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"om approximation is applied to solve KLR, it also faces the time complexity of O(nc2) 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.