Article Dans Une Revue Journal of Complexity Année : 2022

A generative model for fBm with deep ReLU neural networks

Résumé

We provide a large probability bound on the uniform approximation of fractional Brownian motion (BH(t):t[0,1]) with Hurst parameter H, by a deep-feedforward ReLU neural network fed with a N-dimensional Gaussian vector, with bounds on the network construction (number of hidden layers and total number of neurons). Essentially, up to log terms, achieving an uniform error of O(NH) is possible with log(N) hidden layers and O(NlogN) parameters. Our analysis relies, in the standard Brownian motion case (H=1/2), on the Levy construction of BH and in the general fractional Brownian motion case (H1/2), on the Lemarié-Meyer wavelet representation of BH. This work gives theoretical support on new generative models based on neural networks for simulating continuous-time processes.
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Dates et versions

hal-03237854 , version 1 (26-05-2021)
hal-03237854 , version 2 (16-06-2021)
hal-03237854 , version 3 (26-01-2022)
hal-03237854 , version 4 (25-04-2022)

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Michaël Allouche, Stéphane Girard, Emmanuel Gobet. A generative model for fBm with deep ReLU neural networks. Journal of Complexity, 2022, 73, pp.101667. ⟨10.1016/j.jco.2022.101667⟩. ⟨hal-03237854v4⟩
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