Spectral estimators have been broadly applied to statistical network analysis, but they do not incorporate the likelihood information of the network sampling model. This paper proposes a novel surrogate likelihood function for statistical inference of a class of popular network models referred to as random dot product graphs. In contrast to the structurally complicated exact likelihood function, the surrogate likelihood function has a separable structure and is log-concave yet approximates the exact likelihood function well. From the frequentist perspective, we study the maximum surrogate likelihood estimator and establish the accompanying theory. We show its existence, uniqueness, large sample properties, and that it improves upon the baseline spectral estimator with a smaller sum of squared errors. Furthermore, we derive the second-order bias of the proposed estimator and gain insight into why it outperforms some of the existing estimators. A computationally convenient stochastic gradient descent algorithm is designed to find the maximum surrogate likelihood estimator in practice. From the Bayesian perspective, we establish the Bernstein--von Mises theorem of the posterior distribution with the surrogate likelihood function and show that the resulting credible sets have the correct frequentist coverage. The empirical performance of the proposed surrogate-likelihood-based methods is validated through the analyses of simulation examples and two real-world data sets.

We study the sparse high-dimensional Gaussian mixture model when the number of clusters is allowed to grow with the sample size. A minimax lower bound for parameter estimation is established, and we show that a constrained maximum likelihood estimator achieves the minimax lower bound. However, this optimization-based estimator is computationally intractable because the objective function is highly nonconvex and the feasible set involves discrete structures. To address the computational challenge, we propose a computationally tractable Bayesian approach to estimate high-dimensional Gaussian mixtures whose cluster centers exhibit sparsity using a continuous spike-and-slab prior. We further prove that the posterior contraction rate of the proposed Bayesian method is minimax optimal. The mis- clustering rate is obtained as a by-product using tools from matrix perturbation theory. The proposed Bayesian sparse Gaussian mixture model does not require pre-specifying the number of clusters, which can be adaptively estimated. The validity and usefulness of the proposed method is demonstrated through simulation studies and the analysis of a real-world single-cell RNA sequencing data set.

In this paper, we propose to estimate the invariant subspace across heterogeneous multiple networks using a novel bias-corrected joint spectral embedding algorithm. The proposed algorithm recursively calibrates the diagonal bias of the sum of squared network adjacency matrices by leveraging the closed-form bias formula and iteratively updates the subspace estimator using the most recent estimated bias. Correspondingly, we establish a complete recipe for the entrywise subspace estimation theory for the proposed algorithm, including a sharp entrywise subspace perturbation bound and the entrywise eigenvector central limit theorem. Leveraging these results, we settle two multiple network inference problems: the exact community detection in multilayer stochastic block models and the hypothesis testing of the equality of membership profiles in multilayer mixed membership models. Our proof relies on delicate leave-one-out and leave-two-out analyses that are specifically tailored to block-wise symmetric random matrices and a martingale argument that is of fundamental interest for the entrywise eigenvector central limit theorem.