Statistical-computational gap in multiple Gaussian graph alignment

TL;DR

The paper addresses whether a statistical–computational gap exists in aligning multiple Gaussian graphs. It generalizes information-theoretic thresholds for exact and partial recovery as the number of graphs grows with the number of nodes and establishes a low-degree computational barrier that implies polynomial-time methods are no easier with many graphs than with two. The results show that, information-theoretically, more graphs aid recovery, but computationally the problem remains effectively as hard as pairwise alignment up to logarithmic factors, yielding a phase diagram separating regimes of possible and impossible recovery under polynomial-time computation. These findings highlight a broad, bi-dimensional statistical–computational gap in complex graph-structured inference problems and connect to broader themes in LD lower bounds and high-dimensional inference.

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 Massoulié (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 Massoulié (2025), and $p \geq Ω(n/\log(n))$ corresponds to a regime where the problem is as difficult as aligning one single graph with some unknown "signal" graph. Moreover, when $\log p = ω(\log n)$, 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 $ρ$ 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.

Figures (1)

• Figure 1: Statistical-computational landscape of partial recovery in multiple Gaussian graph alignment. The grey region is not fully covered by our results, since our lower bound involve logarithmic factors in $n$ and $p$.

Theorems & Definitions (34)

• Theorem 3.1
• Theorem 3.2
• Corollary 3.3
• Remark 3.1: On the exponential decay
• Theorem 3.4
• Theorem 3.5
• Proposition 4.1
• Theorem 4.2
• Corollary 4.3
• Theorem A.1