The Algorithmic Phase Transition in Correlated Spiked Models

TL;DR

The paper tackles the problem of detecting and recovering correlated signals in paired spiked matrices (Wigner and Wishart). It introduces a counting-based algorithm using edge-decorated cycles that achieves strong detection and weak recovery above a sharp computation threshold F(λ,μ,ρ,γ) > 1, and provides matching lower bounds via low-degree polynomial arguments to indicate a precise computational phase transition at F = 1. By formulating approximate statistics through color-coding, the authors obtain polynomial-time algorithms with provable guarantees, and they contrast their thresholds with classical PLS/CCA benchmarks in multi-dataset inference. The work thus establishes a concrete statistical-computational gap for correlated spiked models and offers a general methodology for leveraging inter-dataset correlations to surpass limits observed when analyzing each dataset separately.

Abstract

We study the computational task of detecting and estimating correlated signals in a pair of spiked matrices $$ X=\tfracλ{\sqrt{n}} xu^{\top}+W, \quad Y=\tfracμ{\sqrt{n}} yv^{\top}+Z $$ where the spikes $x,y$ have correlation $ρ$. Specifically, we consider two fundamental models: (1) Correlated spiked Wigner model with signal-to-noise ratio $λ,μ$; (2) Correlated spiked $n*N$ Wishart (covariance) model with signal-to-noise ratio $\sqrtλ,\sqrtμ$. 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 $$ F(λ,μ,ρ,γ)=\max\Big\{ \frac{ λ^2 }{ γ}, \frac{ μ^2 }{ γ}, \frac{ λ^2 ρ^2 }{ γ-λ^2+λ^2 ρ^2 } + \frac{ μ^2 ρ^2 }{ γ-μ^2+μ^2 ρ^2 } \Big\} \,. $$ We prove our algorithm succeeds for the correlated spiked Wigner model whenever $F(λ,μ,ρ,1)>1$, and succeeds for the correlated spiked Wishart model whenever $F(λ,μ,ρ,\tfrac{n}{N})>1$. 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 $F(λ,μ,ρ,1)<1$ for the correlated spiked Wigner model and when $F(λ,μ,ρ,\tfrac{n}{N})<1$ 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.

Figures (5)

• Figure 1: Phase diagram in the $(\lambda,\mu)$ plane illustrating the thresholds for the subgraph counts method (blue, this work), the PLS method (orange, MZ25+), and the CCA method (green, BHPZ19MY23BG23+). Here we take $\gamma=0.25$ and $\rho=0.99$.
• Figure 2: An unlabeled graph $[H] \in \mathcal{H}(\ell)$ with $\ell=8$. Here $\mathsf{diff}(H)=\{ v_1,v_3,v_5,v_8 \}$.
• Figure 3: An unlabeled graph $[H] \in \mathcal{G}(\ell)$ with $\ell=4$. Here $\mathsf{diff}(H)=\{ v_2,v_4 \}$.
• Figure 4: An unlabeled graph $[H] \in \mathcal{J}(\ell)$ with $\ell=6$. Here $\mathsf{diff}(H)=\{ v_2,v_3,v_4,v_6 \}$.
• Figure 5: An unlabeled graph $[H] \in \mathcal{I}(\ell)$ with $\ell=4$. Here $\mathsf{diff}(H)=\{ v_2,v_4 \}$.

Theorems & Definitions (101)

• Definition 1.1: Spiked Wigner model
• Definition 1.2: Spiked Wishart model
• Theorem
• Theorem
• Definition 1.4: Correlated spikes distribution
• Definition 1.5: Correlated spiked Wigner model
• Definition 1.6: Correlated spiked Wishart model
• Definition 1.7
• Definition 1.8
• Theorem 1.9: Informal