# Structured Optimal Variational Inference for Dynamic Latent Space Models

arXiv: https://arxiv.org/abs/2209.15117 · HTML: https://mlsanity.com/paper/2209.15117/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 70 references that are in the map.
- First posted on arXiv 29 Sep 2022.

## Builds on (most cited first, 20 of 20)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Concentration of tempered posteriors and of their variational approximations](https://mlsanity.com/paper/1706.09293/) (2017)
- [Frequentist Consistency of Variational Bayes](https://mlsanity.com/paper/1705.03439/) (2017)
- [Convergence Rates of Variational Posterior Distributions](https://mlsanity.com/paper/1712.02519/) (2017)
- [Statistical clustering of temporal networks through a dynamic stochastic block model](https://mlsanity.com/paper/1506.07464/) (2015)
- [Latent Space Models for Dynamic Networks](https://mlsanity.com/paper/2005.08808/) (2020)
- [Multilinear tensor regression for longitudinal relational data](https://mlsanity.com/paper/1412.0048/) (2014)
- [Minimax Rates of Community Detection in Stochastic Block Models](https://mlsanity.com/paper/1507.05313/) (2015)
- [On Statistical Optimality of Variational Bayes](https://mlsanity.com/paper/1712.08983/) (2017)
- [Dynamic stochastic blockmodels for time-evolving social networks](https://mlsanity.com/paper/1403.0921/) (2014)
- [Optimal Estimation and Completion of Matrices with Biclustering Structures](https://mlsanity.com/paper/1512.00150/) (2015)
- [Nonparametric Bayes Modeling of Populations of Networks](https://mlsanity.com/paper/1406.7851/) (2014)
- [Conditionally conjugate mean-field variational Bayes for logistic models](https://mlsanity.com/paper/1711.06999/) (2017)
- [Joint Modelling of Multiple Network Views](https://mlsanity.com/paper/1301.3759/) (2013)
- [Bayesian Inference and Testing of Group Differences in Brain Networks](https://mlsanity.com/paper/1411.6506/) (2014)
- [Bayesian Learning of Dynamic Multilayer Networks](https://mlsanity.com/paper/1608.02209/) (2016)
- [Latent Space Approaches to Community Detection in Dynamic Networks](https://mlsanity.com/paper/2005.08276/) (2020)
- [An Eigenmodel for Dynamic Multilayer Networks](https://mlsanity.com/paper/2103.12831/) (2021)
- [Statistical optimality and stability of tangent transform algorithms in logit models](https://mlsanity.com/paper/2010.13039/) (2020)
- [Variational Inference for Latent Space Models for Dynamic Networks](https://mlsanity.com/paper/2105.14093/) (2021)

## Built on by (most cited first, 2 of 2)

- [Factorized Fusion Shrinkage for Dynamic Relational Data](https://mlsanity.com/paper/2210.00091/) (2022)
- [Tensor State Space-based Dynamic Multilayer Network Modeling](https://mlsanity.com/paper/2506.02413/) (2025)

## Abstract (arXiv)

We consider a latent space model for dynamic networks, where our objective is to estimate the pairwise inner products plus the intercept of the latent positions. To balance posterior inference and computational scalability, we consider a structured mean-field variational inference framework, where the time-dependent properties of the dynamic networks are exploited to facilitate computation and inference. Additionally, an easy-to-implement block coordinate ascent algorithm is developed with message-passing type updates in each block, whereas the complexity per iteration is linear with the number of nodes and time points. To certify the optimality, we demonstrate that the variational risk of the proposed variational inference approach attains the minimax optimal rate with only a logarithm factor under certain conditions. To this end, we first derive the minimax lower bound, which might be of independent interest. In addition, we show that the posterior under commonly adopted Gaussian random walk priors can achieve the minimax lower bound with only a logarithm factor. To the best of our knowledge, this is the first such a throughout theoretical analysis of Bayesian dynamic latent space models. Simulations and real data analysis demonstrate the efficacy of our methodology and the efficiency of our algorithm.