  The Ising model is a useful tool for studying complex interactions within a
system. The estimation of such a model, however, is rather challenging,
especially in the presence of high-dimensional parameters. In this work, we
propose efficient procedures for learning a sparse Ising model based on a
penalized composite conditional likelihood with nonconcave penalties.
Nonconcave penalized likelihood estimation has received a lot of attention in
recent years. However, such an approach is computationally prohibitive under
high-dimensional Ising models. To overcome such difficulties, we extend the
methodology and theory of nonconcave penalized likelihood to penalized
composite conditional likelihood estimation. The proposed method can be
efficiently implemented by taking advantage of coordinate-ascent and
minorization--maximization principles. Asymptotic oracle properties of the
proposed method are established with NP-dimensionality. Optimality of the
computed local solution is discussed. We demonstrate its finite sample
performance via simulation studies and further illustrate our proposal by
studying the Human Immunodeficiency Virus type 1 protease structure based on
data from the Stanford HIV drug resistance database. Our statistical learning
results match the known biological findings very well, although no prior
biological information is used in the data analysis procedure.
