Although deep learning-based approximation algorithms have been applied very successfully to numerous problems, at the moment the reasons for their performance are not entirely understood from a mathematical point of view. Recently, estimates for the convergence of the overall error have been obtained in the situation of deep supervised learning, but with an extremely slow rate of convergence. In this note, we partially improve on these estimates. More specifically, we show that the depth of the neural network only needs to increase much slower in order to obtain the same rate of approximation. The results hold in the case of an arbitrary stochastic optimization algorithm with i.i.d. random initializations.
This research concerns the estimation of latent linear or polychoric correlations from fuzzy frequency tables. Fuzzy counts are of particular interest to many disciplines including social and behavioral sciences and are especially relevant when observed data are classified using fuzzy categories—as for socioeconomic studies, clinical evaluations, content analysis, inter-rater reliability analysis—or when imprecise observations are classified into either precise or imprecise categories—as for the analysis of ratings data or fuzzy-coded variables. In these cases, the space of count matrices is no longer defined over naturals and, consequently, the polychoric estimator cannot be used to accurately estimate latent linear correlations. The aim of this contribution is twofold. First, we illustrate a computational procedure based on generalized natural numbers for computing fuzzy frequencies. Second, we reformulate the problem of estimating latent linear correlations from fuzzy counts in the context of expectation–maximization-based maximum likelihood estimation. A simulation study and two applications are used to investigate the characteristics of the proposed method. Overall, the results show that the fuzzy EM-based polychoric estimator is more efficient to deal with imprecise count data as opposed to standard polychoric estimators that may be used in this context.
In this paper, we define signed zero-divisor graphs over commutative rings and investigate the interplay between the algebraic properties of the rings and the combinatorial properties of their corresponding signed zero-divisor graphs. We investigate the structure of signed zero-divisor graphs, the relation between ideals and signed zero-divisor graphs, and the adjacency matrices and the spectra of signed zero-divisor graphs.
In this work, we consider numerical approximations of the phase-field model of diblock copolymer melt confined in Hele–Shaw cell, which is a very complicated coupled nonlinear system consisting of the Darcy equations and the Cahn–Hilliard type equations with the Ohta–Kawaski potential. Through the combination of a novel explicit-Invariant Energy Quadratization approach and the projection method, we develop the first full decoupling, energy stable, and second-order time-accurate numerical scheme. The introduction of two auxiliary variables and the design of two auxiliary ODEs play a vital role in obtaining the full decoupling structure while maintaining energy stability. The scheme is also linear and unconditional energy stable, and the practical implementation efficiency is also very high because it only needs to solve a few elliptic equations with constant coefficients at each time step. We strictly prove that the scheme satisfies the unconditional energy stability and give a detailed implementation process. Numerical experiments further verify the convergence rate, energy stability, and effectiveness of the developed algorithm.
This paper proposes a novel method to compute the diffeomorphic registration of 3D surfaces with point and curve feature landmarks. First the surfaces are mapped to the canonical domain by a curve constrained harmonic map, where the landmark curves are straightened to line segments and their positions and inclining angles are determined intrinsically by the surface geometry and its curve landmarks. Then, the canonical domains are registered by aligning the corresponding point and straight line segments using the dynamic quasiconformal map (DQCM), which introduces the combinatorial diagonal switches to the quasiconformal optimization such that the resultant map is diffeomorphic. The end points of the source curve landmarks are mapped to their corresponding points on the target surface, while the interior points of the source curves can slide on the corresponding target curves, which provides more freedom for the surface registration than the point-based registration methods. Experiments on the real surfaces with point and curve landmarks demonstrate the efficiency, efficacy and robustness of the proposed method.
The method of moving surfaces is an effective tool to implicitize rational parametric surfaces, and it has been extensively studied in the past two decades. An essential step in surface implicitization using the method of moving surfaces is to compute a -basis of a parametric surface with respect to one variable. The -basis is a minimal basis of the syzygy module of a univariate polynomial matrix with special structure defined by the parametric equation of the rational surface. In this paper, we present an efficient algorithm to compute the -basis of a parametric surface with respect to a variable based on the special structure of the corresponding univariate polynomial matrix. Analysis on the computational complexity of the algorithm is also provided. Experiments demonstrate that our algorithm is much faster than the general method to compute the -bases of arbitrary polynomial matrices and outperforms the algorithm based on Gröbner basis computation for relatively low degree rational surfaces.
Assessing the influence of individual observations of the functional linear models is important and challenging, especially when the observations are subject to missingness. In this paper, we introduce three case-deletion diagnostic measures to identify influential observations in functional linear models when the covariate is functional and observations on the scalar response are subject to nonignorable missingness. The nonignorable missing data mechanism is modeled via an exponential tilting semiparametric functional model. A semiparametric imputation procedure is developed to mitigate the effects of missing data. Valid estimations of the functional coefficients are based on functional principal components analysis using the imputed dataset. A smoothed bootstrap sampling method is introduced to estimate the diagnostic probability for each proposed diagnostic measure, which is helpful to unveil which observations have the larger influence on estimation and prediction. Simulation studies and a real data example are conducted to illustrate the finite performance of the proposed methods.
In this paper, the structure of finite groups in which maximal subgroups of some Sylow subgroups have a -soluble or -nilpotent supplement, where is a partition of the set of all prime numbers, is investigated. Some solubility, -solubility and -nilpotency criteria leading to some significant improvements of earlier results are given.