Unsupervised SVMs: On the Complexity of the Furthest Hyperplane Problem

This paper introduces the Furthest Hyperplane Problem (FHP), which is an unsupervised counterpart of Support Vector Machines. Given a set of n points in R^d, 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/θ^2) where θis the optimal separation margin. We show that its exponential dependency on 1/θ^2 is tight, up to sub-polynomial factors, assuming SAT cannot be solved in sub-exponential time. Next, we give an efficient approximation algorithm. For any α∈[0, 1], the algorithm produces a hyperplane whose distance from at least 1 - 3αfraction of the points is at least αtimes 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.