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Science China Mathematics 2022年第1期65卷 203-220页

作者： Marko Huhtanen Pauliina Uusitalo Department of Electrical and Information Engineering University of OuluOulu 90570Finland

Two optimal orthogonalization processes are devised toorthogonalize,possibly approximately,the columns of a very large and possiblysparse matrix A∈C^(n×k).Algorithmically the aim is,at each step,to optimallydecrease nonorthogonality of all the columns of *** process relies on using translated small rank *** is a polynomial orthogonalization process forperforming the Löwdin *** steps rely on using iterative methods combined,preferably,with preconditioning which can have a dramatic effect on how fast thenonorthogonality *** speed of orthogonalization depends on howbunched the singular values of A are,modulo the number of steps *** methods put the steps of the gram-Schmidt orthogonalizationprocess into perspective regardingtheir(lack of)*** constructions are entirely operatortheoretic and can be extended to infinite dimensional Hilbert spaces.

关键词： optimal orthogonalization sparse matrix gram-Schmidt orthogonalization Lowdin orthogonalization polynomial orthogonalization implicit orthogonalization preconditioning gram matrix frame inequality

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A comparative analysis of optimization and generalization properties of two-layer neural network and random feature models under gradient descent dynamics

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Science China Mathematics 2020年第7期63卷 1235-1258页

作者： Weinan E Chao Ma Lei Wu Department of Mathematics Princeton UniversityPrincetonNJ 08544USA The Program in Applied and Computational Mathematics Princeton UniversityNJ 08544USA Beijing Institute of Big Data Research Beijing 100871China

A fairly comprehensive analysis is presented for the gradient descent dynamics for training two-layer neural network models in the situation when the parameters in both layers are *** initialization schemes as well as general regimes for the network width and training data size are *** the overparametrized regime,it is shown that gradient descent dynamics can achieve zero training loss exponentially fast regardless of the quality of the *** addition,it is proved that throughout the training process the functions represented by the neural network model are uniformly close to those of a kernel *** general values of the network width and training data size,sharp estimates of the generalization error are established for target functions in the appropriate reproducing kernel Hilbert space.

关键词： two-layer neural network random feature model gram matrix generalization error

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A Survey of Various Refinements and Generalizations of Hilbert's Inequalities

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Journal of Mathematical Research and Exposition 2005年第2期25卷 227-243页

作者：高明哲徐利治吉首大学师范学院数学与计算机科学系湖南吉首416000 大连理工大学应用数学系辽宁大连116024

Expounded in this survey article is a series of refinements and generalizations of Hilbert's inequalities mostly published during the years 1990 through *** inequalities concerned may be classified into several types (discrete and integral etc.), and various related results obtained respectively by L. C. Hsu, M. Z. Gao, B. C. Yang, J. C. Kuang, Hu Ke and H. Hong *** are described a little more precisely. Moreover, earlier and recent extensions of Hilbert-type inequalities are also stated for reference. And the new trend and the research ways are also brought forward.

关键词： Hilbert's inequalities Hardy-Hilbert's inequalities weight function gram matrix double series.

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