Fast Newton method solving KLR based on Multilevel Circulant Matrix with log-linear complexity

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.