In a previous study by Li et al. (Commun Comput Phys 27(2):379–411, 2020), it is shown that deep neural networks built with rectified power units (RePU) as activation functions can give better approximation for sufficient smooth functions than those built with rectified linear units, by converting polynomial approximations using power series into deep neural networks with optimal complexity and no approximation error. However, in practice, power series approximations are not easy to obtain due to the associated stability issue. In this paper, we propose a new and more stable way to construct RePU deep neural networks based on Chebyshev polynomial approximations. By using a hierarchical structure of Chebyshev polynomial approximation in frequency domain, we obtain efficient and stable deep neural network construction, which we call ChebNet. The approximation of smooth functions by ChebNets is no worse than the approximation by deep RePU nets using power series. On the same time, ChebNets are much more stable. Numerical results show that the constructed ChebNets can be further fine-tuned to obtain much better results than those obtained by tuning deep RePU nets constructed by power series approach. As spectral accuracy is hard to obtain by direct training of deep neural networks, ChebNets provide a practical way to obtain spectral accuracy, it is expected to be useful in real applications that require efficient approximations of smooth functions.
Equivalence assessment via various indices such as relative risk has been widely studied in a matched-pair design with discrete or continuous endpoints over the past years. But existing studies mainly focus on the fully observed or missing at random endpoints. Nonignorable missing endpoints are commonly encountered in a matched-pair design. To this end, this paper proposes several novel methods to assess equivalence of two diagnostics via the difference between two correlated areas under ROC curves (AUCs) in a matched-pair design with nonignorable missing endpoints. An exponential tilting model is utilized to specify the nonignorable missing endpoint mechanism. Three nonparametric approaches and three semiparametric approaches are developed to estimate the difference between two correlated AUCs based on the kernel-regression imputation, inverse probability weighted (IPW), and augmented IPW methods. Under some regularity conditions, we show the consistency and asymptotic normality of the proposed estimators. Simulation studies are conducted to study the performance of the proposed estimators. Empirical results show that the proposed methods outperform the complete-case method. An example from clinical studies is illustrated by the proposed methodologies.
In this paper, we establish three circles theorem for volume of conformal metrics whose scalar curvatures are integrable in a critical (scaling invariant) norm. As applications, we analyze the asymptotic behavior of such metrics near isolated singularities and use it to show the residual terms of the Chern–Gauss–Bonnet formula are integers. Such strong rigidity implies a vanishing theorem on the integral value of the curvature, with application to the bi-Lipschitz equivalence problem for conformal metrics.
The spectrum of the normalized Laplacian matrix of a graph provides a lot of structural information of the graph, and it has applications in numerous areas and in different guises. In this paper, we completely characterize all connected graphs of order with some normalized Laplacian eigenvalue having multiplicity .
We develop the theory of mean random attractors of nonautonomous mean random dynamical systems proposed by Wang (J. Dynam. Differ. Equ., Proc. Amer. Math. Soc., J. Differ. Equ., 2019) in a general setting. Two types of enhanced mean random attractors are introduced by improving the weak compactness and weak attraction in a space of product of finitely many Bochner spaces uniformly over some infinite time-intervals. Then we establish some theoretical results for the existence, topology structures and relations of these enhanced mean random attractors by carefully analyzing the characters and properties of enhanced -limit sets under the theory of weak topology. Our theoretical results are expected to be applied to many types of stochastic models. In particular, we apply these general results to a class of locally monotone and generally coercive stochastic PDEs, a fractional stochastic reaction–diffusion equation defined on , and a stochastic Schrödinger lattice system with a delay diffusion term defined on .
M-dependence is a commonly used assumption in the study of dependent sequences. In this paper, central limit theorems for m-dependent random variables under the sub-linear expectations are established based mainly on the conditions of Zhang (Commun Math Stat, 2023). They can be regarded as the extension of independent Lindeberg central limit theorem, and for proving this, Rosenthal’s inequality for m-dependent random variables is obtained. In particular, we extend the results in Li (Acta Math Appl Sin Engl Ser 31:435–444, 2015) and establish the central limit theorem for m-dependent stationary sequence.
In one-sample mean testing for high-dimensional data, existing tests, e.g., Chen and Qin (Ann Stat 38(2):808–835, 2010) and Wang et al. (J Am Stat Assoc 110(512):1658–1669, 2015), assume that the data are either normally distributed or from a latent factor model. In this paper, we remove these restrictive assumptions and develop a new asymptotic theory, showing that the asymptotic null distribution is a mixture of mixture and normal distributions. With more conditions on the eigenvalues of the covariance matrices, a normal or mixture approximation for the limiting null distribution is derived. The power functions of two test statistic under high-dimensional version local and fixed alternative are also analyzed. A wild bootstrap procedure is proposed to determine the critical values of the mixture of mixture and normal distributions, which is easy to implement and fast to run. Numerical simulations show that our proposed methods control the test’s size more precisely than existing methods using the normal approximation. The merit of the proposed methods is further demonstrated on a real data example.
As main result of the paper, we describe finite non-solvable groups without elements of order 10. In addition, we prove a new general structural theorem on finite non-solvable groups without elements of order 2p for an odd prime p. The theorem reinforces essentially the well-known Vasil’ev theorem on these groups and can be applied to obtain new arithmetical characterizations of finite groups.