论文标题
种植的$ k $ - densest sub-hypergraph问题的统计和计算阈值
Statistical and computational thresholds for the planted $k$-densest sub-hypergraph problem
论文作者
论文摘要
在这项工作中，我们考虑了恢复的问题，该问题是$ d $均匀的超图上种植的$ k $ densest sub-hypergraph。这个基本问题出现在不同的情况下，例如社区检测，平均案例复杂性和神经科学应用，作为Tensor-PCA问题的结构变体。我们提供了紧密的\ emph {信息理论}的上限和下限，可通过最大样品估计器，以及\ emph {algorithMic}界限，以基于近似消息传递算法为基础。该问题表现出在类似的稀疏设置中观察到的典型统计到及时间隙，随着问题的稀疏性的增加而扩大。边界表明，信号结构会影响统计和计算相变的位置，该统计和计算相变的位置未捕获Tensor-PCA模型的已知现有边界。这种效果是由于后一个模型提出的通用植物信号所致。
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.
