通过截断核矩阵边缘元素，加速高斯过程训练
Accelerated training of Gaussian processes using banded square exponential covariances
- 将平方指数协方差矩阵截断为带状结构，保留核心相关性
- 带状近似使逆矩阵与行列式计算复杂度从O(n³)降至O(n²)
- 适用于大规模一维数据，比稀疏变分方法更快
我们提出一种基于平方指数（SE）协方差矩阵中大量离对角线元素接近零的观察，构建了合理的截断方法，生成原协方差矩阵的带状近似。该近似矩阵的逆和行列式可显著降低计算成本，从而高效近似似然函数。我们在一维设置下对所提方法进行了理论分析，证明其能保持原始协方差结构，并在计算效率上优于稀疏高斯过程的变分自由能方法。
原文摘要 · Abstract (English)
We propose a novel approach to computationally efficient GP training based on the observation that square-exponential (SE) covariance matrices contain several off-diagonal entries extremely close to zero. We construct a principled procedure to eliminate those entries to produce a \emph{banded}-matrix approximation to the original covariance, whose inverse and determinant can be computed at a reduced computational cost, thus contributing to an efficient approximation to the likelihood function. We provide a theoretical analysis of the proposed method to preserve the structure of the original covariance in the 1D setting with SE kernel, and validate its computational efficiency against the variational free energy approach to sparse GPs.
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