# Adaptive Estimation of Graphical Models under Total Positivity

arXiv: https://arxiv.org/abs/2210.15471 · HTML: https://mlsanity.com/paper/2210.15471/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 60 references that are in the map.
- First posted on arXiv 27 Oct 2022.

## Builds on (most cited first, 17 of 17)

- [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)
- [Regularized M-estimators with nonconvexity: Statistical and algorithmic theory for local optima](https://mlsanity.com/paper/1305.2436/) (2013)
- [A Unified Framework for Structured Graph Learning via Spectral Constraints](https://mlsanity.com/paper/1904.09792/) (2019)
- [Support recovery without incoherence: A case for nonconvex regularization](https://mlsanity.com/paper/1412.5632/) (2014)
- [Estimation of positive definite M-matrices and structure learning for attractive Gaussian Markov Random fields](https://mlsanity.com/paper/1404.6640/) (2014)
- [Algorithms for Learning Graphs in Financial Markets](https://mlsanity.com/paper/2012.15410/) (2020)
- [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)
- [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)
- [Total positivity in exponential families with application to binary variables](https://mlsanity.com/paper/1905.00516/) (2019)
- [Does the ℓ1-norm Learn a Sparse Graph under Laplacian Constrained Graphical Models?](https://mlsanity.com/paper/2006.14925/) (2020)
- [Maximum Likelihood Estimation for Brownian Motion Tree Models Based on One Sample](https://mlsanity.com/paper/2112.00816/) (2021)
- [Total positivity in multivariate extremes](https://mlsanity.com/paper/2112.14727/) (2021)
- [Locally associated graphical models and mixed convex exponential families](https://mlsanity.com/paper/2008.04688/) (2020)

## Built on by (most cited first, 2 of 2)

- [Learning Large-Scale MTP2 Gaussian Graphical Models via Bridge-Block Decomposition](https://mlsanity.com/paper/2309.13405/) (2023)
- [Network Topology Inference with Sparsity and Laplacian Constraints](https://mlsanity.com/paper/2309.00960/) (2023)

## Abstract (arXiv)

We consider the problem of estimating (diagonally dominant) M-matrices as precision matrices in Gaussian graphical models. These models exhibit intriguing properties, such as the existence of the maximum likelihood estimator with merely two observations for M-matrices citeplauritzen2019maximum,slawski2015estimation and even one observation for diagonally dominant M-matrices citeptruell2021maximum. We propose an adaptive multiple-stage estimation method that refines the estimate by solving a weighted ℓ1-regularized problem at each stage. Furthermore, we develop a unified framework based on the gradient projection method to solve the regularized problem, incorporating distinct projections to handle the constraints of M-matrices and diagonally dominant M-matrices. A theoretical analysis of the estimation error is provided. Our method outperforms state-of-the-art methods in precision matrix estimation and graph edge identification, as evidenced by synthetic and financial time-series data sets.