BBP Phase Transition for a Doubly Sparse Deformed Model

TL;DR

This paper establishes a phase transition phenomenon for a novel doubly sparse deformed Wigner model, showing how spike signals induce outliers and eigenvector correlations in a high-dimensional sparse setting.

Contribution

It introduces a new doubly sparse model and proves a phase transition phenomenon analogous to classical results, extending understanding to supercritical sparsity regimes.

Findings

Spike signals > 1 induce outlier eigenvalues.

Eigenvectors correlate with spike vectors when signals > 1.

Results hold for supercritical sparsity regimes without relating noise and spike sparsities.

Abstract

We prove the equivalent of the Baik, Ben Arous, P\'ech\'e (2004) phenomenon for a novel, doubly sparse model where both the Wigner noise matrix and signal vector(s) are sparse. Specifically, we consider a deformed sub-Gaussian sparse Wigner ensemble with a fixed number of sub-Gaussian spike vectors of the same-order sparsity added. We show that spike vectors with signals greater than one are correlated with the top eigenvectors of the deformed ensemble and that each spike vector of signal greater than one induces an outlier eigenvalue. Notably, our results hold in the supercritical sparsity regime for the Wigner matrix ($q \gg \frac{\log n}{n}$) and for any sparse spike vector with an unbounded number of entries ($np\to \infty$). No further relationship between the sparsities of the noise matrix ($q$) and spike vectors ($p$) is necessary. This generalizes the work of Benaych-Georges and Nadakuditi (2010) and P\'ech\'e (2005).