• DOI:10.48550/arXiv.2301.07825
• Corpus ID: 255999633

XTrace: Making the most of every sample in stochastic trace estimation

@article{Epperly2023XTraceMT,
  title={XTrace: Making the most of every sample in stochastic trace estimation},
  author={Ethan N. Epperly and Joel A. Tropp and Robert J. Webber},
  journal={ArXiv},
  year={2023},
  volume={abs/2301.07825}
}

• Ethan N. Epperly, J. Tropp, R. Webber
• Published 19 January 2023
• Mathematics
• ArXiv

The implicit trace estimation problem asks for an approximation of the trace of a square matrix, accessed via matrix-vector products (matvecs). This paper designs new randomized algorithms, XTrace and XNysTrace, for the trace estimation problem by exploiting both variance reduction and the exchangeability principle. For a fixed budget of matvecs, numerical experiments show that the new methods can achieve errors that are orders of magnitude smaller than existing algorithms, such as the Girard-Hutchinson estimator or the Hutch++ estimator. A theoretical analysis confirms the benefits by offering a precise description of the performance of these algorithms as a function of the spectrum of the input matrix. The paper also develops an exchangeable estimator, XDiag, for approximating the diagonal of a square matrix using matvecs.