Projected Tensor Power Method for Hypergraph Community Recovery

This paper investigates the problem of exact community recovery in the symmetric d-uniform (d ≥ 2) hypergraph stochastic block model (d-HSBM). In this model, a d-uniform hypergraph with n nodes is generated by first partitioning the n nodes into K ≥ 2 equal-sized disjoint communities and then generating hyperedges with a probability that depends on the community memberships of d nodes. Despite the non-convex and discrete nature of the maximum likelihood estimation problem, we develop a simple yet efficient iterative method, called the projected tensor power method, to tackle it. As long as the initialization satisfies a partial recovery condition in the logarithmic degree regime of the problem, we show that our proposed method can exactly recover the hidden community structure down to the information-theoretic limit with high probability. Moreover, our proposed method exhibits a competitive time complexity of O(n log² n/log log n) when the aforementioned initialization condition is met. We also conduct numerical experiments to validate our theoretical findings.