Bounded variation separates weak and strong average Lipschitz

Abstract

We closely examine a notion of average smoothness recently introduced by Ashlagi et al. (JMLR, 2024). The latter defined a {\em weak} and {\em strong} average-Lipschitz seminorm for real-valued functions on general metric spaces. Specializing to the standard metric on the real line, we compare these notions to bounded variation (BV) and discover that the weak notion is strictly weaker than BV while the strong notion strictly stronger. Along the way, we discover that the weak average smooth class is also considerably larger in a certain combinatorial sense, made precise by the fat-shattering dimension.