Robust Vertex Enumeration for Convex Hulls in High Dimensions

We design a fast and robust algorithm named {All Vertex Traingle Algorithm (AVTA)} for detecting the vertices of the convex hull of a set of points in high dimensions. Our proposed algorithm is very general and works for arbitrary convex hulls. In addition to being a fundamental problem in computational geometry and linear programming, vertex enumeration in high dimensions has numerous applications in machine learning. In particular, we apply AVTA to design new practical algorithms for topic models and non-negative matrix factorization. For topic models, our new algorithm leads to significantly better reconstruction of the topic-word matrix than state of the art approaches. Additionally, we provide a robust analysis of AVTA and empirically demonstrate that it can handle larger amounts of noise than existing methods. For non-negative matrix we show that AVTA is competitive with existing methods that are specialized for this task.