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Recycling Randomness with Structure for Sublinear time Kernel Expansions

Krzysztof Choromanski, Vikas Sindhwani
Proceedings of The 33rd International Conference on Machine Learning, pp. 2502–2510, 2016

Abstract

We propose a scheme for recycling Gaussian random vectors into structured matrices to ap- proximate various kernel functions in sublin- ear time via random embeddings. Our frame- work includes the Fastfood construction of Le et al. (2013) as a special case, but also ex- tends to Circulant, Toeplitz and Hankel matri- ces, and the broader family of structured matri- ces that are characterized by the concept of low- displacement rank. We introduce notions of co- herence and graph-theoretic structural constants that control the approximation quality, and prove unbiasedness and low-variance properties of ran- dom feature maps that arise within our frame- work. For the case of low-displacement matri- ces, we show how the degree of structure and randomness can be controlled to reduce statis- tical variance at the cost of increased computa- tion and storage requirements. Empirical results strongly support our theory and justify the use of a broader family of structured matrices for scal- ing up kernel methods using random features.

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