Linearized GMM Kernels and Normalized Random Fourier Features

Abstract

The method of "random Fourier features (RFF)" has become a popular tool for approximating the "radial basis function (RBF)" kernel. The variance of RFF is actually large. Interestingly, the variance can be substantially reduced by a simple normalization step as we theoretically demonstrate. We name the improved scheme as the "normalized RFF (NRFF)". We also propose the "generalized min-max (GMM)" kernel as a measure of data similarity. GMM is positive definite as there is an associated hashing method named "generalized consistent weighted sampling (GCWS)" which linearizes this nonlinear kernel. We provide an extensive empirical evaluation of the RBF kernel and the GMM kernel on more than 50 publicly available datasets. For a majority of the datasets, the (tuning-free) GMM kernel outperforms the best-tuned RBF kernel. We conduct extensive experiments for comparing the linearized RBF kernel using NRFF with the linearized GMM kernel using GCWS. We observe that, to reach a comparable classification accuracy, GCWS typically requires substantially fewer samples than NRFF, even on datasets where the original RBF kernel outperforms the original GMM kernel. The empirical success of GCWS (compared to NRFF) can also be explained from a theoretical perspective. Firstly, the relative variance (normalized by the squared expectation) of GCWS is substantially smaller than that of NRFF, except for the very high similarity region (where the variances of both methods are close to zero). Secondly, if we make a model assumption on the data, we can show analytically that GCWS exhibits much smaller variance than NRFF for estimating the same object (e.g., the RBF kernel), except for the very high similarity region.