Large Precision Matrix Estimation with Unknown Group Structure

The estimation of large precision matrices is crucial in modern multivariate analysis. Traditional sparsity assumptions, while useful, often fall short of accurately capturing the dependencies among features. This paper addresses this limitation by focusing on precision matrix estimation for multivariate data characterized by a flexible yet unknown group structure. We introduce a novel approach that begins with the detection of this unknown group structure, clustering features within the low-dimensional space defined by the leading eigenvectors of the sample covariance matrix. Following this, we employ group-wise multivariate response linear regressions, guided by the identified group memberships, to estimate the precision matrix. We rigorously establish the theoretical foundations of our proposed method for both group detection and precision matrix estimation. The superior numerical performance of our approach is demonstrated through comprehensive simulation experiments and a comparative analysis with established methods in the field. Additionally, we apply our method to a real breast cancer dataset, showcasing its practical utility and effectiveness.