We study the weak recovery problem on 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, and weak recovery asks to recover a non-trivial fraction of the labels. We introduce a multi-terminal version of strong data processing inequalities (SDPIs), which we call the multi-terminal SDPI, and use it to prove a variety of impossibility results for weak recovery. In particular, we prove that weak recovery is impossible below the Kesten-Stigum (KS) threshold if $r=3,4$, or a strength parameter $\lambda$ is at least $\frac 15$. Prior work Pal and Zhu (2021) established that weak recovery in HSBM is always possible above the KS threshold. Consequently, there is no information-computation gap for these cases, 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 KS for $r=3,4$, surprisingly, we demonstrate that for $r\ge 7$ reconstruction is possible also below KS. 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 KS, suggesting that $r=7$ is the correct threshold for onset of the new phase.

翻译：我们研究了具有两个平衡社区的$r$-均匀超图随机块模型（$r$-HSBM）上的弱恢复问题。在HSBM中，若超边的所有顶点共享相同的二元标签，则以更高密度放置超边构建随机图；弱恢复要求恢复非平凡比例的标签。我们引入了一种多终端版本的强数据处理不等式（SDPI），称为多终端SDPI，并利用它证明了弱恢复问题的多种不可能性结果。特别地，我们证明：当$r=3,4$或强度参数$\lambda \geq \frac{1}{5}$时，弱恢复在Kesten-Stigum（KS）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KS阈值以上总是可能的。因此，在这些情形下不存在信息-计算差距，（部分）解决了Angelini等人（2015）提出的猜想。据我们所知，这是关于HSBM弱恢复的首个不可能性结果。如同常见方法，我们将HSBM的不可恢复性问题简化为超树上的相关广播模型（BOHT）的不可重建问题。尽管我们证明当$r=3,4$时BOHT的重建阈值与KS阈值一致，但令人惊讶的是，我们表明当$r\geq 7$时重建在KS阈值以下也可能发生。这揭示了参数$r$的有趣相变，并暗示当$r\geq 7$时，HSBM可能存在信息-计算差距。对于$r=5,6$且度数较大的情形，我们提出了一种在KS阈值以下展示不可重建的方法，表明$r=7$是新相位起始的正确阈值。