Piecewise Linear Homeomorphisms: The Scalar Case

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.