Optimal Rates for Random Fourier Features
Bharath K. Sriperumbudur,Zoltan Szabo
(Submitted on 6 Jun 2015)
Kernel methods represent one of the most powerful tools in machine learning to tackle problems expressed in terms of function values and derivatives due to their capability to represent and model complex relations. While these methods show good versatility, they are computationally intensive and have poor scalability to large data as they require operations on Gram matrices. In order to mitigate this serious computational limitation, recently randomized constructions have been proposed in the literature, which allow the application of fast linear algorithms. Random Fourier features (RFF) are among the most popular and widely applied constructions: they provide an easily computable, low-dimensional feature representation for shift-invariant kernels. Despite the popularity of RFFs, very little is understood theoretically about their approximation quality. In this paper, we provide the first detailed theoretical analysis about the approximation quality of RFFs by establishing optimal (in terms of the RFF dimension) performance guarantees in uniform and $L^r$ ($1\le r<\infty$) norms. We also propose a RFF approximation to derivatives of a kernel with a theoretical study on its approximation quality.
Subjects:Statistics Theory (math.ST); Learning (cs.LG); Functional Analysis (math.FA); Machine Learning (stat.ML)
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