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Gaussian Process is a non-parametric prior which can be understood as adistribution on the function space intuitively. It is known that by introducingappropriate prior to the weights of the neural networks, Gaussian Process canbe obtained by taking the infinite-width limit of the Bayesian neural networksfrom a Bayesian perspective. In this paper, we explore the infinitely wideTensor Networks and show the equivalence of the infinitely wide Tensor Networksand the Gaussian Process. We study the pure Tensor Network and another twoextended Tensor Network structures: Neural Kernel Tensor Network and TensorNetwork hidden layer Neural Network and prove that each one will converge tothe Gaussian Process as the width of each model goes to infinity. (We note herethat Gaussian Process can also be obtained by taking the infinite limit of atleast 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 theinfinite-width cases.) We calculate the mean function (mean vector) and thecovariance function (covariance matrix) of the finite dimensional distributionof the induced Gaussian Process by the infinite-width tensor network with ageneral set-up. We study the properties of the covariance function and derivethe approximation of the covariance function when the integral in theexpectation operator is intractable. In the numerical experiments, we implementthe Gaussian Process corresponding to the infinite limit tensor networks andplot the sample paths of these models. We study the hyperparameters and plotthe sample path families in the induced Gaussian Process by varying thestandard deviations of the prior distributions. As expected, the parameters inthe prior distribution namely the hyper-parameters in the induced GaussianProcess controls the characteristic lengthscales of the Gaussian Process.

Infinitely Wide Tensor Networks as Gaussian Process