The Algorithmic Phase Transition in Correlated Spiked Models

Abstract

We study the computational task of detecting and estimating correlated signals in a pair of spiked matrices $$ X=\tfrac{\lambda}{\sqrt{n}} xu^{\top}+W, \quad Y=\tfrac{\mu}{\sqrt{n}} yv^{\top}+Z $$ where the spikes $x,y$ have correlation $\rho$. Specifically, we consider two fundamental models: (1) Correlated spiked Wigner model with signal-to-noise ratio $\lambda,\mu$; (2) Correlated spiked $n*N$ Wishart (covariance) model with signal-to-noise ratio $\sqrt\lambda,\sqrt\mu$. We propose an efficient detection and estimation algorithm based on counting a specific family of edge-decorated cycles. The algorithm's performance is governed by the function $$ F(\lambda,\mu,\rho,\gamma)=\max\Big\{ \frac{ \lambda^2 }{ \gamma }, \frac{ \mu^2 }{ \gamma }, \frac{ \lambda^2 \rho^2 }{ \gamma-\lambda^2+\lambda^2 \rho^2 } + \frac{ \mu^2 \rho^2 }{ \gamma-\mu^2+\mu^2 \rho^2 } \Big\} \,. $$ We prove our algorithm succeeds for the correlated spiked Wigner model whenever $F(\lambda,\mu,\rho,1)>1$, and succeeds for the correlated spiked Wishart model whenever $F(\lambda,\mu,\rho,\tfrac{n}{N})>1$. Our result shows that an algorithm can leverage the correlation between the spikes to detect and estimate the signals even in regimes where efficiently recovering either $x$ from ${X}$ alone or $y$ from ${Y}$ alone is believed to be computationally infeasible. We complement our algorithmic results with evidence for a matching computational lower bound. In particular, we prove that when $F(\lambda,\mu,\rho,1)<1$ for the correlated spiked Wigner model and when $F(\lambda,\mu,\rho,\tfrac{n}{N})<1$ for the spiked Wishart model, all algorithms based on low-degree polynomials fails to distinguish $({X},{Y})$ with two independent noise matrices. This strongly suggests that $F=1$ is the precise computation threshold for our models.