🤖 AI Summary
This work addresses the problem of detecting and exactly recovering hidden cliques when only the hypergraph adjacency matrix—defined as the co-occurrence counts of node pairs—is observed. To overcome information loss and entry dependencies induced by hyperedge projection, the authors propose a spectral-norm-based detection statistic and a polynomial-time spectral algorithm driven by the leading eigenvector. They establish the first rigorous theoretical guarantees under this observation model. By extending leave-one-out eigenvector analysis to the hypergraph setting, they achieve asymptotically optimal detection and recovery at the √n scale, explicitly characterizing the influence of the background hyperedge probability and demonstrating applicability even in sparse hypergraph regimes.
📝 Abstract
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. 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 the background hyperedge probability. We also extend both results to sparse regimes in which the background 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.