Sparse covariance estimation under Kronecker product structure

We introduce a sparse covariance estimation method for the high dimensional setting when the covariance matrix decomposes as a Kronecker product, i.e., Σ₀ = A₀ ⊗ B₀, and the observations are Gaussian. We propose an ℓ₁ penalized maximum-likelihood approach to solve this problem. The dual formulation motivates an iterative algorithm (penalized flip-flop; FFP) based on a block coordinate-descent approach. Although the ℓ₁-penalized log-likelihood function (objective function) is non-convex in general and non-smooth, we show that FFP converges to a local maximum under relatively mild assumptions. For the fixed dimension case, large-sample statistical consistency is proved and a rate of convergence bound is derived. Simulations show that FFP outperforms its non-penalized counterpart and the naive Glasso algorithm for sparse Kronecker-decomposable covariance matrix.