On the Furthest Hyperplane Problem and Maximal Margin Clustering

Abstract: This paper introduces the Furthest Hyperplane Problem (FHP), which is an unsupervised counterpart of Support Vector Machines. Given a set of n points in Rd, the objective is to produce the hyperplane (passing through the origin) which maximizes the separation margin, that is, the minimal distance between the hyperplane and any input point. To the best of our knowledge, this is the first paper achieving provable results regarding FHP. We provide both lower and upper bounds to this NP-hard problem. First, we give a simple randomized algorithm whose running time is n^O(1/{\theta}^2) where {\theta} is the optimal separation margin. We show that its exponential dependency on 1/{\theta}^2 is tight, up to sub-polynomial factors, assuming SAT cannot be solved in sub-exponential time. Next, we give an efficient approxima- tion algorithm. For any {\alpha} \in [0, 1], the algorithm produces a hyperplane whose distance from at least 1 - 5{\alpha} fraction of the points is at least {\alpha} times the optimal separation margin. Finally, we show that FHP does not admit a PTAS by presenting a gap preserving reduction from a particular version of the PCP theorem.