Geometric Bounds for Persistence

In this paper, we bring a new perspective to persistent homology by incorporating key concepts from metric geometry. For a given compact subset X of a Banach space Y, we study the topological features appearing in family N_∙(X⊂ Y) of nested of neighborhoods of X in Y and give several geometric bounds on their persistence (lifespans). First, we discuss the lifespans of these homology classes in terms of their filling radii in Y. By using this, we relate these lifespans to key invariants in metric geometry such as Urysohn width. Next, we bound these lifespans via ℓ^∞-principal components of the set X, which are also known as Kolmogorov widths. Furthermore, we introduce and study the notion of extinction time of a space X: the critical threshold after which there are no homological features alive in any degree. We also describe novel approaches on how to estimate Čech and Vietoris-Rips extinction times of a set X by relating X to its über-contractible cores and to its tight span, respectively.