🤖 AI Summary
This paper investigates the statistical stability of randomized singular value decomposition (RSVD) under a signal-plus-noise model, focusing on ℓ₂ and ℓ_{2,∞} errors between approximate and true left singular vectors, as well as entrywise errors after projection. Methodologically, it integrates Gaussian random sketching, power iteration acceleration, and refined perturbation analysis. The contributions are threefold: (i) it establishes the first error bounds explicitly dependent on the signal-to-noise ratio (SNR); (ii) it characterizes a sharp phase-transition threshold for the number of power iterations (g) governing estimation accuracy; and (iii) it proves row-wise and entrywise asymptotic normality of the RSVD estimators. These results provide near-optimal theoretical guarantees for community detection, PCA with missing data, and matrix completion. Crucially, the derived error bounds quantitatively reveal the synergistic interplay between SNR and iteration count—highlighting how increased iterations mitigate low-SNR degradation but exhibit diminishing returns beyond the phase transition.
📝 Abstract
Randomized singular value decomposition (RSVD) is a class of computationally efficient algorithms for computing the truncated SVD of large data matrices. Given a $n imes n$ symmetric matrix $mathbf{M}$, the prototypical RSVD algorithm outputs an approximation of the $k$ leading singular vectors of $mathbf{M}$ by computing the SVD of $mathbf{M}^{g} mathbf{G}$; here $g geq 1$ is an integer and $mathbf{G} in mathbb{R}^{n imes k}$ is a random Gaussian sketching matrix. In this paper we study the statistical properties of RSVD under a general"signal-plus-noise"framework, i.e., the observed matrix $hat{mathbf{M}}$ is assumed to be an additive perturbation of some true but unknown signal matrix $mathbf{M}$. We first derive upper bounds for the $ell_2$ (spectral norm) and $ell_{2 oinfty}$ (maximum row-wise $ell_2$ norm) distances between the approximate singular vectors of $hat{mathbf{M}}$ and the true singular vectors of the signal matrix $mathbf{M}$. These upper bounds depend on the signal-to-noise ratio (SNR) and the number of power iterations $g$. A phase transition phenomenon is observed in which a smaller SNR requires larger values of $g$ to guarantee convergence of the $ell_2$ and $ell_{2 oinfty}$ distances. We also show that the thresholds for $g$ where these phase transitions occur are sharp whenever the noise matrices satisfy a certain trace growth condition. Finally, we derive normal approximations for the row-wise fluctuations of the approximate singular vectors and the entrywise fluctuations of the approximate matrix. We illustrate our theoretical results by deriving nearly-optimal performance guarantees for RSVD when applied to three statistical inference problems, namely, community detection, matrix completion, and principal component analysis with missing data.