Abstract We develop new theoretical results on matrix perturbation to shed light the impact of architecture performance a deep network. In particular, we explain analytically what learning practitioners have long observed empirically: parameters some architectures (e.g., residual networks, ResNets, and Dense DenseNets) are easier optimize than others convolutional ConvNets). Building our earlie...

The paper discusses the existence of positive solutions, dead core solutions and pseudodead core solutions of the singular Dirichlet boundary value problem (φ(u)) = λ[ f (t, u, u) + h(t, u, u)], u(0) = u(T ) = A. Here λ is the positive parameter, A > 0, f is singular at the value 0 of its first phase variable and h may be singular at the value 0 of its second phase variable. c © 2007 Elsevier L...

In this paper, we give a regularized directional derivative-based Newton method for solving the inverse singular value problem. The proposed method is also globalized by employing the directional derivative-based Wolfe line search conditions. Under some mild assumptions, The global and quadratic convergence of our method is established. To improve the practical effectiveness, we also propose a ...

The non-polynomial spline method is proposed to solve a singular boundary value problems. Some model problems are solved and the numerical results are compared with B-spline approximation and the exact solution. Results obtained by the method indicate the method is simple and effective. Key–Words: Singular point; Singular two-point boundary value problems; Cubic spline

Abstract We know that to estimate matrix singular values ( especially the largest and the smallest ones ) is an attractive topic in matrix theory and numerical analysis. In this note, we first provide a simple estimate for the smallest singular value σn(A) of n × n positive definite matrix A. Secondly, we obtain some simple estimates for the smallest singular value σn(A) and the largest singula...