论文标题
内核数据包：具有Matérn相关性的高斯过程回归的精确且可扩展的算法
Kernel Packet: An Exact and Scalable Algorithm for Gaussian Process Regression with Matérn Correlations
论文作者
论文摘要
我们开发了一种具有Matérn相关性的一维高斯过程回归的精确且可扩展的算法，其平滑度参数$ν$是半成位。所提出的算法仅需要$ \ MATHCAL {O}（ν^3 N）$操作和$ \ Mathcal {o}（O}（νn）$ storage。这导致了线性成本求解器，因为选择了$ν$是固定的，并且在大多数应用中通常很小。如果使用完整的网格或稀疏网格设计，则提出的方法可以应用于多维问题。所提出的方法基于Matérn相关函数的新型理论。我们发现，这些相关函数的合适重排可以产生紧凑的函数，称为“内核数据包”。使用一组内核数据包作为基础函数会导致导致提出算法的协方差矩阵的稀疏表示。仿真研究表明，在适用的情况下，所提出的算法在计算时间和预测精度中都显着优于现有替代方案。
We develop an exact and scalable algorithm for one-dimensional Gaussian process regression with Matérn correlations whose smoothness parameter $ν$ is a half-integer. The proposed algorithm only requires $\mathcal{O}(ν^3 n)$ operations and $\mathcal{O}(νn)$ storage. This leads to a linear-cost solver since $ν$ is chosen to be fixed and usually very small in most applications. The proposed method can be applied to multi-dimensional problems if a full grid or a sparse grid design is used. The proposed method is based on a novel theory for Matérn correlation functions. We find that a suitable rearrangement of these correlation functions can produce a compactly supported function, called a "kernel packet". Using a set of kernel packets as basis functions leads to a sparse representation of the covariance matrix that results in the proposed algorithm. Simulation studies show that the proposed algorithm, when applicable, is significantly superior to the existing alternatives in both the computational time and predictive accuracy.
