  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.
