On the Furthest Hyperplane Problem and Maximal Margin Clustering

TL;DR

This paper introduces the Furthest Hyperplane Problem (FHP), an unsupervised variant of SVMs focused on maximizing the margin, and provides the first provable bounds, algorithms, and complexity results for this NP-hard problem.

Contribution

It presents the first theoretical analysis of FHP, including bounds, approximation algorithms, and complexity results, establishing its computational difficulty.

Findings

A randomized algorithm with n^{O(1/θ^2)} time complexity.

An approximation algorithm achieving near-optimal margin for most points.

FHP does not admit a PTAS, proven via a gap-preserving reduction.

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.