### Abstract

Language | English |
---|---|

Pages | 198-214 |

Number of pages | 17 |

Journal | Journal of Computational and Applied Mathematics |

Volume | 220 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 15 Oct 2008 |

### Fingerprint

### Keywords

- Gaussian random seeds
- uniformly distributed seeds
- randomized trace estimator
- log-det approximation
- O(N2) operations

### Cite this

*Journal of Computational and Applied Mathematics*,

*220*(1-2), 198-214. https://doi.org/10.1016/j.cam.2007.08.012

}

*Journal of Computational and Applied Mathematics*, vol. 220, no. 1-2, pp. 198-214. https://doi.org/10.1016/j.cam.2007.08.012

**Log-det approximation based on uniformly distributed seeds and its application to Gaussian process regression.** / Zhang, Y.; Leithead, W.E.; Leith, D.J.; Walshe, L.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Log-det approximation based on uniformly distributed seeds and its application to Gaussian process regression

AU - Zhang, Y.

AU - Leithead, W.E.

AU - Leith, D.J.

AU - Walshe, L.

PY - 2008/10/15

Y1 - 2008/10/15

N2 - Maximum likelihood estimation (MLE) of hyperparameters in Gaussian process regression as well as other computational models usually and frequently requires the evaluation of the logarithm of the determinant of a positive-definite matrix (denoted by Chereafter). In general, the exact computation of log del C is of O(N-3) operations where N is the matrix dimension. The approximation of log del C could be developed with O(N-2) operations based on power-series expansion and randomized trace estimator. In this paper, the accuracy and effectiveness of using uniformly distributed seeds for log det C approximation are investigated. The research shows that uniform-seed based approximation is an equally good alternative to Gaussian-seed based approximation, having slightly better approximation accuracy and smaller variance. Gaussian process regression examples also substantiate the effectiveness of such a uniform-seed based log-det approximation scheme.

AB - Maximum likelihood estimation (MLE) of hyperparameters in Gaussian process regression as well as other computational models usually and frequently requires the evaluation of the logarithm of the determinant of a positive-definite matrix (denoted by Chereafter). In general, the exact computation of log del C is of O(N-3) operations where N is the matrix dimension. The approximation of log del C could be developed with O(N-2) operations based on power-series expansion and randomized trace estimator. In this paper, the accuracy and effectiveness of using uniformly distributed seeds for log det C approximation are investigated. The research shows that uniform-seed based approximation is an equally good alternative to Gaussian-seed based approximation, having slightly better approximation accuracy and smaller variance. Gaussian process regression examples also substantiate the effectiveness of such a uniform-seed based log-det approximation scheme.

KW - Gaussian random seeds

KW - uniformly distributed seeds

KW - randomized trace estimator

KW - log-det approximation

KW - O(N2) operations

U2 - 10.1016/j.cam.2007.08.012

DO - 10.1016/j.cam.2007.08.012

M3 - Article

VL - 220

SP - 198

EP - 214

JO - Journal of Computational and Applied Mathematics

T2 - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

IS - 1-2

ER -