A unified precision matrix estimation framework via sparse column-wise inverse operator under weak sparsity

In this paper, we estimate the high dimensional precision matrix under the weak sparsity condition where many entries are nearly zero. We revisit the sparse column-wise inverse operator (SCIO) estimator \cite{liu2015fast} and derive its general error bounds under the weak sparsity condition. A unified framework is established to deal with various cases including the heavy-tailed data, the non-paranormal data, and the matrix variate data. These new methods can achieve the same convergence rates as the existing methods and can be implemented efficiently.