Statistical-computational gap in multiple Gaussian graph alignment
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911294661066752 |
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| author | Even, Bertrand Ganassali, Luca |
| author_facet | Even, Bertrand Ganassali, Luca |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_00610 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Statistical-computational gap in multiple Gaussian graph alignment Even, Bertrand Ganassali, Luca Machine Learning Statistics Theory 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. |
| title | Statistical-computational gap in multiple Gaussian graph alignment |
| topic | Machine Learning Statistics Theory |
| url | https://arxiv.org/abs/2512.00610 |