In this work, we consider the problem of recovery a planted $k$-densest sub-hypergraph on $d$-uniform hypergraphs. This fundamental problem appears in different contexts, e.g., community detection, average-case complexity, and neuroscience applications as a structural variant of tensor-PCA problem. We provide tight \emph{information-theoretic} upper and lower bounds for the exact recovery threshold by the maximum-likelihood estimator, as well as \emph{algorithmic} bounds based on approximate message passing algorithms. The problem exhibits a typical statistical-to-computational gap observed in analogous sparse settings that widen with increasing sparsity of the problem. The bounds show that the signal structure impacts the location of the statistical and computational phase transition that the known existing bounds for the tensor-PCA model do not capture. This effect is due to the generic planted signal prior that this latter model addresses.

翻译：在这项工作中,我们考虑在美元单式高光仪上恢复一个人工种植的美元密度最高次频谱的问题。这个根本问题出现在不同的背景中,例如社区探测、平均情况复杂度和神经科学应用,作为Exronor-PCA问题的结构变体。我们通过最大类似估计值和基于近似电文传递算法的界限,为精确恢复阈值提供紧凑的 emph{信息-理论} 上限和下限。问题显示了在类似稀疏环境中观察到的典型的统计到计算差距,随着问题日益剧烈而扩大。这些界限表明信号结构影响统计和计算阶段过渡的位置,而已知的 Exward-PCA 模型的现有界限并没有被捕捉到。这一效果是由于在后一种模型的地址之前的通用定位信号造成的。