Kernel logistic regression (KLR) is a conventional nonlinear classifier in machine learning. With the explosive growth of data size, the storage and computation of large dense kernel matrices is a major challenge in scaling KLR. Even the nystr\"{o}m approximation is applied to solve KLR, it also faces the time complexity of $O(nc^2)$ and the space complexity of $O(nc)$, where $n$ is the number of training instances and $c$ is the sampling size. In this paper, we propose a fast Newton method efficiently solving large-scale KLR problems by exploiting the storage and computing advantages of multilevel circulant matrix (MCM). Specifically, by approximating the kernel matrix with an MCM, the storage space is reduced to $O(n)$, and further approximating the coefficient matrix of the Newton equation as MCM, the computational complexity of Newton iteration is reduced to $O(n \log n)$. The proposed method can run in log-linear time complexity per iteration, because the multiplication of MCM (or its inverse) and vector can be implemented the multidimensional fast Fourier transform (mFFT). Experimental results on some large-scale binary-classification and multi-classification problems show that the proposed method enables KLR to scale to large scale problems with less memory consumption and less training time without sacrificing test accuracy.

翻译：内核后勤回归( KLR) 是常规的机器学习的非线性分类( KLR) 。 随着数据规模的爆炸性增长, 大型密集内核基质的存储和计算是KLR 的重大挑战。 即使是对KLR采用近似( nystr\ “ { o}m) 近似( KLR) 解决 KLR, 它也面临着美元( nc) 美元( nc) 和美元( nc) 的空间复杂性( 美元) 的复杂度( 美元) 是培训实例的数量, 美元( 美元) 样本大小( $c) 。 在本文中, 我们建议采用快速牛顿方法, 有效地解决大型 KLRR 问题, 利用多级ircultect 矩阵( MCMM) 的存储和计算优势。 具体来说, 将内核气矩阵矩阵的存储空间缩小到 $( n) 美元( ), 并且进一步将 Newton 方方方程式的计算复杂性减少到 $O( n) 。 。 。 拟议的方法可以不以 oralliveral- liveral- liveral