Geometric Problems in Machine Learning

We present some problems with geometric characterizations that arise naturally in practical applications of machine learning. Our motivation comes from a well known machine learning problem, the problem of computing decision trees. Typically one is given a dataset of positive and negative points, and has to compute a decision tree that ts it. The points are in a low dimensional space, and the data are collected experimentally. In most practical solutions heuristic algorithms are used. To compute decision trees quickly, one has to solve optimization problems in one or more dimensions ef-ciently. In this paper we give geometric characterizations for these problems. We present a selection of algorithms for some of them. These algorithms are motivated from practice, and have been in many cases implemented and used as well. In addition, they are theoretically interesting, and typically employ sophisticated geometric techniques. Finally we present future research directions.