Before revision:  Finally we solve this optimization problem using Alternating Direction Method of Multipliers (ADMM). Experiments show that joint multi-family EC analysis can reveal many more native contacts than single-family analysis even for a target family with 4000-5000 non-redundant sequence homologs, which makes many more protein families amenable to co-evolution-based structure and function prediction .
After revision:  Finally we solve this optimization problem using Alternating Direction Method of Multipliers (ADMM). Experiments show that our method can predict contacts much more accurately than existing methods, and that our method performs better on both conserved and family-specific contacts .