arXiv subjects
Publications and source records attributed to Peichang Guo.
The nonlinear eigen-problem $ Ax+F(x)=λx$ is studied where $A$ is an $n\times n$ irreducible Stieltjes matrix. Under certain conditions, this problem has a unique positive solution. We show that, starting from a multiple of the positive eigenvector of $A$, the Newton-like iteration for this problem converges monotonically. Numerical results illustrate the effectiveness of this Newton-like method.
Convolutional neural network is an important model in deep learning. To avoid exploding/vanishing gradient problems and to improve the generalizability of a neural network, it is desirable to have a convolution operation that nearly preserves the norm, or to have the singular values of the transformation matrix corresponding to a convolutional kernel bounded around $1$. We propose a penalty function that can be used in the optimization of a convolutional neural network to constrain the singular values of the transformation matrix around $1$. We derive an algorithm to carry out the gradient descent minimization of this penalty function in terms of convolution kernels. Numerical examples are presented to demonstrate the effectiveness of the method.
We consider the large scale nonsymmetric algebraic Riccati equation arising in transport theory, where the $n\times n$ coefficient matrices $B, C$ are symmetric and low-ranked and $A, E$ are rank one updates of nonsingular diagonal matrices. By introducing a balancing strategy and setting appropriate initial matrices carefully, we can simplify the large-scale structure-preserving doubling algorithm (SDA\_ls) for this special equation. We give modified large-scale structure-preserving doubling algorithm, which can reduce the flop operations of original SDA\_ls by half. Numerical experiments illustrate the effectiveness of our method.
The extinction probability of the Markovian Binary Tree (MBT) is the minimal nonnegative solution of a Quadratic Vector Equation (QVE). In this paper, we present a perturbation analysis for the extinction probability of a supercritical MBT. We derive a perturbation bound for the minimal nonnegative solution of the QVE, and an error bound is also given, which can be used to measure the quality of an approximation solution. Numerical tests show that these bounds are fairly sharp.
In order to determine the stationary distribution for discrete time quasi-birth-death Markov chains, it is necessary to find the minimal nonnegative solution of a quadratic matrix equation. We apply the Newton-Shamanskii method for solving the equation. We show that the sequence of matrices generated by the Newton-Shamanskii method is monotonically increasing and converges to the minimal nonnegative solution of the equation. Numerical experiments show the effectiveness of our method.