Notes on an Elementary Proof for the Stability of Persistence Diagrams

These notes provide a self contained algorithmic proof bottleneck stability of persistence diagrams. The only assumption is familiarity with the standard persistence algorithm. The proof technique itself is a special case of the proof of p-Wasserstein stability for cellular complexes in [1, Section 3]. The proof is further simplified due to the use of bottleneck stability. Similar ideas can also be found in [2]. As input, fix a finite simplicial complex K, endowed with two functions f0, f1, : K → R, which we assume satisfy the following conditions: (1) The functions are piecewise constant, i.e. the function assign a function value to each simplex – so for any simplex σ ∈ K, the notion f0(σ) and f1(σ) make sense. (2) The functions are bounded, i.e. for all σ ∈ K, |f0(σ)| < ∞ and |f1(σ)| < ∞ (3) For any α ∈ R, the sublevel sets f 0 (−∞, α] and f −1 1 (−∞, α] are simplicial complexes. Notice that the above conditions are just needed for the standard persistence algorithm from [3] to be welldefined. The function defines an ordering on the simplices. If each simplex has a unique function value, then the ordering is total (a linear order). Otherwise, we can extend the partial order to a total order. If two simplices have the same function value, they are ordered according to increasing dimension (to ensure that at each step in the ordering is a valid simplicial complex). If they have the same dimension, an arbitrary ordering can be chosen (e.g. lexicographical ordering). Our statement will involve the persistence diagrams of the sub-level set filtrations of f0 and f1, which we denote Dgm(f0) and Dgm(f1) respectively. We recall the following definitions: