Multivariate Tropical Regression And Piecewise-Linear Surface Fitting

In this paper we propose a novel approach for multivariate convex regression by using as approximation model a maximum of hyperplanes, which we represent as a multivariate max-plus tropical polynomial. Our approach uses concepts from tropical geometry and finds an optimal solution for the model parameters (that minimizes a data fitting error norm) by solving systems of max-plus equations using max-plus algebra and projections on weighted lattices. Our method has lower complexity than most other methods for fitting piecewise-linear (PWL) functions and we apply it to optimal PWL regression for fitting max-plus tropical surfaces to arbitrary data that constitute polyhedral shape approximations.