Consider the unsupervised classification problem in random hypergraphs under the non-uniform Hypergraph Stochastic Block Model (HSBM) with two equal-sized communities, where each edge appears independently with some probability depending only on the labels of its vertices. In this paper, the information-theoretic limits on the clustering accuracy and the strong consistency threshold are established, expressed in terms of the generalized Hellinger distance. Below the threshold, it is impossible to assign all vertices to their own communities, and the lower bound of the expected mismatch ratio is derived. On the other hand, the problem space is (sometimes) divided into two disjoint subspaces when above the threshold. When only the contracted adjacency matrix is given, with high probability, one-stage spectral algorithms succeed in assigning every vertex correctly in the subspace far away from the threshold but fail in the other one. Two subsequent refinement algorithms are proposed to improve the clustering accuracy, which attain the lowest possible mismatch ratio, previously derived from the information-theoretical perspective. The failure of spectral algorithms in the second subspace arises from the loss of information induced by tensor contraction. The origin of this loss and possible solutions to minimize the impact are presented. Moreover, different from uniform hypergraphs, strong consistency is achievable by aggregating information from all uniform layers, even if it is impossible when each layer is considered alone.

翻译：考虑非均匀超图随机块模型（HSBM）下具有两个等规模社区的随机超图中的无监督分类问题，其中每条边以某种概率独立出现，该概率仅取决于其顶点的标签。本文建立了聚类精度的信息论极限与强一致性阈值，并以广义Hellinger距离表示。在该阈值以下，无法将所有顶点分配至其所属社区，并推导了期望失配率的下界。另一方面，当超过该阈值时，问题空间（有时）被划分为两个不相交的子空间。当仅给定压缩邻接矩阵时，单阶段谱算法在远离阈值的子空间中能以高概率成功实现每个顶点的正确分配，但在另一子空间中会失败。本文提出了两种后续优化算法以提升聚类精度，这些算法达到了先前从信息论角度推导出的最低可能失配率。谱算法在第二子空间中的失败源于张量压缩引起的信息损失。本文分析了该损失的根源，并提出了最小化其影响的可能解决方案。此外，与均匀超图不同，通过聚合所有均匀层的信息可实现强一致性，即使单独考虑每一层时无法达成该目标。