This paper presents a method for approximate Gaussian process (GP) regression with tensor networks (TNs). A parametric approximation of a GP uses a linear combination of basis functions, where the accuracy of the approximation depends on the total number of basis functions $M$. We develop an approach that allows us to use an exponential amount of basis functions without the corresponding exponential computational complexity. The key idea to enable this is using low-rank TNs. We first find a suitable low-dimensional subspace from the data, described by a low-rank TN. In this low-dimensional subspace, we then infer the weights of our model by solving a Bayesian inference problem. Finally, we project the resulting weights back to the original space to make GP predictions. The benefit of our approach comes from the projection to a smaller subspace: It modifies the shape of the basis functions in a way that it sees fit based on the given data, and it allows for efficient computations in the smaller subspace. In an experiment with an 18-dimensional benchmark data set, we show the applicability of our method to an inverse dynamics problem.

翻译：本文提出了一种利用张量网络（Tensor Networks, TNs）进行近似高斯过程（Gaussian Process, GP）回归的方法。GP的参数化近似采用基函数的线性组合，其近似精度取决于基函数总数 $M$。我们开发了一种方法，允许使用指数数量的基函数而无需承受相应的指数计算复杂度。实现这一点的关键在于采用低秩张量网络。我们首先从数据中寻找一个由低秩张量网络描述的合适低维子空间，然后在该低维子空间中通过求解贝叶斯推断问题来推断模型权重，最后将所得权重投影回原始空间以进行GP预测。本方法的优势在于向更小子空间的投影：它能根据给定数据自适应地调整基函数形态，并在较小子空间中实现高效计算。在18维基准数据集上的实验中，我们展示了该方法在逆动力学问题中的适用性。