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作者：Lehmann, Tobias
作者单位：Leipzig University
摘要：We prove that stochastic replicator dynamics can be interpreted as intrin-sic Brownian motion on the simplex equipped with the Aitchison geometry. As an immediate consequence, we derive three approximation results in the spirit of Wong-Zakai approximation, Donsker's invariance principle and a JKO-scheme. Using the Fokker-Planck equation and Wasserstein-contraction estimates, we also study the long time behavior of the stochastic replicator equation, as an example of a nongradient drift diffusi...
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作者：Schnelli, Kevin; Xu, Yuanyuan
作者单位：Royal Institute of Technology; Institute of Science & Technology - Austria
摘要：We establish a quantitative version of the Tracy-Widom law for the largest eigenvalue of high-dimensional sample covariance matrices. To be precise, we show that the fluctuations of the largest eigenvalue of a sample covariance matrix X*X converge to its Tracy-Widom limit at a rate nearly N-1/3, where X is an M x N random matrix whose entries are independent real or complex random variables, assuming that both M and N tend to in-finity at a constant rate. This result improves the previous esti...
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作者：Gonon, Lukas; Grigoryeva, Lyudmila; Ortega, Juan-pablo
作者单位：University of Munich; University of Warwick; Nanyang Technological University
摘要：This work studies approximation based on single-hidden-layer feed -forward and recurrent neural networks with randomly generated internal weights. These methods, in which only the last layer of weights and a few hy-perparameters are optimized, have been successfully applied in a wide range of static and dynamic learning problems. Despite the popularity of this ap-proach in empirical tasks, important theoretical questions regarding the rela-tion between the unknown function, the weight distribu...
