# Multivariate Analysis for Multiple Network Data via Semi-Symmetric Tensor PCA

arXiv: https://arxiv.org/abs/2202.04719 · HTML: https://mlsanity.com/paper/2202.04719/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 101 references that are in the map.
- First posted on arXiv 9 Feb 2022.

## Builds on (most cited first, 30 of 38)

- [Multilayer Networks](https://mlsanity.com/paper/1309.7233/) (2013)
- [Contact patterns in a high school: a comparison between data collected using wearable sensors, contact diaries and friendship surveys](https://mlsanity.com/paper/1506.03645/) (2015)
- [The Noisy Power Method: A Meta Algorithm with Applications](https://mlsanity.com/paper/1311.2495/) (2013)
- [Tensor SVD: Statistical and Computational Limits](https://mlsanity.com/paper/1703.02724/) (2017)
- [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)
- [Practical sketching algorithms for low-rank matrix approximation](https://mlsanity.com/paper/1609.00048/) (2016)
- [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)
- [Guaranteed Non-Orthogonal Tensor Decomposition via Alternating Rank-1 Updates](https://mlsanity.com/paper/1402.5180/) (2014)
- [High-dimensional changepoint estimation via sparse projection](https://mlsanity.com/paper/1606.06246/) (2016)
- [Factor Models for High-Dimensional Tensor Time Series](https://mlsanity.com/paper/1905.07530/) (2019)
- [Understanding Deflation Process in Over-parametrized Tensor Decomposition](https://mlsanity.com/paper/2106.06573/) (2021)
- [Selecting the number of principal components: estimation of the true rank of a noisy matrix](https://mlsanity.com/paper/1410.8260/) (2014)
- [Provable Sparse Tensor Decomposition](https://mlsanity.com/paper/1502.01425/) (2015)
- [Spectral Learning on Matrices and Tensors](https://mlsanity.com/paper/2004.07984/) (2020)
- [Extension of PCA to Higher Order Data Structures: An Introduction to Tensors, Tensor Decompositions, and Tensor PCA](https://mlsanity.com/paper/1803.00704/) (2018)
- [Two-sample Hypothesis Testing for Inhomogeneous Random Graphs](https://mlsanity.com/paper/1707.00833/) (2017)
- [Univariate Mean Change Point Detection: Penalization, CUSUM and Optimality](https://mlsanity.com/paper/1810.09498/) (2018)
- [Network classification with applications to brain connectomics](https://mlsanity.com/paper/1701.08140/) (2017)
- [Joint Embedding of Graphs](https://mlsanity.com/paper/1703.03862/) (2017)
- [Spectral clustering in the dynamic stochastic block model](https://mlsanity.com/paper/1705.01204/) (2017)
- [Communication-Efficient Distributed SVD via Local Power Iterations](https://mlsanity.com/paper/2002.08014/) (2020)
- [Nonparametric Bayes Modeling of Populations of Networks](https://mlsanity.com/paper/1406.7851/) (2014)
- [Optimal High-order Tensor SVD via Tensor-Train Orthogonal Iteration](https://mlsanity.com/paper/2010.02482/) (2020)
- [Guaranteed Functional Tensor Singular Value Decomposition](https://mlsanity.com/paper/2108.04201/) (2021)
- [CP Factor Model for Dynamic Tensors](https://mlsanity.com/paper/2110.15517/) (2021)
- [On clustering network-valued data](https://mlsanity.com/paper/1606.02401/) (2016)
- [Optimal Change Point Detection and Localization in Sparse Dynamic Networks](https://mlsanity.com/paper/1809.09602/) (2018)

## Built on by (most cited first, 1 of 1)

- [A Subsampled Davis-Kahan Bound for Large-Scale Eigenspace Estimation](https://mlsanity.com/paper/2609.09211/) (2026)

## Abstract (arXiv)

Network data are commonly collected in a variety of applications, representing either directly measured or statistically inferred connections between features of interest. In an increasing number of domains, these networks are collected over time, such as interactions between users of a social media platform on different days, or across multiple subjects, such as in multi-subject studies of brain connectivity. When analyzing multiple large networks, dimensionality reduction techniques are often used to embed networks in a more tractable low-dimensional space. To this end, we develop a framework for principal components analysis (PCA) on collections of networks via a specialized tensor decomposition we term Semi-Symmetric Tensor PCA or SS-TPCA. We derive computationally efficient algorithms for computing our proposed SS-TPCA decomposition and establish statistical efficiency of our approach under a standard low-rank signal plus noise model. Remarkably, we show that SS-TPCA achieves the same estimation accuracy as classical matrix PCA, with error proportional to the square root of the number of vertices in the network and not the number of edges as might be expected. Our framework inherits many of the strengths of classical PCA and is suitable for a wide range of unsupervised learning tasks, including identifying principal networks, isolating meaningful changepoints or outlying observations, and for characterizing the "variability network" of the most varying edges. Finally, we demonstrate the effectiveness of our proposal on simulated data and on an example from empirical legal studies. The techniques used to establish our main consistency results are surprisingly straightforward and may find use in a variety of other network analysis problems.