Hypergraph data are often projected onto a weighted graph by constructing an adjacency matrix whose $(i,j)$ entry counts the number of hyperedges containing both nodes $i$ and $j$. This reduction is computationally convenient, but it can lose information: distinct hypergraphs may induce the same matrix, and the matrix entries are generally dependent because each hyperedge contributes to multiple pairs. We study the planted clique problem under this matrix-only observation model. For detection, we show that a spectral norm test is asymptotically powerful at the $\sqrt{n}$ scale, with explicit dependence on the background hyperedge probability $p$. For recovery, we analyze a polynomial-time spectral method based on the leading eigenvector and prove exact recovery at the canonical $\sqrt{n}$ scale, again with explicit dependence on $p$. We also extend both results to sparse regimes in which the hyperedge probability may depend on \(n\). Our analysis adapts a leave--one--out eigenvector framework to this setting. These results provide rigorous detection and recovery guarantees when only the adjacency matrix is observed.

翻译：超图数据通常通过构造一个加权图进行投影，其邻接矩阵中$(i,j)$项统计同时包含节点$i$和$j$的超边数量。这种降维计算便捷，但可能损失信息：不同的超图可能生成相同的矩阵，且矩阵元素通常存在依赖性，因为每条超边会贡献到多个节点对。我们在此类仅观测矩阵的模型下研究植入团问题。在检测方面，我们证明谱范数检验在$\sqrt{n}$尺度下具有渐近有效性，其性能显式依赖于背景超边概率$p$。在恢复方面，我们分析了一种基于主特征向量的多项式时间谱方法，并证明在标准$\sqrt{n}$尺度下可实现精确恢复，该结果同样显式依赖于$p$。我们还将两种结论推广至超边概率可能随\(n\)变化的稀疏场景。本研究针对该场景适配了留一特征向量分析框架。这些结果为仅观测邻接矩阵时的检测与恢复提供了严格的保证。