Bipath Persistence as Zigzag Persistence

TL;DR

The paper establishes a covering-based reduction of bipath persistence to infinite zigzag persistence, enabling translation of decomposition, computation, and stability results from zigzag persistence to bipath persistence. By constructing a covering map from the infinite zigzag poset $\mathbb{ZZ}$ to a bipath poset $B$ and forming the restricted module $R(M)$, the authors show that the barcode of $R(M)$ completely encodes the bipath arc code of $M$ (up to $\mathbb{Z}$-orbits). This bridge allows transferring the algebraic stability of zigzag (and block-decomposable) modules to bipath persistence, and reduces computation to a finite zigzag slice of size $O(K)$, where $K$ is the maximum chain length in the bipath. The work also introduces the fibered arc code, enriching invariants beyond the fibered barcode and providing finer distinctions between bipath modules, illustrated with concrete examples and comparisons to the interval rank invariant.

Abstract

Persistence modules that decompose into interval modules are important in topological data analysis because we can interpret such intervals as the lifetime of topological features in the data. We can classify the settings in which persistence modules always decompose into intervals, by a recent result of Aoki, Escolar and Tada: these are standard single-parameter persistence, zigzag persistence, and bipath persistence. No other setting offers such guarantees. We show that a bipath persistence module can be decomposed via a closely related infinite zigzag persistence module, understood as a covering. This allows us to translate techniques of zigzag persistence, like recent advancements in its efficient computation by Dey and Hou, to bipath persistence. In addition, and again by the relation with the infinite zigzag, we can define an interleaving and bottleneck distance on bipath persistence. In turn, the algebraic stability of zigzag persistence implies the algebraic stability of bipath persistence.

Figures (10)

• Figure 1: Posets that decompose as an interval
• Figure 2: Illustration of the covering map defined in \ref{['sec:bipath_as_zigzag']}
• Figure 3: Types of intervals of the bipath poset
• Figure 4: Illustration of an arc code
• Figure 5: Illustration of the covering map $\zeta\colon\mathbb{ZZ}\to B$. In this case, $n = m = 4$

Theorems & Definitions (17)

• Definition 2.1
• Lemma 2.2
• Theorem 2.3
• Definition 3.1
• Theorem 3.2: Block isometry theorem
• Lemma 3.3
• Lemma 3.4
• Theorem 3.5: Bipath isometry theorem
• Lemma 3.6
• proof