Grouping pursuit through a regularization solution surface

Extracting grouping structure or identifying homogenous subgroups of predictors in regression is crucial for high-dimensional data analysis. A low-dimensional structure in particular-grouping, when captured in a regression model-enables to enhance predictive performance and to facilitate a model's interpretability. Grouping pursuit extracts homogenous subgroups of predictors most responsible for outcomes of a response. This is the case in gene network analysis, where grouping reveals gene functionalities with regard to progression of a disease. To address challenges in grouping pursuit, we introduce a novel homotopy method for computing an entire solution surface through regularization involving a piecewise linear penalty. This nonconvex and overcomplete penalty permits adaptive grouping and nearly unbiased estimation, which is treated with a novel concept of grouped subdifferentials and difference convex programming for efficient computation. Finally, the proposed method not only achieves high performance as suggested by numerical analysis, but also has the desired optimality with regard to grouping pursuit and prediction as showed by our theoretical results.