Perturbation Analysis of Randomized SVD and its Applications to Statistics
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866909621401157632 |
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| author | Zhang, Yichi Tang, Minh |
| author_facet | Zhang, Yichi Tang, Minh |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_10262 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Perturbation Analysis of Randomized SVD and its Applications to Statistics Zhang, Yichi Tang, Minh Statistics Theory Numerical Analysis Computation Machine Learning 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. |
| title | Perturbation Analysis of Randomized SVD and its Applications to Statistics |
| topic | Statistics Theory Numerical Analysis Computation Machine Learning |
| url | https://arxiv.org/abs/2203.10262 |