Stability and Computation of 2-Parameter Persistent Homology

We show that the standard stability results for union-of-balls and Rips persistent homology have natural analogues in the 2-parameter setting, formulated in terms of the multicover bifiltration and Sheehy’s subdivision bifiltrations. Our results imply that these bifiltrations are robust, i.e., stable to outliers, in a strong sense. We also give similar stability results for degree bifiltrations, which are weaker, but tight. These results raise the question of whether multicover and subdivision bifiltrations can be computed (up to homotopy). I’ll discuss this, focusing in particular on a polynomial-size model of the multicover bifiltration called the rhomboid tiling, introduced by Edelsbrunner and Osang. The talk will cover joint work with Andrew Blumberg, as well as work with René Corbet, Michael Kerber, and Georg Osang.