We consider the problem of inferring a matching hidden in a weighted random $k$-hypergraph. We assume that the hyperedges' weights are random and distributed according to two different densities conditioning on the fact that they belong to the hidden matching, or not. We show that, for $k>2$ and in the large graph size limit, an algorithmic first order transition in the signal strength separates a regime in which a complete recovery of the hidden matching is feasible from a regime in which partial recovery is possible. This is in contrast to the $k=2$ case where the transition is known to be continuous. Finally, we consider the case of graphs presenting a mixture of edges and $3$-hyperedges, interpolating between the $k=2$ and the $k=3$ cases, and we study how the transition changes from continuous to first order by tuning the relative amount of edges and hyperedges.
翻译:我们考虑了在加权随机速率($k$-hyperraphy)中推断隐藏的匹配的问题。我们假设,高端的重量是随机的,根据两种不同的密度分布,取决于它们是否属于隐藏的匹配。我们表明,对于$>2美元和大图形尺寸的极限,信号强度的算法第一顺序过渡将一个制度区分开来,在这种制度中,完全恢复隐藏的匹配是可行的,在这种制度中,可以部分恢复部分匹配。这与知道转型持续发生的美元=2美元的情况相反。最后,我们考虑了显示边缘和3美元-超级屏障混合的图表案例,在2美元和3美元之间相互推算,我们研究如何通过调整边缘和高屏障的相对数量从连续向第一顺序的转变。