We study the problem of weak recovery for the $r$-uniform hypergraph stochastic block model ($r$-HSBM) with two balanced communities. In HSBM a random graph is constructed by placing hyperedges with higher density if all vertices of a hyperedge share the same binary label. By analyzing contraction of a non-Shannon (symmetric-KL) information measure, we prove that for $r=3,4$, weak recovery is impossible below the Kesten-Stigum threshold. Prior work Pal and Zhu (2021) established that weak recovery in HSBM is always possible above the Kesten-Stigum threshold. Consequently, there is no information-computation gap for these $r$, which (partially) resolves a conjecture of Angelini et al. (2015). To our knowledge this is the first impossibility result for HSBM weak recovery. As usual, we reduce the study of non-recovery of HSBM to the study of non-reconstruction in a related broadcasting on hypertrees (BOHT) model. While we show that BOHT's reconstruction threshold coincides with Kesten-Stigum for $r=3,4$, surprisingly, we demonstrate that for $r\ge 7$ reconstruction is possible also below the Kesten-Stigum. This shows an interesting phase transition in the parameter $r$, and suggests that for $r\ge 7$, there might be an information-computation gap for the HSBM. For $r=5,6$ and large degree we propose an approach for showing non-reconstruction below Kesten-Stigum threshold, suggesting that $r=7$ is the correct threshold for onset of the new phase. We admit that our analysis of the $r=4$ case depends on a numerically-verified inequality.

翻译：我们研究了具有两个平衡社区的$r$一致超图随机块模型（$r$-HSBM）中的弱恢复问题。在HSBM中，通过放置超边构建随机图：若超边的所有顶点共享相同的二元标签，则其密度更高。通过分析非香农（对称KL）信息测度的收缩性，我们证明对于$r=3,4$，弱恢复在Kesten-Stigum阈值以下是不可能的。先前Pal和Zhu（2021）的工作表明，HSBM中的弱恢复在Kesten-Stigum阈值以上总是可能的。因此，对于这些$r$值不存在信息-计算间隙，这（部分）解决了Angelini等人（2015）的一个猜想。据我们所知，这是HSBM弱恢复的首个不可能性结果。与常规方法一致，我们将HSBM的不可恢复性研究简化为超树广播（BOHT）模型中相关不可重构性的研究。虽然我们证明了对于$r=3,4$，BOHT的重构阈值与Kesten-Stigum一致，但令人惊讶的是，对于$r\ge 7$，重构在Kesten-Stigum阈值以下也是可能的。这揭示了参数$r$的相变现象，并表明对于$r\ge 7$，HSBM可能存在信息-计算间隙。对于$r=5,6$及大度情形，我们提出了一种在Kesten-Stigum阈值以下证明不可重构的方法，暗示$r=7$是新阶段起始的正确阈值。我们承认，对于$r=4$情况的分析依赖于数值验证的不等式。