发表机构
University of Oxford; Beijing Normal-Hong Kong Baptist University（牛津大学; 北京师范大学-香港浸会大学联合国际学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对随机傅里叶特征仅适用于正定核的局限，提出适用于不定核的有符号随机傅里叶特征，实现了大规模核密度估计的加速，在百万点数据集上验证了其效率与准确性。
AI 中文摘要
核密度估计（KDE）是密度函数最基础的统计估计量之一，其对含N个点的数据集直接实现的计算成本为O(N²)，这对大规模数据集而言是难以承受的。核近似技术可将计算成本降至O(N)，基于从核函数谱密度采样的随机傅里叶特征（RFF）技术，已在机器学习应用中用于加速核估计量，且广受关注。遗憾的是，该技术仅适用于正定核，而KDE中常用的多数核函数（如抛物核）并不满足该性质。为克服这一局限，本文提出有符号随机傅里叶特征（SRFF）技术，它是RFF的推广，适用于逆傅里叶变换绝对可积的不定核。引入该方法的动机是加速通常非正定的多变量紧核的KDE，本文详细阐述了SRFF在乘积核和各向同性核上的实现方式。对于Kuttner-Golubov核类K(xᵢ,xⱼ)=(1-||xᵢ-xⱼ||^α)^β·1_{{||xᵢ-xⱼ||≤1}}（其中xᵢ∈ℝᵈ，xⱼ∈ℝᵈ，α>0，β>0，包含三角核、抛物核、双权核、三权核等KDE关注的核函数作为特例），本文提供了从其有符号谱密度采样的显式接受-拒绝算法。本文在含100万个点的数据集上开展的数值测试，证实了SRFF用于大规模KDE时的计算效率与准确性。
英文摘要
Kernel density estimation (KDE) is one of the most fundamental statistical estimators of density functions. Its direct implementation on a dataset of $N$ points incurs an $\mathcal{O}(N^{2})$ computational cost, which is prohibitive for large-scale datasets. Kernel approximation techniques can be applied to bring the computational cost down to $\mathcal{O}(N)$. The random Fourier features (RFF) technique, based on sampling from the spectral density of the kernel function, has become popular to speed up kernel estimators for machine learning applications. Unfortunately, it is restricted to positive definite kernels, while the majority of kernel functions popular in KDE, such as the parabolic kernel, do not satisfy this property. To overcome this limitation, this article introduces the signed random Fourier features (SRFF) technique. It is a generalization of RFF compatible with indefinite kernels whose inverse Fourier transform is absolutely integrable. The motivation for introducing this method is to speed up KDE in the case of multivariate compact kernels, which are generally not positive definite. We detail how to implement SRFF for both product kernels and isotropic kernels. For the class of Kuttner-Golubov kernels $K(\boldsymbol{x}_{i},\boldsymbol{x}_{j})=(1-\left\Vert \boldsymbol{x}_{i}-\boldsymbol{x}_{j}\right\Vert ^α)^β\mathbf{1}_{\{\left\Vert \boldsymbol{x}_{i}-\boldsymbol{x}_{j}\right\Vert \leq1\}}$ where $\boldsymbol{x}_{i}\in\mathbb{R}^{d}$, $\boldsymbol{x}_{j}\in\mathbb{R}^{d}$, $α>0$, $β>0$, which includes the triangular, parabolic, biweight, triweight, and other kernel functions of interest for KDE as particular examples, we provide an explicit acceptance-rejection algorithm to sample from its signed spectral density. Our numerical tests on a dataset of one million points confirm the computational efficiency and accuracy of SRFF for large-scale KDE.
Comments25 pages, 12 figures
