相关论文: Condition Numbers and Eigenvalue Spectra of Shallo…

In this paper, we investigate condition numbers of eigenvalue problems of matrix polynomials with nonsingular leading coefficients, generalizing classical results of matrix perturbation theory. We provide a relation between the condition…

We derive bounds for the minimum and maximum eigenvalues and the spectral condition number of matrices for isogeometric discretizations of elliptic partial differential equations in an open, bounded, simply connected Lipschitz domain…

Sensitivity of an eigenvalue $\lambda_i$ to the perturbation of matrix elements is controlled by the eigenvalue condition number defined as $\kappa_i = \sqrt{\left< L_i | L_i\right> \left< R_i|R_i \right> }$, where $\left<L_i\right|$ and…

In this work we study the condition number of the least square matrix corresponding to scale free networks. We compute a theoretical lower bound of the condition number which proves that they are ill conditioned. Also, we analyze several…

We describe a randomized Krylov-subspace method for estimating the spectral condition number of a real matrix A or indicating that it is numerically rank deficient. The main difficulty in estimating the condition number is the estimation of…

Conditionspectrum measures the computational stability of solving a linear system. In this paper, ten theorems involving {\epsilon}-conditionspectrum are presented. All these theorems generalize a well known eigenvalue theorem and…

A simple formula is proved to be a tight estimate for the condition number of the full rank linear least squares residual with respect to the matrix of least squares coefficients and scaled 2-norms. The tight estimate reveals that the…

We consider the approximation rates of shallow neural networks with respect to the variation norm. Upper bounds on these rates have been established for sigmoidal and ReLU activation functions, but it has remained an important open problem…

In this article, we study approximation properties of the variation spaces corresponding to shallow neural networks with a variety of activation functions. We introduce two main tools for estimating the metric entropy, approximation rates,…

Since statistical guarantees for neural networks are usually restricted to global optima of intricate objective functions, it is unclear whether these theories explain the performances of actual outputs of neural network pipelines. The goal…

The eigenvalues and eigenvectors of the connectivity matrix of complex networks contain information about its topology and its collective behavior. In particular, the spectral density $\rho(\lambda)$ of this matrix reveals important network…

We study the approximation properties of shallow neural networks with an activation function which is a power of the rectified linear unit. Specifically, we consider the dependence of the approximation rate on the dimension and the…

This paper investigates the approximation properties of shallow neural networks with activation functions that are powers of exponential functions. It focuses on the dependence of the approximation rate on the dimension and the smoothness…

The condition number for eigenvalue computations is a well--studied quantity. But how small can we expect it to be? Namely, which is a perfectly conditioned matrix w.r.t. eigenvalue computations? In this note we answer this question with…

This paper studies approximation by shallow ReLU$^s$ networks, $\sigma_s(t)=\max\{0,t\}^s$, together with their generalization behavior under $\ell_1$ path-norm control. For the $L^p$-type integral spaces…

Parameter space is not function space for neural network architectures. This fact, investigated as early as the 1990s under terms such as ``reverse engineering," or ``parameter identifiability", has led to the natural question of parameter…

The eigenvalues of matrices representing the structure of large-scale complex networks present a wide range of applications, from the analysis of dynamical processes taking place in the network to spectral techniques aiming to rank the…

We present a wavenumber-robust strategy for computing Steklov eigenpairs of the Helmholtz operator $-\Delta -\mu^2$. As the wavenumber $\mu \rightarrow \mu_D$ from below (where $\mu_D^2 $ is a Dirichlet- Laplace eigenvalue of multiplicity…

It is natural to ask: what kinds of matrices satisfy the Restricted Eigenvalue (RE) condition? In this paper, we associate the RE condition (Bickel-Ritov-Tsybakov 09) with the complexity of a subset of the sphere in $\R^p$, where $p$ is the…

We study numerically the spectrum and eigenstate properties of the Google matrix of various examples of directed networks such as vocabulary networks of dictionaries and university World Wide Web networks. The spectra have gapless structure…