GP-HMAT：可扩展的，具有分层低秩矩阵的${O}(n\log(n))$高斯过程回归
GP-HMAT: Scalable, ${O}(n\log(n))$ Gaussian Process Regression with Hierarchical Low-Rank Matrices
摘要：
高斯过程（GP）是一种功能强大且广泛使用的回归技术。GP回归的主要构建块是协方差核，它表征了随机场中对之间的关系。然而，找到最佳核的优化需要几个大规模且通常非结构化的矩阵求逆。我们通过引入一种分层矩阵方法（称为HMAT）来解决这一挑战，该方法可以有效地将矩阵结构以递归方式分解为显着较小的矩阵，其中可以使用直接方法进行求逆。我们的矩阵划分使用一个特定的聚合策略的数据点，这促进了低秩结构的非对角块的分层核矩阵。我们采用了一种随机线性代数方法，在不分解大矩阵的情况下，对低秩非对角块进行矩阵约简。我们提供了分析误差和成本估计的矩阵求逆，调查他们的经验与数值计算，并展示了我们的方法的应用程序的三个数值例子涉及GP回归工程问题和大规模的真实数据集。我们提供了GP-HMAT的计算机实现，适用于GP似然和导数计算的HMAT，以及最后一个数值例子在真实数据集上的实现。我们展示了优越的可扩展性的HMAT方法相比，内置的$\backslash$运营商在MATLAB中的大规模线性解决$\bf{A}\bf{x} = \bf{y}$通过可重复和可验证的实证研究。作为未来的研究，层次半可分（HSS）矩阵的扩展进行了讨论。
英文摘要：
A Gaussian process (GP) is a powerful and widely used regression technique. The main building block of a GP regression is the covariance kernel, which characterizes the relationship between pairs in the random field. The optimization to find the optimal kernel, however, requires several large-scale and often unstructured matrix inversions. We tackle this challenge by introducing a hierarchical matrix approach, named HMAT, which effectively decomposes the matrix structure, in a recursive manner, into significantly smaller matrices where a direct approach could be used for inversion. Our matrix partitioning uses a particular aggregation strategy for data points, which promotes the low-rank structure of off-diagonal blocks in the hierarchical kernel matrix. We employ a randomized linear algebra method for matrix reduction on the low-rank off-diagonal blocks without factorizing a large matrix. We provide analytical error and cost estimates for the inversion of the matrix, investigate them empirically with numerical computations, and demonstrate the application of our approach on three numerical examples involving GP regression for engineering problems and a large-scale real dataset. We provide the computer implementation of GP-HMAT, HMAT adapted for GP likelihood and derivative computations, and the implementation of the last numerical example on a real dataset. We demonstrate superior scalability of the HMAT approach compared to built-in $\backslash$ operator in MATLAB for large-scale linear solves $\bf{A}\bf{x} = \bf{y}$ via a repeatable and verifiable empirical study. An extension to hierarchical semiseparable (HSS) matrices is discussed as future research.
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