Gaussian processes (GPs) for machine learning have been studied systematically over the past two decades. However, kernel design for GPs and the associated hyper-parameters optimization are still difficult, and to a large extent open problems. We consider GP regression for time series modeling and analysis. The underlying stationary kernel is approximated closely by a new grid spectral mixture (GSM) kernel, which is a linear combination of low-rank sub-kernels. In the case where a large number of the involved sub-kernels are used, either the Nyström or the random Fourier feature approximations can be adopted to reduce the required computer storage. The unknown GP hyper-parameters consist of the nonnegative weights of all sub-kernels as well as the noise variance, and they are determined through the maximum-likelihood estimation method. Two optimization methods for solving the unknown hyper-parameters are introduced, including a sequential majorization-minimization (MM) method and a nonlinearly constrained alternating direction method of multipliers (ADMM). Experimental results, based on various time series datasets, corroborate that the proposed GSM kernel-based GP regression model outperforms several benchmarks in terms of prediction accuracy and numerical stability.

中文翻译：

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基于线性多低秩核的时间序列平稳高斯过程回归

在过去的二十年中，已经系统地研究了机器学习的高斯过程（GP）。然而，GPs 的内核设计和相关的超参数优化仍然很困难，并且在很大程度上是开放的问题。我们考虑使用 GP 回归进行时间序列建模和分析。底层的静止内核由一个新的网格频谱混合 (GSM) 内核近似近似，它是低秩子内核的线性组合。在使用大量涉及的子内核的情况下，可以采用 Nyström 或随机傅立叶特征近似来减少所需的计算机存储。未知 GP 超参数由所有子核的非负权重和噪声方差组成，它们通过最大似然估计方法确定。介绍了求解未知超参数的两种优化方法，包括顺序优化最小化 (MM) 方法和乘法器的非线性约束交替方向法 (ADMM)。基于各种时间序列数据集的实验结果证实，所提出的基于 GSM 内核的 GP 回归模型在预测精度和数值稳定性方面优于多个基准。