We consider the problem of jointly learning row-wise and column-wise dependencies of matrix-variate observations, which are modelled separately by two precision matrices. Due to the complicated structure of Kronecker-product precision matrices in the commonly used matrix-variate Gaussian graphical models, a sparser Kronecker-sum structure was proposed recently based on the Cartesian product of graphs. However, existing methods for estimating Kronecker-sum structured precision matrices do not scale well to large scale datasets. In this paper, we introduce DNNLasso, a diagonally non-negative graphical lasso model for estimating the Kronecker-sum structured precision matrix, which outperforms the state-of-the-art methods by a large margin in both accuracy and computational time.

In this paper, we focus on the computation of the nonparametric maximum likelihood estimator (NPMLE) in multivariate mixture models. Our approach discretizes this infinite dimensional convex optimization problem by setting fixed support points for the NPMLE and optimizing over the mixing proportions. We propose an efficient and scalable semismooth Newton based augmented Lagrangian method (ALM). Our algorithm outperforms the state-of-the-art methods (Kim et al., 2020; Koenker and Gu, 2017), capable of handling $n \approx 10^6$ data points with $m \approx 10^4$ support points. A key advantage of our approach is its strategic utilization of the solution's sparsity, leading to structured sparsity in Hessian computations. As a result, our algorithm demonstrates better scaling in terms of $m$ when compared to the mixsqp method (Kim et al., 2020). The computed NPMLE can be directly applied to denoising the observations in the framework of empirical Bayes. We propose new denoising estimands in this context along with their consistent estimates. Extensive numerical experiments are conducted to illustrate the efficiency of our ALM. In particular, we employ our method to analyze two astronomy data sets: (i) Gaia-TGAS Catalog (Anderson et al., 2018) containing approximately $1.4 \times 10^6$ data points in two dimensions, and (ii) a data set from the APOGEE survey (Majewski et al., 2017) with approximately $2.7 \times 10^4$ data points.

Square-root (loss) regularized models have recently become popular in linear regression due to their nice statistical properties. Moreover, some of these models can be interpreted as the distributionally robust optimization counterparts of the traditional least-squares regularized models. In this paper, we give a unified proof to show that any square-root regularized model whose penalty function being the sum of a simple norm and a seminorm can be interpreted as the distributionally robust optimization (DRO) formulation of the corresponding least-squares problem. In particular, the optimal transport cost in the DRO formulation is given by a certain dual form of the penalty. To solve the resulting square-root regularized model whose loss function and penalty function are both nonsmooth, we design a proximal point dual semismooth Newton algorithm and demonstrate its efficiency when the penalty is the sparse group Lasso penalty or the fused Lasso penalty. Extensive experiments demonstrate that our algorithm is highly efficient for solving the square-root sparse group Lasso problems and the square-root fused Lasso problems.