Gaussian Process is a non-parametric prior which can be understood as a distribution on the function space intuitively. It is known that by introducing appropriate prior to the weights of the neural networks, Gaussian Process can be obtained by taking the infinite-width limit of the Bayesian neural networks from a Bayesian perspective. In this paper, we explore the infinitely wide Tensor Networks and show the equivalence of the infinitely wide Tensor Networks and the Gaussian Process. We study the pure Tensor Network and another two extended Tensor Network structures: Neural Kernel Tensor Network and Tensor Network hidden layer Neural Network and prove that each one will converge to the Gaussian Process as the width of each model goes to infinity. (We note here that Gaussian Process can also be obtained by taking the infinite limit of at least one of the bond dimensions $\alpha_{i}$ in the product of tensor nodes, and the proofs can be done with the same ideas in the proofs of the infinite-width cases.) We calculate the mean function (mean vector) and the covariance function (covariance matrix) of the finite dimensional distribution of the induced Gaussian Process by the infinite-width tensor network with a general set-up. We study the properties of the covariance function and derive the approximation of the covariance function when the integral in the expectation operator is intractable. In the numerical experiments, we implement the Gaussian Process corresponding to the infinite limit tensor networks and plot the sample paths of these models. We study the hyperparameters and plot the sample path families in the induced Gaussian Process by varying the standard deviations of the prior distributions. As expected, the parameters in the prior distribution namely the hyper-parameters in the induced Gaussian Process controls the characteristic lengthscales of the Gaussian Process.

中文翻译：

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作为高斯过程的无限宽的张量网络

高斯过程是非参数先验，可以直观地理解为函数空间上的分布。已知通过在神经网络的权重之前引入适当的值，可以通过从贝叶斯的角度考虑贝叶斯神经网络的无限宽度极限来获得高斯过程。在本文中，我们探索了无限宽的张量网络，并证明了无限宽的张量网络和高斯过程的等价性。我们研究了纯的Tensor网络和另外两个扩展的Tensor网络结构：神经核Tensor网络和Tensor网络隐藏层神经网络，并证明了当每个模型的宽度达到无穷大时，每个人都将收敛于高斯过程。（我们在这里注意到，高斯过程也可以通过对张量节点乘积中的键维数$ \ alpha_ {i} $的至少一个取无穷大来获得，并且可以用相同的思想来证明我们用一般设置的无限宽张量网络来计算诱导高斯过程有限维分布的均值函数（均值向量）和协方差函数（协方差矩阵）。我们研究协方差函数的性质，并在期望算符中的积分不可解时推导协方差函数的近似值。在数值实验中，我们实现了与无限极限张量网络相对应的高斯过程，并绘制了这些模型的样本路径。我们研究超参数，并通过改变先验分布的标准偏差，在诱导的高斯过程中绘制样本路径族。如预期的那样，先验分布中的参数（即诱导高斯过程中的超参数）控制着高斯过程的特征长度尺度。