用准蒙特卡洛切片法加速径向核函数求和，提升计算效率。
Fast Summation of Radial Kernels via QMC Slicing
- 通过随机投影到一维子空间结合快速傅里叶求和实现切片计算
- 在标准数据集上相比现有方法速度提升显著，误差可控
- 适合需要高效核方法的机器学习任务，如高维近似与建模
大规模核函数求和是核方法中的关键挑战。本文提出基于切片的方法，利用随机投影将高维问题降至一维子空间，并结合快速傅里叶求和进行加速。我们证明了切片误差的理论界，并提出一种基于球面求积规则的准蒙特卡洛（QMC）投影选择策略。数值实验表明，所提出的QMC切片方法在标准测试数据集上显著优于现有方法，包括(QMC-)随机傅里叶特征、正交傅里叶特征以及非QMC切片方法。
原文摘要 · Abstract (English)
The fast computation of large kernel sums is a challenging task, which arises as a subproblem in any kernel method. We approach the problem by slicing, which relies on random projections to one-dimensional subspaces and fast Fourier summation. We prove bounds for the slicing error and propose a quasi-Monte Carlo (QMC) approach for selecting the projections based on spherical quadrature rules. Numerical examples demonstrate that our QMC-slicing approach significantly outperforms existing methods like (QMC-)random Fourier features, orthogonal Fourier features or non-QMC slicing on standard test datasets.
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