# Spectral Sparsification of Laplacian-Constrained Gaussian and H"usler-Reiss Graphical Models

arXiv: https://arxiv.org/abs/2606.16681 · HTML: https://mlsanity.com/paper/2606.16681/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 26 references that are in the map.
- First posted on arXiv 15 Jun 2026.

## Builds on (most cited first, 11 of 11)

- [A Unified Framework for Structured Graph Learning via Spectral Constraints](https://mlsanity.com/paper/1904.09792/) (2019)
- [Approximate Gaussian Elimination for Laplacians: Fast, Sparse, and Simple](https://mlsanity.com/paper/1605.02353/) (2016)
- [Estimation of positive definite M-matrices and structure learning for attractive Gaussian Markov Random fields](https://mlsanity.com/paper/1404.6640/) (2014)
- [Maximum likelihood estimation in Gaussian models under total positivity](https://mlsanity.com/paper/1702.04031/) (2017)
- [Learning High-dimensional Gaussian Graphical Models under Total Positivity without Adjustment of Tuning Parameters](https://mlsanity.com/paper/1906.05159/) (2019)
- [Fast Projected Newton-like Method for Precision Matrix Estimation under Total Positivity](https://mlsanity.com/paper/2112.01939/) (2021)
- [Structure learning for extremal tree models](https://mlsanity.com/paper/2012.06179/) (2020)
- [Statistical Inference for H"usler-Reiss Graphical Models Through Matrix Completions](https://mlsanity.com/paper/2210.14292/) (2022)
- [Total positivity in multivariate extremes](https://mlsanity.com/paper/2112.14727/) (2021)
- [Improved large-scale graph learning through ridge spectral sparsification](https://mlsanity.com/paper/2604.20078/) (2026)
- [Learning Gaussian Graphical Models under Total Positivity via Spectral Graph Sparsification](https://mlsanity.com/paper/2605.17154/) (2026)

## Abstract (arXiv)

Graph Laplacians encode graph structures in matrix form, and thus facilitate the application of linear algebra to graph theory. In statistics, two related families of probabilistic graphical models can be parameterized by graph Laplacians. The first one is the Laplacian-constrained Gaussian graphical model (LCGGM), which imposes that the (pseudo-)inverse covariance matrix of a Gaussian random vector is a Laplacian matrix. Applications include graph signal processing and network topology learning. The second one is the H"usler-Reiss graphical model, which is considered as an extremal analog of the Gaussian graphical model, and can be used in extremal dependence modeling of floods, heatwaves, and financial losses. For both models, the restriction to positive edge weights in the graph Laplacian gives rise to an approach for graph structure learning that does not require tuning parameters. While these approaches yield a strong model fit in many settings, the resulting graph estimates are typically much denser than the underlying ground truth, limiting interpretability and scalability. In order to improve the accuracy of Laplacian-constrained graph learning, we propose to use spectral graph sparsification as a post-estimation operation. To do so, we replace the original Laplacian estimate by a sparser Laplacian that is spectrally close, and re-fit the model on the resulting graph. We refer to the two resulting methods as Spectral-LCGGM and Spectral-HR. We investigate the properties of the proposed estimators and show several theoretical results on their performance. Furthermore, we demonstrate that the newly proposed methods perform well by running simulations on ErdHos-R'enyi and stochastic block model graphs, and we also showcase their applications to real data.