Speaker
Description
We present a comprehensive matrix perturbation analysis of methods for extracting singular value from approximate subspaces. Given (orthonormal) approximations $\tilde{U}$ and $\tilde{V}$ to the left and right subspaces spanned by the leading singular vectors of a matrix A, we discuss methods to approximate the leading singular values of A and study their accuracy. In particular, we focus our analysis on the generalized Nyström approximation, as surprisingly, it is able to obtain significantly better accuracy than classical methods, namely Rayleigh-Ritz and (one-sided) projected SVD. A key idea of the analysis is to view
the methods as finding the exact singular values of a perturbation of A. In this context, we derive a matrix perturbation result that exploits the structure of such $2\times 2$ block matrix perturbation. We then obtain bounds on the accuracy of the extracted singular values. This leads to sharp bounds that predict well the approximation error trends and explain the difference in the behaviour of these methods. Our theoretical results translate into practical insights on how best to approximate singular values given approximate subspaces. Specifically, when approximations to both left and right subspaces are available, the generalized Nyström approach consistently outperforms other single-pass methods by achieving higher accuracy in the leading singular values, as indicated by our bound analysis.
To validate our theoretical predictions, we conducted numerical experiments. These tests highlight how the derived bounds exhibit tighter estimates than classical bounds, effectively capturing the trends observed empirically. For instance, the bounds reflect a more gradual change in error across the leading singular values.
- Matrix perturbation analysis of methods for extracting singular values from approximate singular subspaces, L.Lazzarino, H. Al Daas, Y. Nakatsukasa, 2024, arXiv:2409.09187
