  Biological networks often change under different environmental and genetic
conditions. Understanding how these networks change becomes an important
problem in biological studies. In this paper, we model the network change as
the difference of two precision matrices and propose a novel loss function for
estimating the precision matrix difference. Under a new irrepresentability
condition, we show that the new loss function with the lasso penalty can give
consistent estimates in high-dimensional setting for sub-Gaussian and
polynomial-tailed distributions. An efficient algorithm is developed based on
the alternating direction method to solve the optimization problem. Simulation
studies and a real data analysis about colorectal cancer show that the proposed
method outperforms other available methods.
