On the Stability of Interval Decomposable Persistence Modules

Abstract

The algebraic stability theorem for persistence modules is a central result in the theory of stability for persistent homology. We introduce a new proof technique which we use to prove a stability theorem for n-dimensional rectangle decomposable persistence modules up to a constant $$2n-1$$ 2 n - 1 that generalizes the algebraic stability theorem, and give an example showing that the bound cannot be improved for $$n=2$$ n = 2 . We then apply the technique to prove stability for block decomposable modules, from which novel results for zigzag modules and Reeb graphs follow. These results are improvements on weaker bounds in previous work, and the bounds we obtain are optimal.

$\ell^p$-Distances on Multiparameter Persistence Modules

“…In fact, our proofs of Theorem 1.7 (i) and (iv) together amount to a novel proof of the isometry theorem. While there already exist several good proofs of this which work in greater generality [3,4,11,34], we feel that the proof given here may be of interest, because it is simple, intuitive, and rests on established facts known to be useful elsewhere. Our proof is similar in some respects to existing proofs of algebraic stability and stability for R-valued functions; specifically, [39,74] use a very similar interpolation strategy, and [30,34,37] use a somewhat similar one.…”

“…The result that d I = d ∞ W on 1-parameter persistence modules is known as the isometry theorem [3,11,21,34,60]; see Theorem 2.16 for a formal statement. The inequality d I ≥ d ∞ W , originally due to Chazal et al [30], is called the algebraic stability theorem.…”

The reflection distance between zigzag persistence modules

“…They were able to prove that the bottleneck distance between the persistence diagrams of the original persistence modules is bounded above by a constant times the value of the distance between them. A recent refinement by Bjerkevik [Bak16] has found the optimal constant for this inequality.…”

“…The Matching Lemma is a useful tool for combining matchings in opposite directions into a single matching. For us, the lemma was crucial for proving our stability result, while Bjerkevik makes use of the lemma in proving his generalization of the algebraic stability theorem [Bak16]. While it may be one of the earliest results in infinite matching theory [Aha91], we believe its applications to stability-type questions in persistence theory make it worth expounding upon here.…”

Algebraic stability theorem for derived categories of zigzag persistence modules

“…It was proved in Botnan and Lesnick (2018) that an AST also holds for purely zigzag persistence modules. Bjerkevik (2016) improved the theorem with a tight bound and provided an isometry theorem for purely zigzag persistence modules. Note that zigzag persistence modules in Botnan and Lesnick (2018); Bjerkevik (2016) are purely zigzag ones in our convention.…”