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

TL;DR

This paper introduces a fast projected Newton-like algorithm for estimating precision matrices in Gaussian distributions with total positivity, significantly improving computational efficiency in high-dimensional settings.

Contribution

The paper proposes a novel two-metric projection method that reduces computational complexity for MTP2 precision matrix estimation, with proven convergence.

Findings

Significant speedup over existing algorithms in synthetic datasets

Effective in high-dimensional precision matrix estimation

Theoretical convergence guarantees provided

Abstract

We study the problem of estimating precision matrices in Gaussian distributions that are multivariate totally positive of order two ($\mathrm{MTP}_2$). 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.