• DOI:10.48550/arXiv.2209.11023
• Corpus ID: 252438898

Randomized low-rank approximation of monotone matrix functions

@article{Persson2022RandomizedLA,
  title={Randomized low-rank approximation of monotone matrix functions},
  author={David Persson and Daniel Kressner},
  journal={ArXiv},
  year={2022},
  volume={abs/2209.11023}
}

• D. Persson, D. Kressner
• Published 22 September 2022
• Computer Science, Mathematics
• ArXiv

This work is concerned with computing low-rank approximations of a matrix function f ( A ) for a large symmetric positive semi-definite matrix A , a task that arises in, e.g., statistical learning and inverse problems. The application of popular randomized methods, such as the randomized singular value decomposition or the Nystr¨om approximation, to f ( A ) requires multiplying f ( A ) with a few random vectors. A significant dis-advantage of such an approach, matrix-vector products with f ( A ) are considerably more expensive than matrix-vector products with A , even when carried out only approximately via, e.g., the Lanczos method. In this work, we present and analyze funNystr¨om, a simple and inexpensive method that constructs a low-rank approximation of f ( A ) directly from a Nystr¨om approximation of A , completely bypassing the need for matrix-vector products with f ( A ). It is sensible to use funNystr¨om whenever f is monotone and satisfies f (0) = 0. Under the stronger assumption that f is operator monotone, which includes the matrix square root A 1 / 2 and the matrix logarithm log( I + A ), we derive probabilistic bounds for the error in the Frobenius, nuclear, and operator norms. These bounds confirm the numerical observation that funNystr¨om tends to return an approximation that compares well with the best low-rank approximation of f ( A ). Our method is also of interest when estimating quantities associated with f ( A ), such as the trace or the diagonal entries of f ( A ). In particular, we propose and analyze funNystr¨om++, a combination of funNystr¨om with the recently developed Hutch++ method for trace estimation.