Column Thresholding for Sparse Spiked Wigner Models: Improved Signal Strength Requirements

We study the sparse spiked Wigner model, where the goal is to recover an $s$-sparse unit vector $\boldsymbol{u} \in \mathbb{R}^d$ from a noisy observation $\boldsymbol{Y} = \beta \boldsymbol{u} \boldsymbol{u}^\top + \boldsymbol{W}$. 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 both estimation and support recovery under the non-uniformity condition $|| \boldsymbol{u} ||_\infty = \Omega(1)$. This condition is not merely technical: it explicitly rules out uniform spikes, for which planted-clique-based hardness results apply, and identifies a concrete class of non-uniform spikes where the required signal strength can be reduced. Building on this initializer, we further develop a truncated power method that iteratively refines the estimate with provable linear convergence.