发表机构
Institute of Applied Physics and Computational Mathematics（北京应用物理与计算数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出自适应多分辨率高斯过程框架，利用自然数据稀疏协方差矩阵实现精确推断，训练成本O(n log²n)，预测成本O(log^d n)，兼顾可扩展性与模型保真度。
AI 中文摘要
高斯过程构成了概率机器学习的基石，然而将其扩展到大规模数据集通常需要在计算效率与模型保真度之间进行权衡。本研究通过提出一种既具可扩展性又保持精确性的自适应多分辨率高斯过程框架来弥合这一差距。我们的关键创新在于利用自适应多分辨率基函数构造自然数据稀疏的协方差矩阵。这些基函数直接锚定于样本，无需辅助点。通过收缩多分辨率基函数的支撑域，矩阵块大小受到限制，从而保证稀疏性。数据稀疏协方差矩阵的逆通过稀疏Cholesky逆算法精确且高效地计算。为进一步改善预测不确定性，我们构造了一个增广基函数。理论分析和数值实验表明，我们的模型实现了精确推断，训练成本为$\mathcal{O}(n \log^2 n)$，预测成本为$\mathcal{O}(\log^d n)$，为可扩展且高保真的高斯过程回归建立了原则性框架。
英文摘要
Gaussian processes constitute a cornerstone of probabilistic machine learning, yet scaling them to large datasets typically forces a trade-off between computational efficiency and model fidelity. This work bridges this gap by presenting an adaptive multi-resolution Gaussian process framework that is both scalable and exact. Our key innovation is constructing a naturally data-sparse covariance matrix with adaptive multi-resolution basis functions. These basis functions are directly anchored to samples, eliminating the need for auxiliary points. By shrinking the support domains of multi-resolution basis, the matrix block sizes are limited, guaranteeing sparsity. The inverse of the data-sparse covariance matrix is computed exactly and efficiently via the sparse Cholesky inverse algorithm. To further improve predictive uncertainties, we construct an augmented basis function. Theoretical analysis and numerical experiments demonstrate that our model achieves exact inference with $\mathcal{O}(n \log^2 n)$ training cost and $\mathcal{O}(\log^d n)$ prediction cost, establishing a principled framework for scalable and high-fidelity Gaussian process regression.
