Perturbation Analysis of Randomized SVD and its Applications to Statistics

• URL: http://arxiv.org/abs/2203.10262v3
• Date: Sun, 25 May 2025 18:22:21 GMT
• Title: Perturbation Analysis of Randomized SVD and its Applications to Statistics
• Authors: Yichi Zhang, Minh Tang,
• Abstract summary: RSVD is a class of computationally efficient algorithms for computing the truncated SVD of large data matrices.<n>In this paper we derive upper bounds for the $ell$ and $ell_2,infty$ distances between the exact left singular vectors $widehatmathbfU$ of $widehatmathbfM$.<n>We apply our theoretical results to settings where $widehatmathbfM$ is an additive perturbation of some unobserved signal matrix $mathbfM$.
• Score: 8.731676546744353
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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. We apply our theoretical results to settings where $\widehat{\mathbf{M}}$ is an additive perturbation of some unobserved signal matrix $\mathbf{M}$. In particular, we obtain the nearly-optimal convergence rate and asymptotic normality for RSVD on three inference problems, namely, subspace estimation and community detection in random graphs, noisy matrix completion, and PCA with missing data.

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