Gaussian Sobolev
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Optimality of quasi-Monte Carlo methods and suboptimality of the sparse-grid Gauss--Hermite rule in Gaussian Sobolev spaces
Announce Type: replace Abstract: Optimality of several quasi-Monte Carlo methods and suboptimality of the sparse-grid quadrature based on the univariate Gauss--Hermite rule is proved in the Sobolev spaces of mixed dominating smoothness of order $\alpha$, where the optimality is in the sense of worst-case convergence rate. For sparse-grid Gauss--Hermite quadrature, lower and upper bounds are established, with rates coinciding up to a logarithmic factor. The dominant rate is found to be only...
Quantifying Uncertainty In Wide Two-Layer Neural Networks: On The Law Of The Limiting Fluctuation Process
Announce Type: new Abstract: Uncertainty quantification in neural networks prediction is a main issue for usual applications. Our approach seeks at reducing computation costs by directly evaluating uncertainty using PDE's information on the asymptotic variance, rather than the deep ensemble method which may be seen as a Monte Carlo estimation of the prediction, requiring the training of multiple networks. We thus study the law of the limiting process describing the random fluctuations around...