Finding statistically significant high-order interactions in predictive modeling is important but challenging task because the possible number of high-order interactions is extremely large (e.g., $> 10^{17}$). In this paper we study feature selection and statistical inference for sparse high-order interaction models. Our main contribution is to extend recently developed selective inference framework for linear models to high-order interaction models by developing a novel algorithm for efficiently characterizing the selection event for the selective inference of high-order interactions. We demonstrate the effectiveness of the proposed algorithm by applying it to an HIV drug response prediction problem.

In this paper we study predictive pattern mining problems where the goal is to construct a predictive model based on a subset of predictive patterns in the database. Our main contribution is to introduce a novel method called safe pattern pruning (SPP) for a class of predictive pattern mining problems. The SPP method allows us to efficiently find a superset of all the predictive patterns in the database that are needed for the optimal predictive model. The advantage of the SPP method over existing boosting-type method is that the former can find the superset by a single search over the database, while the latter requires multiple searches. The SPP method is inspired by recent development of safe feature screening. In order to extend the idea of safe feature screening into predictive pattern mining, we derive a novel pruning rule called safe pattern pruning (SPP) rule that can be used for searching over the tree defined among patterns in the database. The SPP rule has a property that, if a node corresponding to a pattern in the database is pruned out by the SPP rule, then it is guaranteed that all the patterns corresponding to its descendant nodes are never needed for the optimal predictive model. We apply the SPP method to graph mining and item-set mining problems, and demonstrate its computational advantage.