Statistical-computational gap in multiple Gaussian graph alignment

TL;DR

This paper explores the limits of statistical and computational methods in aligning multiple Gaussian graphs, revealing a gap where efficient algorithms struggle despite the problem's statistical feasibility.

Contribution

It generalizes existing thresholds for Gaussian graph alignment to regimes where the number of graphs grows with nodes, and establishes the first computational barrier in the low-degree framework.

Findings

Identifies regimes where the problem is statistically solvable but computationally hard.

Shows the computational difficulty of aligning multiple graphs is comparable to aligning two graphs.

Highlights a divergence between partial and exact recovery thresholds when the number of graphs grows rapidly.

Abstract

We investigate the existence of a statistical-computational gap in multiple Gaussian graph alignment. We first generalize a previously established informational threshold from Vassaux and Massouli\'e (2025) to regimes where the number of observed graphs $p$ may also grow with the number of nodes $n$: when $p \leq O(n/\log(n))$, we recover the results from Vassaux and Massouli\'e (2025), and $p \geq \Omega(n/\log(n))$ corresponds to a regime where the problem is as difficult as aligning one single graph with some unknown "signal" graph. Moreover, when $\log p = \omega(\log n)$, the informational thresholds for partial and exact recovery no longer coincide, in contrast to the all-or-nothing phenomenon observed when $\log p=O(\log n)$. Then, we provide the first computational barrier in the low-degree framework for (multiple) Gaussian graph alignment. We prove that when the correlation $\rho$ is less than $1$, up to logarithmic terms, low degree non-trivial estimation fails. Our results suggest that the task of aligning $p$ graphs in polynomial time is as hard as the problem of aligning two graphs in polynomial time, up to logarithmic factors. These results characterize the existence of a statistical-computational gap and provide another example in which polynomial-time algorithms cannot handle complex combinatorial bi-dimensional structures.