种植$k$ -denominant子超图问题的统计和计算阈值
Statistical and computational thresholds for the planted $k$-densest sub-hypergraph problem
摘要：
在这项工作中，我们考虑了恢复一个种植的$k$ -$d$ -均匀超图上最稠密的子超图的问题。这个基本问题出现在不同的背景下，例如，社区发现、平均情况复杂性和作为张量PCA问题的结构变体的神经科学应用。我们提供严格的\emph{information-theoretic}上限和下限的确切恢复阈值的最大似然估计，以及\emph{algorithmic}边界的基础上近似的消息传递算法。这个问题表现出一个典型的几何计算差距，观察类似的稀疏设置，扩大与问题的稀疏性。边界表明，信号结构影响的统计和计算相变的位置，已知的现有的张量PCA模型的边界不捕捉。这种效应是由于后一种模型处理的通用植入信号。
英文摘要：
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.
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