# Fast Projected Newton-like Method for Precision Matrix Estimation under Total Positivity

arXiv: https://arxiv.org/abs/2112.01939 · HTML: https://mlsanity.com/paper/2112.01939/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 63 references that are in the map.
- First posted on arXiv 3 Dec 2021.

## Builds on (most cited first, 13 of 13)

- [A Unified Framework for Structured Graph Learning via Spectral Constraints](https://mlsanity.com/paper/1904.09792/) (2019)
- [Learning Some Popular Gaussian Graphical Models without Condition Number Bounds](https://mlsanity.com/paper/1905.01282/) (2019)
- [Structured Graph Learning Via Laplacian Spectral Constraints](https://mlsanity.com/paper/1909.11594/) (2019)
- [Estimation of positive definite M-matrices and structure learning for attractive Gaussian Markov Random fields](https://mlsanity.com/paper/1404.6640/) (2014)
- [Large-scale Log-determinant Computation through Stochastic Chebyshev Expansions](https://mlsanity.com/paper/1503.06394/) (2015)
- [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)
- [Decomposition by Partial Linearization: Parallel Optimization of Multi-Agent Systems](https://mlsanity.com/paper/1302.0756/) (2013)
- [Speeding Up Latent Variable Gaussian Graphical Model Estimation via Nonconvex Optimizations](https://mlsanity.com/paper/1702.08651/) (2017)
- [Total positivity in exponential families with application to binary variables](https://mlsanity.com/paper/1905.00516/) (2019)
- [A proximal Newton framework for composite minimization: Graph learning without Cholesky decompositions and matrix inversions](https://mlsanity.com/paper/1301.1459/) (2013)

## Built on by (most cited first, 8 of 8)

- [Learning Graphical Factor Models with Riemannian Optimization](https://mlsanity.com/paper/2210.11950/) (2022)
- [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)
- [Adaptive Estimation of Graphical Models under Total Positivity](https://mlsanity.com/paper/2210.15471/) (2022)
- [Graph Learning Across Data Silos](https://mlsanity.com/paper/2301.06662/) (2023)
- [Leveraging Low-rank Factorizations of Conditional Correlation Matrices in Graph Learning](https://mlsanity.com/paper/2506.10628/) (2025)
- [Learning Gaussian Graphical Models under Total Positivity via Spectral Graph Sparsification](https://mlsanity.com/paper/2605.17154/) (2026)
- [Spectral Sparsification of Laplacian-Constrained Gaussian and H"usler-Reiss Graphical Models](https://mlsanity.com/paper/2606.16681/) (2026)

## Abstract (arXiv)

We study the problem of estimating precision matrices in Gaussian distributions that are multivariate totally positive of order two (MTP2). The precision matrix in such a distribution is an M-matrix. This problem can be formulated as a sign-constrained log-determinant program. Current algorithms are designed using the block coordinate descent method or the proximal point algorithm, which becomes computationally challenging in high-dimensional cases due to the requirement to solve numerous nonnegative quadratic programs or large-scale linear systems. To address this issue, we propose a novel algorithm based on the two-metric projection method, incorporating a carefully designed search direction and variable partitioning scheme. Our algorithm substantially reduces computational complexity, and its theoretical convergence is established. Experimental results on synthetic and real-world datasets demonstrate that our proposed algorithm provides a significant improvement in computational efficiency compared to the state-of-the-art methods.