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Partial correlation is a useful connectivity measure for brain networks, especially, when it is needed to remove the confounding effects in highly correlated networks. Since it is difficult to estimate the exact partial correlation under the small-n large-p situation, a sparseness constraint is generally introduced. In this paper, we consider the sparse linear regression model with a l1-norm penalty, also known as the least absolute shrinkage and selection operator (LASSO), for estimating sparse brain connectivity. LASSO is a well-known decoding algorithm in the compressed sensing (CS). The CS theory states that LASSO can reconstruct the exact sparse signal even from a small set of noisy measurements. We briefly show that the penalized linear regression for partial correlation estimation is related to CS. It opens a new possibility that the proposed framework can be used for a sparse brain network recovery. As an illustration, we construct sparse brain networks of 97 regions of interest (ROIs) obtained from FDG-PET imaging data for the autism spectrum disorder (ASD) children and the pediatric control (PedCon) subjects. As validation, we check the network reproducibilities by leave-one-out cross validation and compare the clustered structures derived from the brain networks of ASD and PedCon.