Sparse Partial Correlation Estimation With Scaled Lasso and Its GPU-Parallel Algorithm

Sparse partial correlation estimation is a popular topic in high-dimensional data analysis, where nonzero partial correlation represents the conditional dependency between two corresponding variables given the other variables. In the Gaussian graphical model, many methods have been developed using the l(1) regularization to achieve sparsity on conditional dependency. Most of the existing methods impose l(1) penalty on the off-diagonal entries of the precision matrix. This approach may fail to identify the conditional dependencies with partial correlations of moderate magnitudes when the corresponding elements of the precision matrix are relatively small. In this study, we propose a two-stage procedure to estimate sparse partial correlations using scaled Lasso. The proposed procedure resolves the non-convexity of partial correlation estimation by using a consistent estimator of the diagonal elements of the precision matrix from scaled Lasso. Moreover, we develop an efficient algorithm for the proposed method using graphics processing units based on the iterative shrinkage algorithm. Our numerical study shows that the proposed method performs better than the existing methods in terms of edge recovery and the estimation of the partial correlations under the Frobenius norm.