Covariance estimation for high-dimensional datasets is a fundamental problem in modern day statistics with numerous applications. In these high dimensional datasets, the number of variables p is typically larger than the sample size n. A popular way of tackling this challenge is to induce sparsity in the covariance matrix, its inverse or a relevant transformation. In particular, methods inducing sparsity in the Cholesky parameter of the inverse covariance matrix can be useful as they are guaranteed to give a positive definite estimate of the covariance matrix. Also, the estimated sparsity pattern corresponds to a Directed Acyclic Graph (DAG) model for Gaussian data.

In this talk, we propose a new penalized likelihood method for sparse estimation of the inverse covariance Cholesky parameter that aims to overcome some of the shortcomings of current methods, but retains their respective strengths. We obtain a jointly convex formulation for our objective function, which leads to convergence guarantees, even when p > n. We establish high-dimensional estimation and graph selection consistency, and also demonstrate finite sample performance on simulated/real data.