Column Thresholding: Bridging the Computational-Statistical Gap in the Sparse Spiked Wigner Model

Jian-Feng Cai, Zhuozhi Xian, Jiaxi Ying

Keywords: spiked Wigner model, spectral method, diagonal thresholding, truncated power method

Abstract: We study the sparse spiked Wigner model, where the goal is to recover an $s$-sparse unit vector $\mathbf{u} \in \mathbb{R}^d$ from a noisy matrix observation $\mathbf{Y} = \beta \mathbf{u} \mathbf{u}^\top + \mathbf{W}$. While the information-theoretic threshold is $\beta = \widetilde{\Omega}(\sqrt{s})$, existing polynomial-time algorithms require $\beta = \widetilde{\Omega}(s)$, yielding a substantial computational-statistical gap. We propose a column thresholding method that attains the $\widetilde{\Omega}(\sqrt{s})$ scaling for estimation and support recovery under the condition $||\mathbf{u}||_\infty = \Omega(1)$. Building on this initializer, we further develop a truncated power method that iteratively refines the estimate with provable linear convergence. Experiments validate our theoretical guarantees and demonstrate superior performance in estimation accuracy, support recovery, and computational efficiency.

Primary Area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)

Submission Number: 20474