Improved Bounds for the Nyström Method With Application to Kernel Classification

Rong Jin, Tianbao Yang, Mehrdad Mahdavi, Yu-Feng Li, Zhi-Hua Zhou

Abstract: We develop two approaches for analyzing the approximation error bound for the Nyström method that approximates a positive semidefinite (PSD) matrix by sampling a small set of columns, one based on a concentration inequality for integral operators, and one based on random matrix theory. We show that the approximation error, measured in the spectral norm, can be improved from O(N/√m) to O(N/m <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1-ρ</sup> ) in the case of large eigengap, where N is the total number of data points, m is the number of sampled data points, and ρ ∈ (0, 1/2) is a positive constant that characterizes the eigengap. When the eigenvalues of the kernel matrix follow a p-power law, our analysis based on random matrix theory further improves the bound to O(N/m <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p-1</sup> ) under an incoherence assumption. We present a kernel classification approach based on the Nyström method and derive its generalization performance using the improved bound. We show that when the eigenvalues of the kernel matrix follow a p-power law, we can reduce the number of support vectors to N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2p/(p2</sup> - 1), which is sublinear in N when p > 1+√2, without seriously sacrificing its generalization performance.