不要让克罗内克者困扰：高维不完整网格上的高斯过程
Don't Get Your Kroneckers in a Twist: Gaussian Processes on High-Dimensional Incomplete Grids
AI总结：
CUTS-GPR通过快速核矩阵-向量乘法实现高维高斯过程回归，展现线性或近线性可扩展性，用于高维势能面的贝叶斯建模。
摘要：
我们介绍了CUTS-GPR，一种在高维设置中进行数值精确高斯过程回归（GPR）的新方法。CUTS-GPR的关键组件是一个极快的核矩阵-向量乘法，其规模与训练数据量$N$呈近线性或线性关系，与维度$D$呈低阶多项式关系。这通过结合加法核与不完整网格并利用由此产生的核矩阵结构来实现。我们通过运行数十亿数据点和数千维的基准测试来展示矩阵-向量乘法的可扩展性。对于$N = 447265$和$D = 24$，完整的GPR计算，包括超参数优化，在几小时内完成。我们证明了CUTS-GPR能够实现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。
英文摘要：
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.
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