We consider the problem of improving kernel approximation via randomized
feature maps. These maps arise as Monte Carlo approximation to integral
representations of kernel functions and scale up kernel methods for larger
datasets. Based on an efficient numerical integration technique, we propose a
unifying approach that reinterprets the previous random features methods and
extends to better estimates of the kernel approximation. We derive the
convergence behaviour and conduct an extensive empirical study that supports
our hypothesis.

通过基于 Monte Carlo 近似的、通过积分表示核函数并扩展到更好的核近似估计的数值积分技术，我们提出了一个统一的方法来改进核逼近的随机特征方法，并得出了其收敛行为，并进行了大量实证研究，支持了我们的假设。

基于 Quadrature 的特征用于核逼近

Quadrature-based features for kernel approximation

Kernel approximation using randomized feature maps has recently gained a lot
of interest. In this work, we identify that previous approaches for polynomial
kernel approximation create maps that are rank deficient, and therefore do not
utilize the capacity of the projected feature space effectively. To address
this challenge, we propose compact random feature maps (CRAFTMaps) to
approximate polynomial kernels more concisely and accurately. We prove the
error bounds of CRAFTMaps demonstrating their superior kernel reconstruction
performance compared to the previous approximation schemes. We show how
structured random matrices can be used to efficiently generate CRAFTMaps, and
present a single-pass algorithm using CRAFTMaps to learn non-linear multi-class
classifiers. We present experiments on multiple standard data-sets with
performance competitive with state-of-the-art results.

提出了一种基于随机矩阵的紧凑型随机特征映射（CRAFTMaps）来更加简洁准确地近似多项式核，证明了其比以前的近似方案有优越的核重建性能，并演示了如何使用 CRAFTMaps 学习非线性多类分类器。