Computational thresholds in high-dimensional statistics: the case of graph alignment
Laurent Massoulié
TL;DR
This work addresses graph alignment in high-dimensional statistics by studying the recoverability of an unknown permutation $π^*$ from two correlated adjacency matrices under sparse Erdős-Rényi and correlated Gaussian Wigner models. It develops and analyzes multiple polynomial-time approaches: a local message-passing algorithm (MP-Align) for sparse graphs, a fast spectral method (EIG1) for Gaussian matrices, and a convex relaxation based on the Birkhoff polytope, each with distinct computational thresholds. Key findings include a threshold $s^*(λ)$ for MP-Align tied to a correlated-tree model, a limiting threshold $ ilde{α}$ linked to Otter's constant for tree correlation testing, a $σ$-dependent convex/linear-algebraic threshold between $n^{-1}$ and $n^{-1/2}$, and a sharper $σ$-scaling result for spectral methods. These results delineate when polynomial-time algorithms can achieve partial or near-exact recovery and point to concrete directions for proving hardness in the remaining regimes, advancing understanding of computational barriers in high-dimensional graph alignment.
Abstract
In this article we consider the graph alignment problem from the perspective of high-dimensional statistics: we aim to estimate an unknown permutation $π^*$ from the observation of two correlated random adjacency matrices $A_1$, $A_2$. We establish the following computational thresholds. For $A_1$, $A_2$ the adjacency matrices of two correlated Erdős-Rényi random graphs ${\mathcal {G}}(n,p)$ in the sparse regime with average degree $λ:=np= O(1)$ and edge correlation parameter $s\in(0,1)$, we identify a critical threshold $s^*(λ)$ for $s$ above which a message-passing, local algorithm succeeds at alignment, and below which no local algorithm succeeds. This result crucially depends on an associated model of correlated random trees. We then consider the case where $A_1$, $A_2$ are two correlated Gaussian Wigner matrices with correlation parameter $s=1/\sqrt{1+σ^2}$ for some noise parameter $σ$. For a fast spectral algorithm, we identify the critical scaling for noise parameter $σ$ at which the fraction of entries of $π^*$ correctly recovered goes from $1-o(1)$ to $o(1)$. We next consider the convex relaxation approach which obtains the doubly stochastic matrix $X$ that minimizes $\|X A_1 -A_2 X\|_F$. We obtain upper and lower bounds on the critical noise parameter $σ$ at which a simple post-processing of $X$ correctly recovers a fraction $1-o(1)$ of entries of $π^*$. We finally identify promising future directions on i) computational thresholds for spectral methods and convex relaxation methods of practical interest, and ii) impossibility results for broad classes of algorithms, notably low degree polynomial algorithms and local search algorithms.