Persistent homology quantifies the stability of critical points of a scalar function: a critical point with large persistence cannot be eliminated by a small perturbation. Motivated by the fact that noise can give rise to spurious critical points, we consider the converse problem: within a prescribed tolerance from some input function, minimize the number of critical points.

For functions on surfaces, we solve this problem using a novel combination of persistent homology and discrete Morse theory. We construct a function that achieves the lower bound on the number of critical points dictated by the stability of persistence diagrams. Such a function can be constructed in linear time after persistence pairs have been identified.

Moreover, we obtain not only a single solution but a whole a convex set of optimal solutions. Within this set, a convex functional can be minimized using convex optimization methods, guaranteeing that the minimum number of critical points is still maintained.

(Joint work with Carsten Lange and Max Wardetzky. Reference: http://dx.doi.org/10.1007/s00454-011-9350-z)