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    Large-width functional asymptotics for deep Gaussian neural networks

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    In this paper, we consider fully connected feed-forward deep neural networks where weights and biases are independent and identically distributed according to Gaussian distributions. Extending previous results (Matthews et al., 2018a;b; Yang, 2019) we adopt a function-space perspective, i.e. we look at neural networks as infinite-dimensional random elements on the input space RI\mathbb{R}^I. Under suitable assumptions on the activation function we show that: i) a network defines a continuous Gaussian process on the input space RI\mathbb{R}^I; ii) a network with re-scaled weights converges weakly to a continuous Gaussian process in the large-width limit; iii) the limiting Gaussian process has almost surely locally γ\gamma-H\"older continuous paths, for 0<γ<10 < \gamma <1. Our results contribute to recent theoretical studies on the interplay between infinitely wide deep neural networks and Gaussian processes by establishing weak convergence in function-space with respect to a stronger metric
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