相关论文: On the discrepancy principle

A decision maker typically (i) incorporates training data to learn about the relative effectiveness of treatments, and (ii) chooses an implementation mechanism that implies an ``optimal'' predicted outcome distribution according to some…

This paper investigates the use of methods from partial differential equations and the Calculus of variations to study learning problems that are regularized using graph Laplacians. Graph Laplacians are a powerful, flexible method for…

We consider the on-line predictive version of the standard problem of linear regression; the goal is to predict each consecutive response given the corresponding explanatory variables and all the previous observations. We are mainly…

Convergence results are stated for the variational iteration method applied to solve an initial value problem for a system of ordinary differential equations.

Neural networks have become ubiquitous tools for solving signal and image processing problems, and they often outperform standard approaches. Nevertheless, training neural networks is a challenging task in many applications. The prevalent…

The superposition principle is fundamental to linear wave systems, ensuring that their physical behaviour is independent of the chosen basis representation. While this principle underpins many analytical techniques, including modal…

We shall study special regularity properties of solutions to some nonlinear dispersive models. The goal is to show how regularity on the initial data is transferred to the solutions. This will depend on the spaces where regularity is…

Regularization is a popular technique to solve the overfitting problem of machine learning algorithms. Most regularization technique relies on parameter selection of the regularization coefficient. Plug-in method and cross-validation…

In an incoherent dictionary, most signals that admit a sparse representation admit a unique sparse representation. In other words, there is no way to express the signal without using strictly more atoms. This work demonstrates that sparse…

The problem of characterising the accuracy of, and disturbance caused by a joint measurement of position and momentum is investigated. In a previous paper the problem was discussed in the context of the unbiased measurements considered by…

This paper develops some objective priors for certain parameters of the bivariate normal distribution. The parameters considered are the regression coefficient, the generalized variance, and the ratio of the conditional variance of one…

In this note we prove a Wiener criterion of regularity of boundary points for the Dirichlet problem related to $X$-elliptic operators in divergence form enjoying the doubling condition and the Poincar\'e inequality. As a step towards this…

In this paper we study a general class of nonlinear elliptic problems in divergence form. First, we prove that the solutions to these problems satisfy a convexity property when the given domain is strictly convex. Then, making use of this…

We show how a number of well-known uncertainty principles for the Fourier transform, such as the Heisenberg uncertainty principle, the Donoho--Stark uncertainty principle, and Meshulam's non-abelian uncertainty principle, have little to do…

There are two distinct definitions of 'P-value' for evaluating a proposed hypothesis or model for the process generating an observed dataset. The original definition starts with a measure of the divergence of the dataset from what was…

The goal of the paper is to develop a method that will combine the use of variational techniques with regularization methods in order to study existence and multiplicity results for the periodic and the Dirichlet problem associated to the…

We study variational regularisation methods for inverse problems with imperfect forward operators whose errors can be modelled by order intervals in a partial order of a Banach lattice. We carry out analysis with respect to existence and…

In a multiple linear regression model, the algebraic formula of the decomposition theorem explains the relationship between the univariate regression coefficient and partial regression coefficient using geometry. It was found that…

We propose an alternative variational principle whose critical point is the algebraic plane curve associated to a matrix model (the spectral curve, i.e. the large $N$ limit of the resolvent). More generally, we consider a variational…

We present a new variational principle for linking models of beams and deformable solids, providing also its mathematical analysis. Despite the apparent differences between the two types of governing equations, it will be shown that the…