Apex Representatives

TL;DR

By embedding the zigzag into a Mayer–Vietoris pyramid generated by a level-set of a real-valued function, the paper develops apex representatives that yield barcode representations for the input zigzag. An efficient Lift-Cycle construction lifts ordinary-persistence cycles to apex representatives, which then map back to the zigzag, giving compatible, apex-based bases. The lifting cost for a $p$-cycle is $O(p m \log m)$ and zigzag representatives can be recovered in $O(\log m + C)$ time, where $C$ is the output size. This framework unifies zigzag persistence with level-set and extended-persistence theory, enabling scalable, exact barcode representations for sequential data.

Abstract

Given a zigzag filtration, we want to find its barcode representatives, i.e., a compatible choice of bases for the homology groups that diagonalize the linear maps in the zigzag. To achieve this, we convert the input zigzag to a levelset zigzag of a real-valued function. This function generates a Mayer-Vietoris pyramid of spaces, which generates an infinite strip of homology groups. We call the origins of indecomposable (diamond) summands of this strip their apexes and give an algorithm to find representative cycles in these apexes from ordinary persistence computation. The resulting representatives map back to the levelset zigzag and thus yield barcode representatives for the input zigzag. Our algorithm for lifting a $p$-dimensional cycle from ordinary persistence to an apex representative takes $O(p \cdot m \log m)$ time. From this we can recover zigzag representatives in time $O(\log m + C)$, where $C$ is the size of the output.

Figures (4)

• Figure 1: A zigzag on the top. A total complex $K$ on the left. The support of each simplex in a zigzag, $T(\sigma)$, shown as an interval. The corresponding prism $\overline{K}$ on the bottom. A 1-cycle in $K[11,17]$ and its lift in $\overline{K}[11,17]$ are highlighted in red. The lifted cycle is an apex representative; its slices (pairs of vertices in red) are representatives of a bar in the original zigzag.
• Figure 2: Left: Mayer--Vietoris Pyramid arranges spaces $\mathbb{X}[b,d], \mathbb{X}(b,d], \mathbb{X}[b,d), \mathbb{X}(b,d)$ in a way where every diamond in the pyramid belongs to the Mayer--Vietoris long exact sequence. Right: Four types of flush diamond indecomposables in the infinite strip of homology groups; the apex of each diamond is marked.
• Figure 3: A $\Delta$-complex $K$, with the times $T(\sigma)$ of its simplices illustrated, together with the corresponding prism $\overline{K}$ and its projection $\overline{f}$ onto the second coordinate.
• Figure 4: Coefficients of some $(p-1)$-dimensional simplex $\sigma$ in cycle $w$ in \ref{['alg:lift-cycle']} and in the boundary of its cofaces $\tau_1, \tau_2, \tau_3, \tau_4$. The coefficients in the cycle change exactly when \ref{['alg:lift-cycle']} encounters the chosen times $t_{\tau_i}$ of the cofaces. \ref{['alg:lift-cycle-reinterpreted']} takes the $\sigma$-centric view and instead of keeping track of the full cycle $w$ computes the coefficients of $\sigma$ on the different intervals. We note that in this example $\sigma \notin w_\textrm{init}$, but $\sigma \in w_\textrm{final}$. In particular, we are lifting a relative cycle.

Theorems & Definitions (28)

• Lemma 1: Lemma 1 in NM24
• Remark 2
• Corollary 3
• Remark
• Remark 4
• Remark
• Remark
• Claim 5
• Claim 6
• Remark