Anisotropy and non-stationarity in statistical learning, and its influence on the convergence of function approximation

在给定一组已知训练点的情况下，高斯过程已被证明是重建函数的一种强大而灵活的工具，并在机器学习、优化和数据同化中得到了应用。然而，当重建的函数是非平稳的或各向异性的时候，它们可能是有限的。深高斯过程使用分层过程来构造，其中高斯过程的输入本身就是高斯过程，其目的是给出一种更灵活的函数重建方法。我们根据已知训练点的数量来考察这些深高斯过程的收敛速度。我们还表明，在重构非平稳和各向异性函数时，深度高斯过程回归比标准高斯过程回归获得了更好的结果。

Gaussian processes have proved to be a powerful and flexible tool in the reconstruction of functions given a set of known training points, with applications in machine learning, optimisation and data assimilation. However, they can be limited when the functions being reconstructed are of a non-stationary or anisotropic nature. Deep Gaussian processes, constructed using a hierarchical process where the inputs to a Gaussian process are themselves Gaussian processes, aim to give a more flexible approach to function reconstruction. We look at convergence rates of these deep Gaussian processes in terms of the number of known training points. We also show that deep Gaussian process regression achieves considerably better results than standard Gaussian process regression when reconstructing non-stationary and anisotropic functions.