Understand
Randomized singular value decomposition (RSVD) is a class of computationally efficient algorithms for computing the truncated SVD of large data matrices.
- Given an $m \times n$ matrix $\widehat{{\mathbf M}}$, the prototypical RSVD algorithm outputs an approximation of the $k$ leading left singular vectors of $\widehat{\mathbf{M}}$ by computing the SVD of $\widehat{\mathbf{M}} (\widehat{{\mathbf M}}^{\top} \widehat{\mathbf{M}})^{g} \mathbf G$; here $g \geq 1$ is an integer and $\mathbf G \in \mathbb{R}^{n \times \widetilde{k}}$ is a random Gaussian sketching matrix with $\widetilde{k} \geq k$.
- In this paper we derive upper bounds for the $\ell_2$ and $\ell_{2,\infty}$ distances between the exact left singular vectors $\widehat{\mathbf{U}}$ of $\widehat{\mathbf{M}}$ and its approximation $\widehat{\mathbf{U}}_g$ (obtained via RSVD), as well as entrywise error bounds when $\widehat{\mathbf{M}}$ is projected onto $\widehat{\mathbf{U}}_g \widehat{\mathbf{U}}_g^{\top}$.
- These bounds depend on the singular values gap and number of power iterations $g$, and smaller gap requires larger values of $g$ to guarantee the convergences of the $\ell_2$ and $\ell_{2,\infty}$ distances.