# Learning Cartesian Product Graphs with Laplacian Constraints

arXiv: https://arxiv.org/abs/2402.08105 · HTML: https://mlsanity.com/paper/2402.08105/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 74 references that are in the map.
- First posted on arXiv 12 Feb 2024.

## Builds on (most cited first, 25 of 25)

- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [A Survey on Knowledge Graphs: Representation, Acquisition and Applications](https://mlsanity.com/paper/2002.00388/) (2020)
- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [How to learn a graph from smooth signals](https://mlsanity.com/paper/1601.02513/) (2016)
- [Connecting the Dots: Identifying Network Structure via Graph Signal Processing](https://mlsanity.com/paper/1810.13066/) (2018)
- [Network Topology Inference from Spectral Templates](https://mlsanity.com/paper/1608.03008/) (2016)
- [A Unified Framework for Structured Graph Learning via Spectral Constraints](https://mlsanity.com/paper/1904.09792/) (2019)
- [Learning heat diffusion graphs](https://mlsanity.com/paper/1611.01456/) (2016)
- [Learning Sparse Graphs Under Smoothness Prior](https://mlsanity.com/paper/1609.03448/) (2016)
- [Learning Graphs from Smooth and Graph-Stationary Signals with Hidden Variables](https://mlsanity.com/paper/2111.05588/) (2021)
- [Estimation of positive definite M-matrices and structure learning for attractive Gaussian Markov Random fields](https://mlsanity.com/paper/1404.6640/) (2014)
- [Joint Inference of Multiple Graphs from Matrix Polynomials](https://mlsanity.com/paper/2010.08120/) (2020)
- [Tensor Graphical Lasso (TeraLasso)](https://mlsanity.com/paper/1705.03983/) (2017)
- [Product Manifold Learning](https://mlsanity.com/paper/2010.09908/) (2020)
- [Identifying the Topology of Undirected Networks from Diffused Non-stationary Graph Signals](https://mlsanity.com/paper/1801.03862/) (2018)
- [Multi-way Graph Signal Processing on Tensors: Integrative analysis of irregular geometries](https://mlsanity.com/paper/2007.00041/) (2020)
- [Sparse Inverse Covariance Matrix Estimation Using Quadratic Approximation](https://mlsanity.com/paper/1306.3212/) (2013)
- [The Sylvester Graphical Lasso (SyGlasso)](https://mlsanity.com/paper/2002.00288/) (2020)
- [Product Graph Learning from Multi-domain Data with Sparsity and Rank Constraints](https://mlsanity.com/paper/2012.08090/) (2020)
- [Stationary time-vertex signal processing](https://mlsanity.com/paper/1611.00255/) (2016)
- [Learning Product Graphs Underlying Smooth Graph Signals](https://mlsanity.com/paper/2002.11277/) (2020)
- [Learning product graphs from multidomain signals](https://mlsanity.com/paper/1911.07411/) (2019)
- [Learning Product Graphs from Spectral Templates](https://mlsanity.com/paper/2211.02893/) (2022)
- [EiGLasso for Scalable Sparse Kronecker-Sum Inverse Covariance Estimation](https://mlsanity.com/paper/2105.09872/) (2021)
- [Graph Laplacian Learning with Exponential Family Noise](https://mlsanity.com/paper/2306.08201/) (2023)

## Built on by (most cited first, 3 of 3)

- [Learning Kronecker-Structured Graphs from Smooth Signals](https://mlsanity.com/paper/2505.09822/) (2025)
- [From topology learning to graph generation: A unifying perspective](https://mlsanity.com/paper/2609.02286/) (2026)
- [Graph Learning with Spectral Connectivity Priors for Scarce Data](https://mlsanity.com/paper/2609.27278/) (2026)

## Abstract (arXiv)

Graph Laplacian learning, also known as network topology inference, is a problem of great interest to multiple communities. In Gaussian graphical models (GM), graph learning amounts to endowing covariance selection with the Laplacian structure. In graph signal processing (GSP), it is essential to infer the unobserved graph from the outputs of a filtering system. In this paper, we study the problem of learning Cartesian product graphs under Laplacian constraints. The Cartesian graph product is a natural way for modeling higher-order conditional dependencies and is also the key for generalizing GSP to multi-way tensors. We establish statistical consistency for the penalized maximum likelihood estimation (MLE) of a Cartesian product Laplacian, and propose an efficient algorithm to solve the problem. We also extend our method for efficient joint graph learning and imputation in the presence of structural missing values. Experiments on synthetic and real-world datasets demonstrate that our method is superior to previous GSP and GM methods.