# Tensor Graphical Lasso (TeraLasso)

arXiv: https://arxiv.org/abs/1705.03983 · HTML: https://mlsanity.com/paper/1705.03983/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 May 2017.

## Built on by (most cited first, 16 of 16)

- [Methods for Recovering Conditional Independence Graphs: A Survey](https://mlsanity.com/paper/2211.06829/) (2022)
- [EiGLasso for Scalable Sparse Kronecker-Sum Inverse Covariance Estimation](https://mlsanity.com/paper/2105.09872/) (2021)
- [SG-PALM: a Fast Physically Interpretable Tensor Graphical Model](https://mlsanity.com/paper/2105.12271/) (2021)
- [Scalable Bigraphical Lasso: Two-way Sparse Network Inference for Count Data](https://mlsanity.com/paper/2203.07912/) (2022)
- [Neural Graph Revealers](https://mlsanity.com/paper/2302.13582/) (2023)
- [Learning Cartesian Product Graphs with Laplacian Constraints](https://mlsanity.com/paper/2402.08105/) (2024)
- [Learning Sparse High-Dimensional Matrix-Valued Graphical Models From Dependent Data](https://mlsanity.com/paper/2404.19073/) (2024)
- [Jointly Modeling and Clustering Tensors in High Dimensions](https://mlsanity.com/paper/2104.07773/) (2021)
- [Online Graph Topology Learning from Matrix-valued Time Series](https://mlsanity.com/paper/2107.08020/) (2021)
- [Multiway Ensemble Kalman Filter](https://mlsanity.com/paper/2112.04322/) (2021)
- [Mode-wise Principal Subspace Pursuit and Matrix Spiked Covariance Model](https://mlsanity.com/paper/2307.00575/) (2023)
- [DNNLasso: Scalable Graph Learning for Matrix-Variate Data](https://mlsanity.com/paper/2403.02608/) (2024)
- [Knowledge Propagation over Conditional Independence Graphs](https://mlsanity.com/paper/2308.05857/) (2023)
- [Making Multi-Axis Gaussian Graphical Models Scalable to Millions of Cells](https://mlsanity.com/paper/2407.19892/) (2024)
- [Graphical Modelling without Independence Assumptions for Uncentered Data](https://mlsanity.com/paper/2408.02393/) (2024)
- [Learning Kronecker-Structured Graphs from Smooth Signals](https://mlsanity.com/paper/2505.09822/) (2025)

## Abstract (arXiv)

This paper introduces a multi-way tensor generalization of the Bigraphical Lasso (BiGLasso), which uses a two-way sparse Kronecker-sum multivariate-normal model for the precision matrix to parsimoniously model conditional dependence relationships of matrix-variate data based on the Cartesian product of graphs. We call this generalization the bf Tensor gbf raphical Lasso (TeraLasso). We demonstrate using theory and examples that the TeraLasso model can be accurately and scalably estimated from very limited data samples of high dimensional variables with multiway coordinates such as space, time and replicates. Statistical consistency and statistical rates of convergence are established for both the BiGLasso and TeraLasso estimators of the precision matrix and estimators of its support (non-sparsity) set, respectively. We propose a scalable composite gradient descent algorithm and analyze the computational convergence rate, showing that the composite gradient descent algorithm is guaranteed to converge at a geometric rate to the global minimizer of the TeraLasso objective function. Finally, we illustrate the TeraLasso using both simulation and experimental data from a meteorological dataset, showing that we can accurately estimate precision matrices and recover meaningful conditional dependency graphs from high dimensional complex datasets.