Bounded variation separates weak and strong average Lipschitz

TL;DR

This paper investigates the relationship between weak and strong average Lipschitz notions and bounded variation on the real line, revealing that weak is weaker and strong is stronger than BV, with implications for function class complexity.

Contribution

It establishes a precise comparison between average Lipschitz seminorms and bounded variation, clarifying their hierarchy and size in terms of fat-shattering dimension.

Findings

Weak average Lipschitz is strictly weaker than bounded variation.

Strong average Lipschitz is strictly stronger than bounded variation.

Weak class has larger combinatorial complexity, measured by fat-shattering dimension.

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.