Planted matching problems on random hypergraphs

Abstract

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.

arXiv comment13 pages, 12 figures

References in corpus (16)

• Message Passing Algorithms for Compressed Sensing
• Networks beyond pairwise interactions: structure and dynamics
• The dynamics of message passing on dense graphs, with applications to compressed sensing
• Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications
• Gibbs States and the Set of Solutions of Random Constraint Satisfaction Problems
• Statistical physics of inference: Thresholds and algorithms
• Maximum Weight Matching via Max-Product Belief Propagation
• Hiding Quiet Solutions in Random Constraint Satisfaction Problems
• Belief-Propagation for Weighted b-Matchings on Arbitrary Graphs and its Relation to Linear Programs with Integer Solutions
• Inference in particle tracking experiments by passing messages between images
• Typology of phase transitions in Bayesian inference problems
• Random multi-index matching problems
• Marvels and Pitfalls of the Langevin Algorithm in Noisy High-dimensional Inference
• A frozen glass phase in the multi-index matching problem
• Recovery thresholds in the sparse planted matching problem
• The planted $k$-factor problem