Sparsity in Max-Plus Algebra and Applications in Multivariate Convex Regression

In this paper, we study concepts of sparsity in the max-plus algebra and apply them to the problem of multivariate convex regression. We show how to efficiently find sparse (containing many −∞ elements) approximate solutions to max-plus equations by leveraging notions from submodular optimization. Subsequently, we propose a novel method for piecewise-linear surface fitting of convex multivariate functions, with optimality guarantees for the model parameters and an approximately minimum number of affine regions.