Nonconcave penalized composite conditional likelihood estimation of sparse Ising models

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.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1017 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

catena-Poly[[(1,10-phenanthroline)zinc]-μ-2,2′-oxydibenzoato]

In the title one-dimensional coordination polymer, [Zn(C14H8O5)(C12H8N2)]n, the ZnII ion is in a distorted octa­hedral coordination geometry with four O atoms from two carboxyl­ate groups in bidentate chelating modes and two N atoms from a 1,10-phenanthroline ligand. The two terminal carboxyl­ate groups bind the ZnII ions, leading to a chain along the c axis. Adjacent chains are further linked by inter­molecular π–π inter­actions with a shortest centroid–centroid distance of 3.586 (3) Å, forming a two-dimensional supra­molecular architecture with (6,3)-network topology