducing kernel Hilbert space (RKHS) for SAQR. We establish nonasymptotic oracle inequalities for the excess risk of the proposed estimator without any coherent conditions. If additional assumptions including an extension of the restricted eigenvalue condition are satisfied, the proposed method enjoys sharp oracle rates without the light tail requirement. In particular, the proposed estimator achieves the minimax lower bounds established for sparse additive mean regression. As a by-product, we also establish the concentration inequality for estimating the population mean when the general Lipschitz loss is involved. The practical effectiveness of the new method is demonstrated by competitive numerical results.