Generalized inverses are ubiquitous in matrix algebra and its applications. Not all Moore-Penrose properties are needed to ensure that a generalized inverse solves key problems, like least squares. So there is the opportunity to find sparser generalized inverses that do the jobs. I will present theoretical and computational results on various approaches to this, in particular approximation algorithms and convex relaxation. Joint works with: Marcia Fampa, Luze Xu and Gabriel Ponte.