Title: Column Thresholding: Bridging the Computational-Statistical Gap in the Sparse Spiked Wigner Model
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
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 bridges this gap under a mild condition, achieving both constant estimation error bound and exact support recovery. We further develop a truncated power method that iteratively refines the column thresholding estimate with provable linear convergence. Extensive experiments validate our theoretical guarantees and demonstrate superior performance in estimation accuracy, support recovery, and computational efficiency across diverse settings.