# Latent Space Model for Higher-order Networks and Generalized Tensor Decomposition

arXiv: https://arxiv.org/abs/2106.16042 · HTML: https://mlsanity.com/paper/2106.16042/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 75 references that are in the map.
- First posted on arXiv 30 Jun 2021.

## Builds on (most cited first, 30 of 33)

- [Consistency of spectral clustering in stochastic block models](https://mlsanity.com/paper/1312.2050/) (2013)
- [Higher-order organization of complex networks](https://mlsanity.com/paper/1612.08447/) (2016)
- [Multilayer Networks](https://mlsanity.com/paper/1309.7233/) (2013)
- [Statistical inference on random dot product graphs: a survey](https://mlsanity.com/paper/1709.05454/) (2017)
- [What are higher-order networks?](https://mlsanity.com/paper/2104.11329/) (2021)
- [Tensor SVD: Statistical and Computational Limits](https://mlsanity.com/paper/1703.02724/) (2017)
- [Rate-Optimal Perturbation Bounds for Singular Subspaces with Applications to High-Dimensional Statistics](https://mlsanity.com/paper/1605.00353/) (2016)
- [Inference for multiple heterogeneous networks with a common invariant subspace](https://mlsanity.com/paper/1906.10026/) (2019)
- [An Optimal Statistical and Computational Framework for Generalized Tensor Estimation](https://mlsanity.com/paper/2002.11255/) (2020)
- [Statistical clustering of temporal networks through a dynamic stochastic block model](https://mlsanity.com/paper/1506.07464/) (2015)
- [Community Detection on Mixture Multi-layer Networks via Regularized Tensor Decomposition](https://mlsanity.com/paper/2002.04457/) (2020)
- [Latent Space Models for Dynamic Networks](https://mlsanity.com/paper/2005.08808/) (2020)
- [On Polynomial Time Methods for Exact Low Rank Tensor Completion](https://mlsanity.com/paper/1702.06980/) (2017)
- [Spectral and matrix factorization methods for consistent community detection in multi-layer networks](https://mlsanity.com/paper/1704.07353/) (2017)
- [Community Detection for Hypergraph Networks via Regularized Tensor Power Iteration](https://mlsanity.com/paper/1909.06503/) (2019)
- [Generalized Low-rank plus Sparse Tensor Estimation by Fast Riemannian Optimization](https://mlsanity.com/paper/2103.08895/) (2021)
- [Consistency of Spectral Hypergraph Partitioning under Planted Partition Model](https://mlsanity.com/paper/1505.01582/) (2015)
- [Coauthorship and Citation Networks for Statisticians](https://mlsanity.com/paper/1410.2840/) (2014)
- [Detecting Overlapping Communities in Networks Using Spectral Methods](https://mlsanity.com/paper/1412.3432/) (2014)
- [Stochastic Block Model for Hypergraphs: Statistical limits and a semidefinite programming approach](https://mlsanity.com/paper/1807.02884/) (2018)
- [Learning from Binary Multiway Data: Probabilistic Tensor Decomposition and its Statistical Optimality](https://mlsanity.com/paper/1811.05076/) (2018)
- [Spectral clustering in the dynamic stochastic block model](https://mlsanity.com/paper/1705.01204/) (2017)
- [Community Detection in General Hypergraph via Graph Embedding](https://mlsanity.com/paper/2103.15035/) (2021)
- [Fast Change Point Detection on Dynamic Social Networks](https://mlsanity.com/paper/1705.07325/) (2017)
- [Community detection in the sparse hypergraph stochastic block model](https://mlsanity.com/paper/1904.05981/) (2019)
- [Community Detection in Networks with Node Features](https://mlsanity.com/paper/1509.01173/) (2015)
- [Change Point Estimation in a Dynamic Stochastic Block Model](https://mlsanity.com/paper/1812.03090/) (2018)
- [Optimal Change Point Detection and Localization in Sparse Dynamic Networks](https://mlsanity.com/paper/1809.09602/) (2018)
- [Link prediction for partially observed networks](https://mlsanity.com/paper/1301.7047/) (2013)
- [Stochastic Block Transition Models for Dynamic Networks](https://mlsanity.com/paper/1411.5404/) (2014)

## Built on by (most cited first, 13 of 13)

- [Optimal Estimation and Computational Limit of Low-rank Gaussian Mixtures](https://mlsanity.com/paper/2201.09040/) (2022)
- [Optimal Clustering by Lloyd Algorithm for Low-Rank Mixture Model](https://mlsanity.com/paper/2207.04600/) (2022)
- [Computationally Efficient and Statistically Optimal Robust High-Dimensional Linear Regression](https://mlsanity.com/paper/2305.06199/) (2023)
- [Online Tensor Learning: Computational and Statistical Trade-offs, Adaptivity and Optimal Regret](https://mlsanity.com/paper/2306.03372/) (2023)
- [Optimal Clustering of Discrete Mixtures: Binomial, Poisson, Block Models, and Multi-layer Networks](https://mlsanity.com/paper/2311.15598/) (2023)
- [rMultiNet: An R Package For Multilayer Networks Analysis](https://mlsanity.com/paper/2302.04437/) (2023)
- [Tensor Methods in High Dimensional Data Analysis: Opportunities and Challenges](https://mlsanity.com/paper/2405.18412/) (2024)
- [Statistical Inference in Tensor Completion: Optimal Uncertainty Quantification and Statistical-to-Computational Gaps](https://mlsanity.com/paper/2410.11225/) (2024)
- [Efficient Estimation for Longitudinal Networks via Adaptive Merging](https://mlsanity.com/paper/2211.07866/) (2022)
- [Bidirectional Brain Image Translation using Transfer Learning from Generic Pre-trained Models](https://mlsanity.com/paper/2501.12488/) (2025)
- [A Transfer Learning Framework for Multilayer Networks via Model Averaging](https://mlsanity.com/paper/2506.12455/) (2025)
- [Perfect Clustering in Very Sparse Diverse Multiplex Networks](https://mlsanity.com/paper/2507.19423/) (2025)
- [T-GINEE: A Tensor-Based Multilayer Graph Representation Learning](https://mlsanity.com/paper/2605.28300/) (2026)

## Abstract (arXiv)

We introduce a unified framework, formulated as general latent space models, to study complex higher-order network interactions among multiple entities. Our framework covers several popular models in recent network analysis literature, including mixture multi-layer latent space model and hypergraph latent space model. We formulate the relationship between the latent positions and the observed data via a generalized multilinear kernel as the link function. While our model enjoys decent generality, its maximum likelihood parameter estimation is also convenient via a generalized tensor decomposition procedure.We propose a novel algorithm using projected gradient descent on Grassmannians. We also develop original theoretical guarantees for our algorithm. First, we show its linear convergence under mild conditions. Second, we establish finite-sample statistical error rates of latent position estimation, determined by the signal strength, degrees of freedom and the smoothness of link function, for both general and specific latent space models. We demonstrate the effectiveness of our method on synthetic data. We also showcase the merit of our method on two real-world datasets that are conventionally described by different specific models in producing meaningful and interpretable parameter estimations and accurate link prediction. We demonstrate the effectiveness of our method on synthetic data. We also showcase the merit of our method on two real-world datasets that are conventionally described by different specific models in producing meaningful and interpretable parameter estimations and accurate link prediction.