Piecewise Linear Homeomorphisms: The Scalar Case
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The class of piecewise linear homeomorphisms (PLH) provides a convenient functional representation for many applications wherein an approximation to data is required that is invertible in closed form. In this paper we introduce the graph intersection (GI) algorithm for "learning" piecewise linear scalar functions in two settings: "approximation," where an "oracle" outputs accurate functional values in response to input queries; and "estimation," where only a fixed discrete data base of input-output pairs is available. We provide a local convergence result for the approximation version of the GI algorithm as well as a study of its numerical performance in the estimation setting. We conclude that PLH offers accuracy closed to that of a neural net while requiring, via our GI algorithm, far shorter training time and preserving desired invariant properties unlike any other presently popular basis family.
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Copyright 2000 IEEE. Reprinted from Proceedings of the IEEE-INNS-ENNS International Joint Conference on Neural Networks, Volume 3, 2000, pages 259-264. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the University of Pennsylvania's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to pubs-permissions@ieee.org. By choosing to view this document, you agree to all provisions of the copyright laws protecting it. NOTE: At the time of publication, author Daniel Koditschek was affiliated with the University of Michigan. Currently, he is a faculty member in the Department of Electrical and Systems Engineering at the University of Pennsylvania.