Sparse variational approximations are popular methods for scaling up inference and learning in Gaussian processes to larger datasets. For $N$ training points, exact inference has $O(N^3)$ cost; with $M \ll N$ features, state of the art sparse variational methods have $O(NM^2)$ cost. Recently, methods have been proposed using more sophisticated features; these promise $O(M^3)$ cost, with good performance in low dimensional tasks such as spatial modelling, but they only work with a very limited class of kernels, excluding some of the most commonly used. In this work, we propose integrated Fourier features, which extends these performance benefits to a very broad class of stationary covariance functions. We motivate the method and choice of parameters from a convergence analysis and empirical exploration, and show practical speedup in synthetic and real world spatial regression tasks.

翻译：稀疏变分近似是将高斯过程的推理和学习扩展到更大数据集的流行方法。对于$N$个训练点，精确推理的计算成本为$O(N^3)$；当使用$M \ll N$个特征时，最先进的稀疏变分方法的计算成本为$O(NM^2)$。近期，有研究提出了基于更复杂特征的方法，这些方法承诺$O(M^3)$的计算成本，且在空间建模等低维任务中表现良好，但它们仅适用于非常有限的一类核函数，排除了部分最常用的核函数。在本工作中，我们提出了集成傅里叶特征，该方法将上述性能优势扩展至非常广泛的一类平稳协方差函数。我们通过收敛性分析和经验探索来论证该方法及其参数选择，并在合成与真实世界的空间回归任务中展示了实际的加速效果。