Publication
JMLR
Paper

Hierarchically compositional kernels for scalable nonparametric learning

Abstract

We propose a novel class of kernels to alleviate the high computational cost of large-scale nonparametric learning with kernel methods. The proposed kernel is defined based on a hierarchical partitioning of the underlying data domain, where the Nyström method (a globally low-rank approximation) is married with a locally lossless approximation in a hierarchical fashion. The kernel maintains (strict) positive-definiteness. The corresponding kernel matrix admits a recursively off-diagonal low-rank structure, which allows for fast linear algebra computations. Suppressing the factor of data dimension, the memory and arithmetic complexities for training a regression or a classifier are reduced from O(n2) and O(n3) to O(nr) and O(nr2), respectively, where n is the number of training examples and r is the rank on each level of the hierarchy. Although other randomized approximate kernels entail a similar complexity, empirical results show that the proposed kernel achieves a matching performance with a smaller r. We demonstrate comprehensive experiments to show the effective use of the proposed kernel on data sizes up to the order of millions.

Date

01 Aug 2017

Publication

JMLR

Authors

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