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 we term “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 w ell as a study of its numerical performance (compared to truncated Taylor series approximation and to Neural Nets) in the estimation setting. We conclude that PLH i) offers nearly the accuracy of a Neural Net while ii) requiring, via our GI algorithm, the far shorter (several orders of magnitude less) training time typical of Taylor series approximants and iii) preserving desired invariant properties unlike any other presently popular basis family.