Block-wise MCEM for latent space hypergraphs

The Monte Carlo Expectation-Maximization (MCEM) framework is a widely used approach for maximum likelihood estimation in latent variable models.A notable example in network science is the MCEM algorithm of \cite{hoff_2002}, developed for network data with latent covariates.In this section, we propose a block-wise MCEM algorithm tailored to our latent space model for hypergraphs. Our algorithm incorporates two key features that enhance both computational efficiency and estimation accuracy.First, we introduce a fast likelihood evaluation method for hypergraph data that reduces the computational cost by a factor of approximately $\frac{1}{m}$, where $m$ denotes the number of observed hyperedges.Second, we adopt a component-wise Metropolis-within-Gibbs sampler \cite{ascolani_2024} in the simulated E-step of MCEM.Unlike the full scan approach of \cite{hoff_2002}, which updates the entire latent covariate matrix in each iteration,our method updates only a randomly selected subblock of the latent covariate matrix, significantly improving sampling efficiency and estimation accuracy.Hypergraph Data in Section 5 is also provided in ./data/.