Toward a Sparsity Theory on Weighted Lattices

This paper studies issues of sparse representation in nonlinear vector spaces. In particular, we focus on complete weighted lattices Maragos (Math. Control Signals Syst 29: 21, 2017), a class of nonlinear spaces that generalizes mathematical morphology and max-plus algebra. We show how one can obtain sparse solutions to equations that arise in such spaces and discuss the computational hardness of the problem. Then, the focus shifts to max-plus algebra and, in particular, to sparse approximate solutions to max-plus equations. The developed theoretical tools allow us to make structured arguments about the pruning of a special class of neural networks, called morphological neural networks.