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断与学习扩展到更大数据集上的常用方法。对于 $N$ 个训练点，精确推断的计算复杂度为 $O(N^3)$；而采用 $M \ll N$ 个特征时，最先进的稀疏变分方法复杂度为 $O(NM^2)$。近期，有方法提出使用更复杂的特征，将复杂度降至 $O(M^3)$，并在空间建模等低维任务中展现出良好性能。然而，这些方法仅适用于非常有限的一类核函数，排除了某些最常用的核函数。本文提出集成傅里叶特征，将上述性能优势扩展到非常广泛的一类平稳协方差函数。我们通过收敛性分析和实证探索来论证该方法及参数选择的合理性，并在合成与真实空间回归任务中展示实际加速效果。