# Learning Gaussian Graphical Models under Total Positivity via Spectral Graph Sparsification

arXiv: https://arxiv.org/abs/2605.17154 · HTML: https://mlsanity.com/paper/2605.17154/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 22 references that are in the map.
- First posted on arXiv 16 May 2026.

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

- [A Unifying Framework for Spectrum-Preserving Graph Sparsification and Coarsening](https://mlsanity.com/paper/1902.09702/) (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)
- [Covariance Matrix Estimation under Total Positivity for Portfolio Selection](https://mlsanity.com/paper/1909.04222/) (2019)
- [Fast Projected Newton-like Method for Precision Matrix Estimation under Total Positivity](https://mlsanity.com/paper/2112.01939/) (2021)
- [Learning Large-Scale MTP2 Gaussian Graphical Models via Bridge-Block Decomposition](https://mlsanity.com/paper/2309.13405/) (2023)
- [Total positivity in exponential families with application to binary variables](https://mlsanity.com/paper/1905.00516/) (2019)
- [Locally associated graphical models and mixed convex exponential families](https://mlsanity.com/paper/2008.04688/) (2020)
- [Improved large-scale graph learning through ridge spectral sparsification](https://mlsanity.com/paper/2604.20078/) (2026)

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

- [Spectral Sparsification of Laplacian-Constrained Gaussian and H"usler-Reiss Graphical Models](https://mlsanity.com/paper/2606.16681/) (2026)

## Abstract (arXiv)

Many practical data analysis tasks reduce to learning, from observed samples, how a collection of variables depend on each other. A widely used approach is to fit a Gaussian graphical model, which represents the dependence structure as a graph connecting the variables. In a number of important applications, such as financial returns, gene co-expression, and climate or network analysis, the dependencies tend to be positive: variables move together rather than offset each other. Encoding this positivity through the constraint of multivariate total positivity of order two (MTP2) yields an attractive estimator that produces accurate fits with no tuning required. The resulting graphs are, however, typically much denser than the underlying ground-truth model, which makes them hard to interpret and slow to use in any downstream task that operates on the graph. In this work, we propose a novel highly-scalable approach for learning Gaussian graphical models from data using spectral sparsification; we call it Spectral-MTP2. Spectral graph sparsification is a fundamental method which aims to preserve meaningful properties of a dense graph with a sparser subgraph. We theoretically and empirically investigate and validate our method, and show that learning Gaussian Graphical Models under MTP2 using spectral sparsification preserves MTP2 and approximates well the original model in terms of Kullback-Leibler divergence and Gaussian log-likelihood. In simulations and applications to equity returns and gene expression, we observe that Spectral-MTP2 retains most of the fit quality of the denser MTP2 baseline, while producing substantially sparser and more interpretable graphs.