A Structure-Guided Gauss-Newton Method for Shallow ReLU Neural Network

In this paper, we propose a structure-guided Gauss-Newton (SgGN) method for solving least squares problems using a shallow ReLU neural network. The method effectively takes advantage of both the least squares structure and the neural network structure of the objective function. By categorizing the weights and biases of the hidden and output layers of the network as nonlinear and linear parameters, respectively, the method iterates back and forth between the nonlinear and linear parameters. The nonlinear parameters are updated by a damped Gauss-Newton method and the linear ones are updated by a linear solver. Moreover, at the Gauss-Newton step, a special form of the Gauss-Newton matrix is derived for the shallow ReLU neural network and is used for efficient iterations. It is shown that the corresponding mass and Gauss-Newton matrices in the respective linear and nonlinear steps are symmetric and positive definite under reasonable assumptions. Thus, the SgGN method naturally produces an effective search direction without the need of additional techniques like shifting in the Levenberg-Marquardt method to achieve invertibility of the Gauss-Newton matrix. The convergence and accuracy of the method are demonstrated numerically for several challenging function approximation problems, especially those with discontinuities or sharp transition layers that pose significant challenges for commonly used training algorithms in machine learning.

Making the Nyström method highly accurate for low-rank approximations

The Nystr\"om method is a convenient heuristic method to obtain low-rank approximations to kernel matrices in nearly linear complexity. Existing studies typically use the method to approximate positive semidefinite matrices with low or modest accuracies. In this work, we propose a series of heuristic strategies to make the Nystr\"om method reach high accuracies for nonsymmetric and/or rectangular matrices. The resulting methods (called high-accuracy Nystr\"om methods) treat the Nystr\"om method and a skinny rank-revealing factorization as a fast pivoting strategy in a progressive alternating direction refinement process. Two refinement mechanisms are used: alternating the row and column pivoting starting from a small set of randomly chosen columns, and adaptively increasing the number of samples until a desired rank or accuracy is reached. A fast subset update strategy based on the progressive sampling of Schur complements is further proposed to accelerate the refinement process. Efficient randomized accuracy control is also provided. Relevant accuracy and singular value analysis is given to support some of the heuristics. Extensive tests with various kernel functions and data sets show how the methods can quickly reach prespecified high accuracies in practice, sometimes with quality close to SVDs, using only small numbers of progressive sampling steps.