## Abstract

Nyström approximation is a fast randomized method that rapidly solves kernel ridge regression (KRR) problems through subsampling the n-by-n empirical kernel matrix appearing in the objective function. However, the performance of such a sub-sampling method heavily relies on correctly estimating the statistical leverage scores for forming the sampling distribution, which can be as costly as solving the original KRR. In this work, we propose a linear time (modulo poly-log terms) algorithm to accurately approximate the statistical leverage scores in the stationary-kernel-based KRR with theoretical guarantees. Particularly, by analyzing the first-order condition of the KRR objective, we derive an analytic formula, which depends on both the input distribution and the spectral density of stationary kernels, for capturing the non-uniformity of the statistical leverage scores. Numerical experiments demonstrate that with the same prediction accuracy our method is orders of magnitude more efficient than existing methods in selecting the representative sub-samples in the Nyström approximation.

Original language | English (US) |
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Pages (from-to) | 2935-2943 |

Number of pages | 9 |

Journal | Proceedings of Machine Learning Research |

Volume | 130 |

State | Published - 2021 |

Externally published | Yes |

Event | 24th International Conference on Artificial Intelligence and Statistics, AISTATS 2021 - Virtual, Online, United States Duration: Apr 13 2021 → Apr 15 2021 |

## ASJC Scopus subject areas

- Artificial Intelligence
- Software
- Control and Systems Engineering
- Statistics and Probability