José Vinícius de M. Cardoso

Jiaxi Ying, José Vinícius de M. Cardoso, Daniel P. Palomar

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 \citep{lauritzen2019maximum,slawski2015estimation} and even one observation for diagonally dominant M-matrices \citep{truell2021maximum}. We propose an adaptive multiple-stage estimation method that refines the estimate by solving a weighted $\ell_1$-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.

Sandeep Kumar, Jiaxi Ying, José Vinícius de M. Cardoso et al.

Graph learning from data represents a canonical problem that has received substantial attention in the literature. However, insufficient work has been done in incorporating prior structural knowledge onto the learning of underlying graphical models from data. Learning a graph with a specific structure is essential for interpretability and identification of the relationships among data. Useful structured graphs include the multi-component graph, bipartite graph, connected graph, sparse graph, and regular graph. In general, structured graph learning is an NP-hard combinatorial problem, therefore, designing a general tractable optimization method is extremely challenging. In this paper, we introduce a unified graph learning framework lying at the integration of Gaussian graphical models and spectral graph theory. To impose a particular structure on a graph, we first show how to formulate the combinatorial constraints as an analytical property of the graph matrix. Then we develop an optimization framework that leverages graph learning with specific structures via spectral constraints on graph matrices. The proposed algorithms are provably convergent, computationally efficient, and practically amenable for numerous graph-based tasks. Extensive numerical experiments with both synthetic and real data sets illustrate the effectiveness of the proposed algorithms. The code for all the simulations is made available as an open source repository.

Jian-Feng Cai, José Vinícius de M. Cardoso, Daniel P. Palomar et al.

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.

Jiaxi Ying, José Vinícius de M. Cardoso, Daniel P. Palomar

We consider the problem of learning a sparse graph under the Laplacian constrained Gaussian graphical models. This problem can be formulated as a penalized maximum likelihood estimation of the Laplacian constrained precision matrix. Like in the classical graphical lasso problem, recent works made use of the $\ell_1$-norm regularization with the goal of promoting sparsity in Laplacian constrained precision matrix estimation. However, we find that the widely used $\ell_1$-norm is not effective in imposing a sparse solution in this problem. Through empirical evidence, we observe that the number of nonzero graph weights grows with the increase of the regularization parameter. From a theoretical perspective, we prove that a large regularization parameter will surprisingly lead to a complete graph, i.e., every pair of vertices is connected by an edge. To address this issue, we introduce the nonconvex sparsity penalty, and propose a new estimator by solving a sequence of weighted $\ell_1$-norm penalized sub-problems. We establish the non-asymptotic optimization performance guarantees on both optimization error and statistical error, and prove that the proposed estimator can recover the edges correctly with a high probability. To solve each sub-problem, we develop a projected gradient descent algorithm which enjoys a linear convergence rate. Finally, an extension to learn disconnected graphs is proposed by imposing additional rank constraint. We propose a numerical algorithm based on based on the alternating direction method of multipliers, and establish its theoretical sequence convergence. Numerical experiments involving synthetic and real-world data sets demonstrate the effectiveness of the proposed method.