Persistence-sensitive simplification of functions on 2-manifolds

We continue the study of topological persistence [5] by investigat- ing the problem of simplifying a function f in a way that removes topological noise as determined by its persistence diagram [2]. To state our results, we call a function g an ε-simplification of another function f if ∥f − g∥∞ ≤ ε, and the persistence diagrams of g are the same as those of f except all points within L1-distance at most ε from the diagonal have been removed. We prove that for func- tions f on a 2-manifold such ε-simplification exists, and we give an algorithm to construct them in the piecewise linear case.