## Abstract

We study the problem of estimating the trace of a matrix A that can only be accessed through matrix-vector multiplication. We introduce a new randomized algorithm, Hutch^{++}, which computes a (1±ε) approximation to tr(A) for any positive semidefinite (PSD) A using just O(1/ε) matrix-vector products. This improves on the ubiquitous Hutchinson’s estimator, which requires O(1/ε^{2}) matrix-vector products. Our approach is based on a simple technique for reducing the variance of Hutchinson’s estimator using a low-rank approximation step, and is easy to implement and analyze. Moreover, we prove that, up to a logarithmic factor, the complexity of Hutch++ is optimal amongst all matrix-vector query algorithms, even when queries can be chosen adaptively. We show that it significantly outperforms Hutchinson’s method in experiments. While our theory requires A to be positive semidefinite, empirical gains extend to applications involving non-PSD matrices, such as triangle estimation in networks.

Original language | English (US) |
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Title of host publication | 4th Symposium on Simplicity in Algorithms, SOSA 2021 |

Editors | Valerie King, Hung Viet Le |

Publisher | Society for Industrial and Applied Mathematics Publications |

Pages | 142-155 |

Number of pages | 14 |

ISBN (Electronic) | 9781713827122 |

State | Published - 2021 |

Event | 4th Symposium on Simplicity in Algorithms, SOSA 2021, co-located with SODA 2021 - Alexandria, United States Duration: Jan 11 2021 → Jan 12 2021 |

### Publication series

Name | 4th Symposium on Simplicity in Algorithms, SOSA 2021 |
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### Conference

Conference | 4th Symposium on Simplicity in Algorithms, SOSA 2021, co-located with SODA 2021 |
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Country/Territory | United States |

City | Alexandria |

Period | 1/11/21 → 1/12/21 |

## ASJC Scopus subject areas

- Computational Theory and Mathematics
- Computational Mathematics
- Numerical Analysis
- Theoretical Computer Science

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