Sparse Network Lasso for Local High-dimensional Regression

We introduce the sparse network lasso, which is suited for interpreting models in addition to having high predicting power, for high dimensionality $d$ and small sample size $n$ types of problems. More specifically, we consider a function that consists of local models, where each local model is sparse. We introduce sample-wise network regularization and sample-wise exclusive group sparsity (a.k.a., $\ell_{1,2}$ norm) to introduce diversity into the local models, with different chosen feature sets interpreted as different local models. This would help to interpret not only features but also local models (i.e., samples) in practical problems. The proposed method is a convex method, and thus, it can find an optimal solution. Moreover, we propose a simple yet efficient iterative least-squares based optimization procedure for the sparse network lasso, which does not need a tuning parameter, and is guaranteed to converge to an optimal solution. The solution is empirically shown to outperform straightforward alternatives in predicting outputs for both simulated and molecular biological personalized medicine data.