We derive upper bounds on the Wasserstein distance (), with respect to sup-norm, between any continuous valued random field indexed by the -sphere and the Gaussian, based on Stein's method. We develop a novel Gaussian smoothing technique that allows us to transfer a bound in a smoother metric to the distance. The smoothing is based on covariance functions constructed using powers of Laplacian operators, designed so that the associated Gaussian process has a tractable Cameron-Martin or Reproducing Kernel Hilbert Space. This feature enables us to move beyond one dimensional interval-based index sets that were previously considered in the literature. Specializing our general result, we obtain the first bounds on the Gaussian random field approximation of wide random neural networks of any depth and Lipschitz activation functions at the random field level. Our bounds are explicitly expressed in terms of the widths of the network and moments of the random weights. We also obtain tighter bounds when the activation function has three bounded derivatives.

中文翻译：

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通过 Stein 方法进行高斯随机场逼近及其在广泛随机神经网络中的应用


基于 Stein 的方法，我们推导出由 球面和高斯索引的任何连续值随机场之间相对于超范数的 Wasserstein 距离 () 的上限。我们开发了一种新颖的高斯平滑技术，使我们能够将更平滑度量的界限转移到距离。平滑基于使用拉普拉斯算子的幂构造的协方差函数，其设计使得相关的高斯过程具有易于处理的卡梅伦-马丁或再现核希尔伯特空间。此功能使我们能够超越先前文献中考虑的基于一维区间的索引集。专门化我们的一般结果，我们获得了任意深度的宽随机神经网络的高斯随机场近似的第一个界限以及随机场级别的 Lipschitz 激活函数。我们的边界明确地用网络的宽度和随机权重的矩来表示。当激活函数具有三个有界导数时，我们还会获得更严格的界限。