Fast Projected Newton-like Method for Precision Matrix Estimation under Total Positivity
This work addresses a computational bottleneck for researchers and practitioners in statistics and machine learning dealing with high-dimensional Gaussian models, though it is incremental as it builds on existing methods.
The authors tackled the problem of estimating precision matrices under total positivity constraints, which is computationally challenging in high dimensions, and proposed a novel algorithm that significantly improves computational efficiency compared to state-of-the-art methods, as demonstrated in experiments.
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.