高效处理高维不完整网格的高斯过程回归新方法
Don't Get Your Kroneckers in a Twist: Gaussian Processes on High-Dimensional Incomplete Grids
- 用加性核与不完整网格结构加速核矩阵-向量乘法
- 百万级数据点、千维场景下全模型计算数小时内完成
- 适用于高维势能面建模，助力计算化学难题解决
我们提出 CUTS-GPR，一种在高维场景下实现数值精确高斯过程回归（GPR）的新方法。其核心是极快的核矩阵-向量乘法，具有近线性甚至线性随训练数据量 N 的增长特性，且维度 D 的复杂度为低阶多项式。该效果通过结合加性核与不完整网格，并利用由此产生的核矩阵结构实现。我们通过实测验证了该乘法的可扩展性：在数十亿数据点和数千维场景下仍保持高效。对于 N = 447,265 且 D = 24 的情况，包括超参数优化在内的完整 GPR 计算可在数小时内完成。实验表明，CUTS-GPR 可有效支持高维势能面的贝叶斯建模——这是计算化学中长期存在的挑战。
原文摘要 · Abstract (English)
We introduce CUTS-GPR, a new method for performing numerically exact Gaussian process regression (GPR) in high-dimensional settings. The key component of CUTS-GPR is an extremely fast kernel matrix-vector product, which exhibits near-linear or even linear scaling with the amount of training data, $N$, and low-order polynomial scaling with dimensionality, $D$. This is obtained by combining an additive kernel with an incomplete grid and exploiting the resulting structure of the kernel matrix. We demonstrate the scalability of the matrix-vector product by running benchmarks with billions of data points and thousands of dimensions. Full GPR calculations, including hyperparameter optimization, are completed in a matter of hours for $N = 447 265$ and $D = 24$. We demonstrate that our CUTS-GPR enables Bayesian modeling of high-dimensional potential energy surfaces - a longstanding challenge in computational chemistry.
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