Computational thresholds in high-dimensional statistics: the case of graph alignment

In this article we consider the graph alignment problem from the perspective of high-dimensional statistics: we aim to estimate an unknown permutation $\pi^*$ 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 Erd\H{o}s-R\'enyi random graphs ${\mathcal {G}}(n,p)$ in the sparse regime with average degree $\lambda:=np= O(1)$ and edge correlation parameter $s\in(0,1)$, we identify a critical threshold $s^*(\lambda)$ 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 $s=1/\sqrt{1+\sigma^2}$ for some noise parameter $\sigma$. For a fast spectral algorithm, we identify the critical scaling for noise parameter $\sigma$ at which the fraction of entries of $\pi^*$ 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 $\sigma$ at which a simple post-processing of $X$ correctly recovers a fraction $1-o(1)$ of entries of $\pi^*$. 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.