Datadependent compression of random features for largescale kernel approximation
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Proceedings of Machine Learning Research, PMLR 89:18221831, 2019.
Abstract
Kernel methods offer the flexibility to learn complex relationships in modern, large data sets while enjoying strong theoretical guarantees on quality. Unfortunately, these methods typically require cubic running time in the data set size, a prohibitive cost in the large data setting. Random feature maps (RFMs) and the Nystroöm method both consider low rank approximations to the kernel matrix as a potential solution. But, in order to achieve desirable theoretical guarantees, the former may require a prohibitively large number of features J+, and the latter may be prohibitively expensive for highdimensional problems. We propose to combine the simplicity and generality of RFMs with a datadependent feature selection scheme to achieve desirable theoretical approximation properties of Nyström with just $O(\log J+)$ features. Our key insight is to begin with a large set of random features, then reduce them to a small number of weighted features in a datadependent, computationally efficient way, while preserving the statistical guarantees of using the original large set of features. We demonstrate the efficacy of our method with theory and experimentsincluding on a data set with over 50 million observations. In particular, we show that our method achieves small kernel matrix approximation error and better test set accuracy with provably fewer random features than stateoftheart methods.
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