Learning feature interactions can be the key for multivariate predictive modeling. ReLU-activated neural networks create piecewise linear prediction models, and other nonlinear activation functions lead to models with only high-order feature interactions. Recent methods incorporate candidate polynomial terms of fixed orders into deep learning, which is subject to the issue of combinatorial explosion, or learn the orders that are difficult to adapt to different regions of the feature space. We propose a Polyhedron Attention Module (PAM) to create piecewise polynomial models where the input space is split into polyhedrons which define the different pieces and on each piece the hyperplanes that define the polyhedron boundary multiply to form the interactive terms, resulting in interactions of adaptive order to each piece. PAM is interpretable to identify important interactions in predicting a target. Theoretic analysis shows that PAM has stronger expression capability than ReLU-activated networks. Extensive experimental results demonstrate the superior classification performance of PAM on massive datasets of the click-through rate prediction and PAM can learn meaningful interaction effects in a medical problem.

Data in large-scale genetic studies of complex human diseases, such as substance use disorders, are often incomplete. Despite great progress in genotype imputation, e.g., the IMPUTE2 method, considerably less progress has been made in inferring phenotypes. We designed a novel approach to integrate individuals' comorbid conditions with their genotype data to infer missing (unreported) diagnostic criteria of a disorder. The premise of our approach derives from correlations among symptoms and the shared biological bases of concurrent disorders such as co-dependence on cocaine and opioids. We describe a matrix completion method to construct a bi-linear model based on the interactions of genotypes and known symptoms of related disorders to infer unknown values of another set of symptoms or phenotypes. An efficient stochastic and parallel algorithm based on the linearized alternating direction method of multipliers was developed to solve the proposed optimization problem. Empirical evaluation of the approach in comparison with other advanced data matrix completion methods via a case study shows that it both significantly improves imputation accuracy and provides greater computational efficiency.