A peak norm is defined for Lp spaces of E-valued Bochner integrable functions, where E is a Banach space, and best approximations from a sun to elements of the space are characterized. Applications are given to some f...A peak norm is defined for Lp spaces of E-valued Bochner integrable functions, where E is a Banach space, and best approximations from a sun to elements of the space are characterized. Applications are given to some families of simultaneous best approximation problems.展开更多
The aim of the paper is to estimate the density functions or distribution functions measured by Wasserstein metric, a typical kind of statistical distances, which is usually required in the statistical learning. Based...The aim of the paper is to estimate the density functions or distribution functions measured by Wasserstein metric, a typical kind of statistical distances, which is usually required in the statistical learning. Based on the classical Bernstein approximation, a scheme is presented. To get the error estimates of the scheme, the problem turns to estimating the L1 norm of the Bernstein approximation for monotone C-1 functions, which was rarely discussed in the classical approximation theory. Finally, we get a probability estimate by the statistical distance.展开更多
The objective of this paper is to quantify the complexity of rank and nuclear norm constrained methods for low rank matrix estimation problems. Specifically, we derive analytic forms of the degrees of freedom for thes...The objective of this paper is to quantify the complexity of rank and nuclear norm constrained methods for low rank matrix estimation problems. Specifically, we derive analytic forms of the degrees of freedom for these types of estimators in several common settings. These results provide efficient ways of comparing different estimators and eliciting tuning parameters. Moreover, our analyses reveal new insights on the behavior of these low rank matrix estimators. These observations are of great theoretical and practical importance. In particular, they suggest that, contrary to conventional wisdom, for rank constrained estimators the total number of free parameters underestimates the degrees of freedom, whereas for nuclear norm penalization, it overestimates the degrees of freedom. In addition, when using most model selection criteria to choose the tuning parameter for nuclear norm penalization, it oftentimes suffices to entertain a finite number of candidates as opposed to a continuum of choices. Numerical examples are also presented to illustrate the practical implications of our results.展开更多
Recently,physics-informed neural networks(PINNs)have been shown to be a simple and efficient method for solving PDEs empirically.However,the numerical analysis of PINNs is still incomplete,especially why over-paramete...Recently,physics-informed neural networks(PINNs)have been shown to be a simple and efficient method for solving PDEs empirically.However,the numerical analysis of PINNs is still incomplete,especially why over-parameterized PINNs work remains unknown.This paper presents the first convergence analysis of the overparameterized PINNs for the Laplace equations with Dirichlet boundary conditions.We demonstrate that the convergence rate can be controlled by the weight norm,regardless of the number of parameters in the network.展开更多
文摘A peak norm is defined for Lp spaces of E-valued Bochner integrable functions, where E is a Banach space, and best approximations from a sun to elements of the space are characterized. Applications are given to some families of simultaneous best approximation problems.
基金Supported by 973-Project of China(2006cb303102)the National Science Foundation of China(11461161006,11201079)
文摘The aim of the paper is to estimate the density functions or distribution functions measured by Wasserstein metric, a typical kind of statistical distances, which is usually required in the statistical learning. Based on the classical Bernstein approximation, a scheme is presented. To get the error estimates of the scheme, the problem turns to estimating the L1 norm of the Bernstein approximation for monotone C-1 functions, which was rarely discussed in the classical approximation theory. Finally, we get a probability estimate by the statistical distance.
基金supported by National Science Foundation of USA (Grant No. DMS1265202)National Institutes of Health of USA (Grant No. 1-U54AI117924-01)
文摘The objective of this paper is to quantify the complexity of rank and nuclear norm constrained methods for low rank matrix estimation problems. Specifically, we derive analytic forms of the degrees of freedom for these types of estimators in several common settings. These results provide efficient ways of comparing different estimators and eliciting tuning parameters. Moreover, our analyses reveal new insights on the behavior of these low rank matrix estimators. These observations are of great theoretical and practical importance. In particular, they suggest that, contrary to conventional wisdom, for rank constrained estimators the total number of free parameters underestimates the degrees of freedom, whereas for nuclear norm penalization, it overestimates the degrees of freedom. In addition, when using most model selection criteria to choose the tuning parameter for nuclear norm penalization, it oftentimes suffices to entertain a finite number of candidates as opposed to a continuum of choices. Numerical examples are also presented to illustrate the practical implications of our results.
基金supported by the National Key Research and Development Program of China(No.2023YFA1000103)the National Natural Science Foundation of China(No.123B2019,No.12125103,No.U24A2002,No.12371424,No.12371441)by the Fundamental Research Funds for the Central Universities.
文摘Recently,physics-informed neural networks(PINNs)have been shown to be a simple and efficient method for solving PDEs empirically.However,the numerical analysis of PINNs is still incomplete,especially why over-parameterized PINNs work remains unknown.This paper presents the first convergence analysis of the overparameterized PINNs for the Laplace equations with Dirichlet boundary conditions.We demonstrate that the convergence rate can be controlled by the weight norm,regardless of the number of parameters in the network.
