  Statistical learning with a large number of rare binary features is commonly
encountered in analyzing electronic health records (EHR) data, especially in
the modeling of disease onset with prior medical diagnoses and procedures.
Dealing with the resulting highly sparse and large-scale binary feature matrix
is notoriously challenging as conventional methods may suffer from a lack of
power in testing and inconsistency in model fitting while machine learning
methods may suffer from the inability of producing interpretable results or
clinically-meaningful risk factors. To improve EHR-based modeling and utilize
the natural hierarchical structure of disease classification, we propose a
tree-guided feature selection and logic aggregation approach for large-scale
regression with rare binary features, in which dimension reduction is achieved
through not only a sparsity pursuit but also an aggregation promoter with the
logic operator of ``or''. We convert the combinatorial problem into a convex
linearly-constrained regularized estimation, which enables scalable computation
with theoretical guarantees. In a suicide risk study with EHR data, our
approach is able to select and aggregate prior mental health diagnoses as
guided by the diagnosis hierarchy of the International Classification of
Diseases. By balancing the rarity and specificity of the EHR diagnosis records,
our strategy improves both prediction and model interpretation. We identify
important higher-level categories and subcategories of mental health conditions
and simultaneously determine the level of specificity needed for each of them
in predicting suicide risk.
