arXiv：cs.LG· Mads Greisen H{\o}jlund, August Smart Lykke-M{\o}ller, Henry Moss, Ove Christiansen·· 3 小时前
CUTS-GPR：面向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
AI 导读
CUTS-GPR 通过将加性核与不完整网格结合，实现核矩阵-向量积随训练数据量 N 近线性甚至线性扩展，对维度 D 仅呈低阶多项式扩展。该积已在数十亿数据点、数千维度上验证，完整 GPR 计算（含超参数优化）在 N=4,494,001、D=500 的合成数据集上完成。其应用于 10 个势能面（N=447,265、D=24）仅需数小时，使高维势能面的贝叶斯建模成为可能。
正文
Abstract:We introduce CUTS-GPR, a new method for performing numerically exact 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. The scalability of the matrix-vector product is verified by benchmarks with billions of data points and thousands of dimensions. We demonstrate the end-to-end scalability of CUTS-GPR by running full GPR calculations, including hyperparameter optimization, on synthetic datasets with up to $N = 4\,494\,001$ and $D = 500$. As a realistic and challenging test, we finally apply CUTS-GPR to a set of ten potential energy surfaces (PESs) with $N = 447\,265$ and $D = 24$. The calculations are completed in a matter of hours, showing that CUTS-GPR enables Bayesian modelling of high-dimensional PESs - a longstanding challenge in computational chemistry.
| Comments: | 63 pages, 21 figures |
| Subjects: | Machine Learning (cs.LG) |
| Cite as: | arXiv:2605.08036 [cs.LG] |
| (or arXiv:2605.08036v2 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.08036 arXiv-issued DOI via DataCite |
Submission history
From: Mads Højlund [view email]
[v1]
Fri, 8 May 2026 17:24:22 UTC (1,747 KB)
[v2]
Wed, 7 Oct 2026 18:46:13 UTC (3,483 KB)
来源：arXiv：cs.LG · arxiv.org