We consider multivariate splines and show that they have a random feature expansion as infinitely wide neural networks with one-hidden layer and a homogeneous activation function which is the power of the rectified linear unit. We show that the associated function space is a Sobolev space on a Euclidean ball, with an explicit bound on the norms of derivatives. This link provides a new random feature expansion for multivariate splines that allow efficient algorithms. This random feature expansion is numerically better behaved than usual random Fourier features, both in theory and practice. In particular, in dimension one, we compare the associated leverage scores to compare the two random expansions and show a better scaling for the neural network expansion.

翻译：我们考虑多元样条，并证明它们具有作为无限宽神经网络的随机特征展开形式，该网络为单隐藏层且激活函数是修正线性单元的幂次齐次函数。我们证明相关函数空间是欧几里得球上的索伯列夫空间，并给出导数范数的显式界。这一联系为多元样条提供了新的随机特征展开，从而支持高效算法。无论在理论上还是实践中，这种随机特征展开在数值表现上均优于通常的随机傅里叶特征。特别地，在一维情形下，我们通过比较两种随机展开的关联杠杆评分，表明神经网络展开具有更优的缩放特性。