Arbitrary-norm separating planefn1fn1This material is based on research supported by NSF Grant CCR-9322479 and AFOSR Grant F49620-97-1-0326.

  • Authors:

  • O. L. Mangasarian

  • Affiliations:

  • Computer Sciences Department, University of Wisconsin, 1210 West Dayton Street, Madison, WI 53706, USA

  • Venue:
  • Operations Research Letters
  • Year:
  • 1999

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Abstract

A plane separating two point sets in n-dimensional real space is constructed such that it minimizes the sum of arbitrary-norm distances of misclassified points to the plane. In contrast to previous approaches that used surrogates for distance-minimization, the present work is based on a precise norm-dependent explicit closed form for the projection of a point on a plane. This projection is used to formulate the separating-plane problem as a minimization of a convex function on a unit sphere in a norm dual to that of the arbitrary norm used. For the 1-norm, the problem can be solved in polynomial time by solving 2n linear programs or by solving a bilinear program. For a general p-norm, the minimization problem can be transformed via an exact penalty formulation to minimizing the sum ofa convex function and a bilinear function on a convex set. For the one and infinity norms, a finite successive linearization algorithm can be used for solving the exact penalty formulation.