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

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. The problem can be formulated as a sign-constrained log-determinant program. The existing algorithms designed for solving this problem are based on the block coordinate descent method, which are computationally prohibitive in high-dimensional cases, because of the need to solve a large number of nonnegative quadratic programs. We propose a novel algorithm based on the two-metric projection method, with a well-designed search direction and variable partitioning scheme. Our algorithm reduces the computational complexity significantly in solving this problem, and its theoretical convergence is established. Experiments involving synthetic and real-world data demonstrate that our proposed algorithm is significantly more efficient, from a computational time perspective, than the state-of-the-art methods.