Computing Optimal Persistent Cycles for Levelset Zigzag on Manifold-like Complexes

TL;DR

This work tackles the problem of representing and computing optimal topological features across a levelset zigzag filtration induced by a piecewise-linear function on manifold-like complexes. It introduces levelset persistent cycles as sequences of $p$-cycles that track feature evolution between consecutive critical values, and formulates a polynomial-time framework that reduces optimal cycle computation to minimum cuts on dual graphs of weak $(p+1)$-pseudomanifolds. The authors develop algorithms for all four levelset zigzag interval types, proving their correctness and outlining two-phase strategies to handle monkey saddles, with complexity $O(n^2)$ dominated by max-flow. They also establish equivalence with classical levelset zigzag filtrations and demonstrate practical performance via implementations and experiments on triangular meshes, where the computed cycles capture intra-interval feature variations with good quality. The contributions advance practical, rigorous representations for zigzag persistence in levelset settings and provide a scalable toolkit for levelset-aware topological data analysis.

Abstract

In standard persistent homology, a persistent cycle born and dying with a persistence interval (bar) associates the bar with a concrete topological representative, which provides means to effectively navigate back from the barcode to the topological space. Among the possibly many, optimal persistent cycles bring forth further information due to having guaranteed quality. However, topological features usually go through variations in the lifecycle of a bar which a single persistent cycle may not capture. Hence, for persistent homology induced from PL functions, we propose levelset persistent cycles consisting of a sequence of cycles that depict the evolution of homological features from birth to death. Our definition is based on levelset zigzag persistence which involves four types of persistence intervals as opposed to the two types in standard persistence. For each of the four types, we present a polynomial-time algorithm computing an optimal sequence of levelset persistent $p$-cycles for the so-called weak $(p+1)$-pseudomanifolds. Given that optimal cycle problems for homology are NP-hard in general, our results are useful in practice because weak pseudomanifolds do appear in applications. Our algorithms draw upon an idea of relating optimal cycles to min-cuts in a graph that was exploited earlier for standard persistent cycles. Notice that levelset zigzag poses non-trivial challenges for the approach because a sequence of optimal cycles instead of a single one needs to be computed in this case. We show some empirical evidence that optimal cycles produced by our implemented software have nice quality.

Figures (11)

• Figure 1: Evolution of a homological feature across different critical points.
• Figure 2: A critical value $\alpha_i$ across which the 0th and 2nd homology stays the same; $f$ is defined on a 3D domain and $s_{i-1}$, $s_{i}$ are two regular values with $s_{i-1}<\alpha_i<s_{i}$. The levelset $f^{-1}(s_{i-1})$ is a 2-sphere where two antipodal points are getting close and eventually pinch in $f^{-1}(\alpha_{i})$. Crossing the critical value, $f^{-1}(s_{i})$ becomes a torus.
• Figure 3: A torus with the height function $f$ taken over the horizontal line. The 1st levelset barcode is $\{(\alpha^1_1,\alpha^1_4),[\alpha^1_2,\alpha^1_3]\}$. We list the first half of $\mathcal{L}^\mathsf{c}_1(f)$ but excluding $\mathbb{X}^1_{(0,1)}=\varnothing$; the remaining half is symmetric. An empty dot indicates the point is not included in the space.
• Figure 4: Finer triangulation makes the discrete levelset filtration equivalent with the continuous one.
• Figure 5: A sequence of levelset persistent 1-cycles for an open-open interval $(\alpha^1_1,\alpha^1_4)$; the complex (assuming the torus to be finely triangulated), the function, and the 1st critical values are the same as in \ref{['fig:torus']}.

Theorems & Definitions (49)

• Definition 1
• Definition 2: $p$-th homologically critical value
• Remark 1
• Definition 3: $p$-th levelset zigzag
• Remark 2
• Remark 3
• Definition 4: Simplex-wise levelset filtration
• Proposition 1
• Remark 4
• proof