高斯过程回归中的logdet近似算法及数值实验

Approximate Implementation of Logarithm of Matrix Determinant in Gaussian Process Regression and Numerical Experiments

在高斯过程以及其他空间回归模型中,超参数的最大似然估计(MLE)通常需要求矩阵行列式对数的估计,简称logdet。提出了一种基于幂级数展开的结构,用于广义正定矩阵logdet的近似计算,并给出三种新的补偿方案,进一步提高近似值的精确度和计算效率,所提logdet的近似实现方案仅需50N2次操作。大量的数值实验,包括对随机产生的正定矩阵、随机产生的协方差矩阵以及两个高斯过程回归实例产生的协方差矩阵序列的检验,都已证实所提方案的可行性。

Maximum likelihood estimation （MLE） of hyperparameters in Gaussian processes as well as other spatial regression models usually requires the evaluation of the logarithm of the matrix determinant, in short, log det. A power-series expansion based framework was proposed for approximating the log det of general positive-definite matrices. Three novel compensation schemes were given to further improve the approximation accuracy and computational efficiency. The proposed log det approximation required only 50（ N 2） operations. The proposed scheme was substantiated by a large number of numerical experiments, including tests on randomly-generated positive-definite matrices, randomly-generated covariance matrices and sequences of covariance matrices generated online in two Gaussian process regression examples.