We consider the problem of fitting a regression model that is both flexible and interpretable. We propose two procedures for this task: the Fused Lasso Additive Model (FLAM), which is an additive model of piecewise constant fits; and Convex Regression with Interpretable Sharp Partitions (CRISP), which extends FLAM to allow for non-additivity. Both FLAM and CRISP are the solutions to convex...