Piecewise affine regression via recursive multiple least squares and multicategory discrimination

Abstract

In nonlinear regression choosing an adequate model structure is often a challenging problem. While simple models (such as linear functions) may not be able to capture the underlying relationship among the variables, over-parametrized models described by a large set of nonlinear basis functions tend to overfit the training data, leading to poor generalization on unseen data. Piecewise-affine (PWA) models can describe nonlinear and possible discontinuous relationships while maintaining simple local affine regressor-to-output mappings, with extreme flexibility when the polyhedral partitioning of the regressor space is learned from data rather than fixed a priori. In this paper, we propose a novel and numerically very efficient two-stage approach for PWA regression based on a combined use of (i) recursive multi-model least-squares techniques for clustering and fitting linear functions to data, and (ii) linear multi-category discrimination, either offline (batch) via a Newton-like algorithm for computing a solution of unconstrained optimization problems with objective functions having a piecewise smooth gradient, or online (recursive) via averaged stochastic gradient descent.