We consider the community detection problem in a sparse $q$-uniform hypergraph $G$, assuming that $G$ is generated according to the Hypergraph Stochastic Block Model (HSBM). We prove that a spectral method based on the non-backtracking operator for hypergraphs works with high probability down to the generalized Kesten-Stigum detection threshold conjectured by Angelini et al. (2015). We characterize the spectrum of the non-backtracking operator for the sparse HSBM and provide an efficient dimension reduction procedure using the Ihara-Bass formula for hypergraphs. As a result, community detection for the sparse HSBM on $n$ vertices can be reduced to an eigenvector problem of a $2n\times 2n$ non-normal matrix constructed from the adjacency matrix and the degree matrix of the hypergraph. To the best of our knowledge, this is the first provable and efficient spectral algorithm that achieves the conjectured threshold for HSBMs with $r$ blocks generated according to a general symmetric probability tensor.

翻译：我们考虑稀疏的$q$一致超图$G$中的社区检测问题，假设$G$是根据超图随机块模型（HSBM）生成的。我们证明，基于超图非回溯算子的谱方法能以高概率达到Angelini等人（2015）猜想的广义Kesten-Stigum检测阈值。我们刻画了稀疏HSBM中非回溯算子的谱特征，并利用超图的Ihara-Bass公式提出了一种高效的降维方法。由此，稀疏HSBM在$n$个顶点上的社区检测问题可简化为一个由超图邻接矩阵和度矩阵构造的$2n\times 2n$非正规矩阵的特征向量问题。据我们所知，这是首个可证明且高效的谱算法，能够达到由一般对称概率张量生成的包含$r$个块的HSBM的猜想检测阈值。