Algorithm xxx: Spherical Triangle Algorithm: A Fast Oracle for Convex Hull Membership Queries

Abstract

The Convex Hull Membership (CHM) tests whether \(p \in conv(S) \) , where \(p \) and the \(n \) points of \(S \) lie in \(\mathbb { R}^m \) . CHM finds applications in Linear Programming, Computational Geometry and Machine Learning. The Triangle Algorithm (TA), previously developed, in \(O(1/\varepsilon ^2) \) iterations computes \(p^{\prime } \in conv(S) \) , either an \(\varepsilon \) - approximate solution , or a witness certifying \(p \not\in conv(S) \) . We first prove the equivalence of exact and approximate versions of CHM and Spherical -CHM, where \(p=0 \) and \(\Vert v\Vert =1 \) for each \(v \) in \(S \) . If for some \(M \ge 1 \) every non-witness with \(\Vert p^{\prime }\Vert \gt \varepsilon \) admits \(v \in S \) satisfying \(\Vert p^{\prime } - v\Vert \ge \sqrt {1+\varepsilon /M} \) , we prove the number of iterations improves to \(O(M/\varepsilon) \) and \(M \le 1/\varepsilon \) always holds. Equivalence of CHM and Spherical-CHM implies Minimum Enclosing Ball (MEB) algorithms can be modified to solve CHM. However, we prove \((1+ \varepsilon) \) -approximation in MEB is \(\Omega (\sqrt {\varepsilon }) \) -approximation in Spherical-CHM. Thus even \(O(1/\varepsilon) \) iteration MEB algorithms are not superior to Spherical-TA. Similar weakness is proved for MEB core sets. Spherical-TA also results a variant of the All Vertex Triangle Algorithm (AVTA) for computing all vertices of \(conv(S) \) . Substantial computations on distinct problems demonstrate that TA and Spherical-TA generally achieve superior efficiency over algorithms such as Frank-Wolfe, MEB and LP-Solver.