Related papers: A Structure-Guided Gauss-Newton Method for Shallow…
We introduced the least-squares ReLU neural network (LSNN) method for solving the linear advection-reaction problem with discontinuous solution and showed that the method outperforms mesh-based numerical methods in terms of the number of…
In this paper, we investigate the convergence performance of a cooperative diffusion Gauss-Newton (GN) method, which is widely used to solve the nonlinear least squares problems (NLLS) due to the low computation cost compared with Newton's…
An important question in deep learning is how higher-order optimization methods affect generalization. In this work, we analyze a stochastic Gauss-Newton (SGN) method with Levenberg-Marquardt damping and mini-batch sampling for training…
This article focuses on solving parametric transmission problems in one and two spatial dimensions. These problems belong to a class of partial differential equations that arise in the modeling of physical systems with heterogeneous…
The paper is devoted to the solution of a weighted nonlinear least-squares problem for low-rank signal estimation, which is related to Hankel structured low-rank approximation problems. A modified weighted Gauss-Newton method, which uses…
In this paper, a Gauss-Seidel method with oblique direction (GSO) is proposed for finding the least-squares solution to a system of linear equations, where the coefficient matrix may be full rank or rank deficient and the system is…
We consider the optimization problem associated with fitting two-layer ReLU networks with respect to the squared loss, where labels are assumed to be generated by a target network. Focusing first on standard Gaussian inputs, we show that…
We present Exact Gauss-Newton (EGN), a stochastic second-order optimization algorithm that combines the generalized Gauss-Newton (GN) Hessian approximation with low-rank linear algebra to compute the descent direction. Leveraging the…
This paper presents a unified Least-Squares framework for solving nonlinear partial differential equations by recasting the governing system as a residual minimisation problem. A Least-Squares functional is formulated and the corresponding…
The least-squares ReLU neural network (LSNN) method was introduced and studied for solving linear advection-reaction equation with discontinuous solution in \cite{Cai2021linear,cai2023least}. The method is based on an equivalent…
This work focuses on developing and motivating a stochastic version of a wellknown inverse problem methodology. Specifically, we consider the iteratively regularized Gauss-Newton method, originally proposed by Bakushinskii for…
We studied the least-squares ReLU neural network (LSNN) method for solving linear advection-reaction equation with discontinuous solution in [Cai, Zhiqiang, Jingshuang Chen, and Min Liu. ``Least-squares ReLU neural network (LSNN) method for…
Shortest path search is a core operation in graph-based applications, yet existing methods face important limitations. Classical algorithms such as Dijkstra's and A* become inefficient as graphs grow more complex, while index-based…
The Gaussian-radial-basis function neural network (GRBFNN) has been a popular choice for interpolation and classification. However, it is computationally intensive when the dimension of the input vector is high. To address this issue, we…
In this paper, we develop a variant of the well-known Gauss-Newton (GN) method to solve a class of nonconvex optimization problems involving low-rank matrix variables. As opposed to the standard GN method, our algorithm allows one to handle…
This paper addresses the challenge of solving large-scale nonlinear equations with H\"older continuous Jacobians. We introduce a novel Incremental Gauss--Newton (IGN) method within explicit superlinear convergence rate, which outperforms…
We develop a novel randomized conjugate gradient least squares (RCGLS) method for solving least-squares problems, in which iterative sketching is employed at each step to reduce the dimension and hence the computational cost. In particular,…
Neural networks are trained by optimizing multi-dimensional sets of fitting parameters on non-convex loss landscapes. Low-loss regions of the landscapes correspond to the parameter sets that perform well on the training data. A key issue in…
We present a stochastic inexact Gauss-Newton method for the solution of nonlinear least-squares. To reduce the computational cost with respect to the classical method, at each iteration the proposed algorithm approximately minimizes the…
We develop and analyze stochastic inexact Gauss-Newton methods for nonlinear least-squares problems and for nonlinear systems ofequations. Random models are formed using suitable sampling strategies for the matrices involved in the…