Regularization and feature selection for networked features

In the standard formalization of supervised learning problems, a datum is represented as a vector of features without prior knowledge about relationships among features. However, for many real world problems, we have such prior knowledge about structure relationships among features. For instance, in Microarray analysis where the genes are features, the genes form biological pathways. Such prior knowledge should be incorporated to build a more accurate and interpretable model, especially in applications with high dimensionality and low sample sizes. Towards an efficient incorporation of the structure relationships, we have designed a classification model where we use an undirected graph to capture the relationship of features. In our method, we combine both L1 norm and Laplacian based L2 norm regularization with logistic regression. In this approach, we enforce model sparsity and smoothness among features to identify a small subset of grouped features. We have derived efficient optimization algorithms based on coordinate decent for the new formulation. Using comprehensive experimental study, we have demonstrated the effectiveness of the proposed learning methods.