We introduce so-called functional input neural networks defined on a possibly infinite dimensional weighted space with values also in a possibly infinite dimensional output space. To this end, we use an additive family as hidden layer maps and a non-linear activation function applied to each hidden layer. Relying on Stone-Weierstrass theorems on weighted spaces, we can prove a global universal approximation result for generalizations of continuous functions going beyond the usual approximation on compact sets. This then applies in particular to approximation of (non-anticipative) path space functionals via functional input neural networks. As a further application of the weighted Stone-Weierstrass theorem we prove a global universal approximation result for linear functions of the signature. We also introduce the viewpoint of Gaussian process regression in this setting and show that the reproducing kernel Hilbert space of the signature kernels are Cameron-Martin spaces of certain Gaussian processes. This paves the way towards uncertainty quantification for signature kernel regression.

本文介绍了一种在可能是无限维的加权空间中定义的所谓功能输入神经网络，其值也可能在可能是无限维的输出空间中。通过在隐藏层映射中使用加性家族和应用于每个隐藏层的非线性激活函数，我们可以证明该模型具有全局通用逼近能力，适用于连续函数的一般化，超越紧致集上的通常逼近。这特别适用于通过功能性输入神经网络来近似（非先决性）路径空间函数。作为加权 Stone-Weierstrass 定理的进一步应用，我们证明了 signature 的线性函数具有全局通用逼近能力。我们还在该设置中介绍了 Gaussian 过程回归的视点，并表明 signature 核的再现核希尔伯特空间是某些 Gaussian 过程的 Cameron-Martin 空间，这为 signature 核回归的不确定性量化铺平了道路。

加权空间上函数输入映射的全局通用逼近