Related papers: Generalization Error Estimates of Machine Learning…

We deal with an initial-boundary value problem for the generalized time-dependent Schr\"odinger equation with variable coefficients in an unbounded $n$--dimensional parallelepiped ($n\geq 1$). To solve it, the Crank-Nicolson in time and the…

The sparse generalized eigenvalue problem arises in a number of standard and modern statistical learning models, including sparse principal component analysis, sparse Fisher discriminant analysis, and sparse canonical correlation analysis.…

We present the first systematic work for deriving a posteriori error estimates for general non-polynomial basis functions in an interior penalty discontinuous Galerkin (DG) formulation for solving eigenvalue problems associated with second…

We derive generalization error bounds for the training of two-layer neural networks without assuming boundedness of the loss function, using Wasserstein distance estimates on the discrepancy between a probability distribution and its…

We study gradient-based regularization methods for neural networks. We mainly focus on two regularization methods: the total variation and the Tikhonov regularization. Applying these methods is equivalent to using neural networks to solve…

The aim of this paper is to provide uniform estimates for the eigenvalue spacings of one-dimensional semiclassical Schr\"odinger operators with singular potentials on the half-line. We introduce a new development of semiclassical measures…

We estimate the error of the Deep Ritz Method for linear elliptic equations. For Dirichlet boundary conditions, we estimate the error when the boundary values are imposed through the boundary penalty method. Our results apply to arbitrary…

Generalization error bounds are critical to understanding the performance of machine learning models. In this work, building upon a new bound of the expected value of an arbitrary function of the population and empirical risk of a learning…

A method for approximating sixth-order ordinary differential equations is proposed, which utilizes a deep learning feedforward artificial neural network, referred to as a neural solver. The efficacy of this unsupervised machine learning…

We consider the problem of learning the evolution operator for the time-dependent Schr\"{o}dinger equation, where the Hamiltonian may vary with time. Existing neural network-based surrogates often ignore fundamental properties of the…

The ability of machine learning (ML) algorithms to generalize well to unseen data has been studied through the lens of information theory, by bounding the generalization error with the input-output mutual information (MI), i.e., the MI…

Many machine learning tasks can be formulated as Regularized Empirical Risk Minimization (R-ERM), and solved by optimization algorithms such as gradient descent (GD), stochastic gradient descent (SGD), and stochastic variance reduction…

We revisit the method of Carleman linearization for systems of ordinary differential equations with polynomial right-hand sides. This transformation provides an approximate linearization in a higher-dimensional space through the exact…

In this paper, we propose a novel sparse recovery method based on the generalized error function. The penalty function introduced involves both the shape and the scale parameters, making it very flexible. The theoretical analysis results in…

Practitioners in diverse fields such as healthcare, economics and education are eager to apply machine learning to improve decision making. The cost and impracticality of performing experiments and a recent monumental increase in electronic…

We derive analytic expressions of the recursive solutions to the Schr\"{o}dinger's equation by means of a cutoff potential technique for one-dimensional piecewise constant potentials. These solutions provide a method for accurately…

Unconditional guaranteed lower and upper eigenvalue bounds are mandatory for the understanding of the Schr\"odinger eigenvalue spectrum and its spectral gaps. While upper eigenvalue bounds are naturally induced by conforming…

This paper develops a new class of algorithms for general linear systems and eigenvalue problems. These algorithms apply fast randomized sketching to accelerate subspace projection methods, such as GMRES and Rayleigh--Ritz. This approach…

When using a finite difference method to solve an initial--boundary--value problem, the truncation error is often of lower order at a few grid points near boundaries than in the interior. Normal mode analysis is a powerful tool to analyze…

Generalization error bounds are essential for comprehending how well machine learning models work. In this work, we suggest a novel method, i.e., the Auxiliary Distribution Method, that leads to new upper bounds on expected generalization…