Tensor Networks (TNs) have recently been used to speed up kernel machines by constraining the model weights, yielding exponential computational and storage savings. In this paper we prove that the outputs of Canonical Polyadic Decomposition (CPD) and Tensor Train (TT)-constrained kernel machines recover a Gaussian Process (GP), which we fully characterize, when placing i.i.d. priors over their parameters. We analyze the convergence of both CPD and TT-constrained models, and show how TT yields models exhibiting more GP behavior compared to CPD, for the same number of model parameters. We empirically observe this behavior in two numerical experiments where we respectively analyze the convergence to the GP and the performance at prediction. We thereby establish a connection between TN-constrained kernel machines and GPs.

翻译：最近，张量网络（TN）已被用于通过约束模型权重加速核机，从而在计算和存储上实现指数级节省。本文证明，当对其参数施加独立同分布先验时，典型多项式分解（CPD）和张量列（TT）约束核机的输出恢复为高斯过程（GP），我们对其进行了完整刻画。我们分析了CPD和TT约束模型的收敛性，并展示了在相同模型参数数量下，TT相比CPD如何产生更具GP行为的模型。我们在两个数值实验中实证观察到这一行为，分别分析了向GP的收敛性和预测性能。由此，我们建立了张量网络约束核机与高斯过程之间的联系。