# Laplacian Constrained Precision Matrix Estimation: Existence and High Dimensional Consistency

arXiv: https://arxiv.org/abs/2111.00590 · HTML: https://mlsanity.com/paper/2111.00590/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 51 references that are in the map.
- First posted on arXiv 31 Oct 2021.

## Builds on (most cited first, 16 of 16)

- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [Learning graphs from data: A signal representation perspective](https://mlsanity.com/paper/1806.00848/) (2018)
- [Connecting the Dots: Identifying Network Structure via Graph Signal Processing](https://mlsanity.com/paper/1810.13066/) (2018)
- [Graph signal processing for machine learning: A review and new perspectives](https://mlsanity.com/paper/2007.16061/) (2020)
- [A Unified Framework for Structured Graph Learning via Spectral Constraints](https://mlsanity.com/paper/1904.09792/) (2019)
- [Graph Signal Processing for Geometric Data and Beyond: Theory and Applications](https://mlsanity.com/paper/2008.01918/) (2020)
- [Estimation of positive definite M-matrices and structure learning for attractive Gaussian Markov Random fields](https://mlsanity.com/paper/1404.6640/) (2014)
- [Graph Learning from Filtered Signals: Graph System and Diffusion Kernel Identification](https://mlsanity.com/paper/1803.02553/) (2018)
- [Maximum likelihood estimation in Gaussian models under total positivity](https://mlsanity.com/paper/1702.04031/) (2017)
- [Learning Graphs with Monotone Topology Properties and Multiple Connected Components](https://mlsanity.com/paper/1705.10934/) (2017)
- [Covariance Matrix Estimation under Total Positivity for Portfolio Selection](https://mlsanity.com/paper/1909.04222/) (2019)
- [Confidence intervals for high-dimensional inverse covariance estimation](https://mlsanity.com/paper/1403.6752/) (2014)
- [An efficient algorithm for graph Laplacian optimization based on effective resistances](https://mlsanity.com/paper/2004.08451/) (2020)
- [Topology design for stochastically-forced consensus networks](https://mlsanity.com/paper/1506.03437/) (2015)
- [Learning Graph Laplacian with MCP](https://mlsanity.com/paper/2010.11559/) (2020)

## Built on by (most cited first, 4 of 4)

- [Covariance Density Neural Networks](https://mlsanity.com/paper/2505.11139/) (2025)
- [Joint Graph and Vertex Importance Learning](https://mlsanity.com/paper/2303.08552/) (2023)
- [Variable-size Symmetry-based Graph Fourier Transforms for image compression](https://mlsanity.com/paper/2411.15824/) (2024)
- [Co-Hub Node Based Multiview Graph Learning with Theoretical Guarantees](https://mlsanity.com/paper/2512.12435/) (2025)

## Abstract (arXiv)

This paper considers the problem of estimating high dimensional Laplacian constrained precision matrices by minimizing Stein's loss. We obtain a necessary and sufficient condition for existence of this estimator, that consists on checking whether a certain data dependent graph is connected. We also prove consistency in the high dimensional setting under the symmetrized Stein loss. We show that the error rate does not depend on the graph sparsity, or other type of structure, and that Laplacian constraints are sufficient for high dimensional consistency. Our proofs exploit properties of graph Laplacians, the matrix tree theorem, and a characterization of the proposed estimator based on effective graph resistances. We validate our theoretical claims with numerical experiments.