Bridging Theory and Practice: Statistical Inference for Latent Space Models of Networks

TL;DR

This paper develops a unified framework for statistical inference in latent space models of networks, bridging the gap between theoretical guarantees and practical algorithms.

Contribution

It relaxes existing theoretical constraints and establishes explicit connections between practical algorithms and maximum likelihood estimators.

Findings

Broadened applicability of asymptotic theory by relaxing spectral-multiplicity constraints.

Developed adaptive criteria to connect algorithm outputs with MLE without unknown parameters.

Provided theoretical guarantees for gradient-based algorithms in network latent space models.

Abstract

Latent space models have been widely adopted in modeling network data. Developing statistical inference for estimated model parameters enables quantifying associated uncertainty and is pivotal for downstream tasks. Despite recent progress on statistical inference of maximum likelihood estimation, crucial gaps remain between asymptotic theoretical guarantees and practical use. Specifically, how are the oracle maximum likelihood estimators related to the solutions produced by algorithms in practice? Can rigorous guarantees be established for existing algorithms without unnecessary restrictions? To address these fundamental questions, we develop a unified analytical framework that bridges theory and practice of statistical inference for latent space models. First, for the maximum likelihood estimation, we relax the spectral-multiplicity constraint in the existing asymptotic theory to broaden the applicability. Second, we overcome the dependence on unknown true parameters in prior algorithmic analyses by developing novel adaptive criteria and theoretical tools. For the widely used algorithm based on the projected gradient descent and the singular value thresholding, we explicitly connect their outputs to the maximum likelihood estimator without relying on unknown information. Our results provide a solid foundation for practically useful and statistically principled statistical inference in network analysis.