A differentiability framework for zigzag persistent homology via linear interpolation

1 INTRODUCTION

• Zigzag persistence tracks features across complexes connected by inclusions in either direction, but no gradient passes through its barcode into a learning dynamic.Standard persistent homology instead takes a filtration of one fixed complex examined at all scales.Source: §1 · p. 1
• Standard barcode stability is tied to filtration perturbations, while zigzag stability is generally stated at module level, leaving no general filtering-function framework for zigzag optimization.The cited filtering-function stability result applies only to levelset zigzag.Source: §1 · p. 1
• The framework unrolls complex sequences into simplex-wise zigzag filtrations, then obtains those sequences by thresholding time-dependent filtering values on a fixed complex.Barcode endpoints are assigned the real times when linearly interpolated filtering values cross the threshold, yielding smooth local lifts and differentials by the chain rule.Source: §1 · pp. 1–2
• The diagram map is locally Lipschitz outside an explicit exclusion set, but this does not provide stochastic subgradient convergence guarantees; optimization is tested in two experiments.The experiments minimize sensor-network coverage defect persistence and classify dynamic graphs using a learnable zigzag persistence feature map.Source: §1 · p. 2

2.1 ZIGZAG PERSISTENT HOMOLOGY

• A zigzag filtration is a finite sequence of simplicial complexes with inclusions in either direction, and it is simplex-wise when adjacent complexes differ by one simplex.Applying homology gives a zigzag persistence module that decomposes into interval modules, whose birth-death pairs form the persistence diagram.Source: §2.1 · p. 2
• The paper treats finite sequences of subcomplexes with empty endpoints and distinct consecutive complexes as primary objects, even when consecutive complexes are not nested.It unrolls each sequence into a simplex-wise zigzag filtration; the resulting diagrams agree regardless of the unrolling choice, and omitting intermediate complexes removes artificial discontinuities and shrinks the non-locally-Lipschitz region.Source: §2.1 · pp. 2–3

2.2 DIFFERENTIABILITY OF MAPS TO AND FROM THE SPACE OF PERSISTENCE DIAGRAMS

• The persistence-diagram space is a quotient of ordered diagram coordinates by permutations, allowing diagrams locally to be represented by Euclidean vectors.For a differentiable diagram-valued map, a local lift into ordered coordinates supplies a differential; differentiable maps out of diagrams compose with it by the chain rule.Source: §2.2 · p. 3
• The chain rule composes the differential of a diagram-valued map with that of a differentiable map from diagrams, independently of the chosen lift.This framework is the tool the paper combines with zigzag persistent homology.Source: §2 · p. 2, §2.2 · p. 3

3.1 PERSISTENCE DIAGRAM-VALUED MAPS FROM SEQUENCES OF COMPLEXES

• Sequences are unrolled into valid simplex-wise zigzag filtrations by ordering simultaneous face and coface events, then persistence pairings map event indices to event times.Within a cell of fixed combinatorial structure, the diagram map acts as a permuted coordinate projection and admits a local lift when the parameterized sequence remains in that cell.Source: §3.1 · pp. 3–4

3.2 TEMPORAL CONTINUITY VIA LINEAR INTERPOLATION

• Thresholding time-dependent filtering values gives fixed-time complex sequences, while linear interpolation assigns each entry or exit event its real threshold-crossing time.Within a cell, the crossing-time map is smooth; under the stated neighborhood and regularity conditions, this yields a differentiable local lift and an explicit differential.Source: §3.2 · pp. 5–6

3.3 OBJECTIVE OPTIMIZATION

• For definable filtering-function parametrizations, the persistence diagram map and a differentiable diagram loss are differentiable almost everywhere, with differentials obtained by the chain rule.The result applies outside a measure-zero set in parameter space.Source: §3.3 · p. 6
• The diagram map is locally Lipschitz outside an explicit null set, but the authors cannot guarantee a lower bound on how much of parameter space avoids that set.Consequently, stochastic subgradient convergence guarantees do not follow directly, although the paper reports that the lack of local Lipschitzness does not appear to hinder convergence in its examples.Source: §3.3 · p. 7

3.4 APPLICATIONS

• Upper-star and Vietoris–Rips extension maps produce valid filtering functions from vertex values or moving point clouds, respectively.Both extensions are semialgebraic, so the almost-everywhere differentiability result applies and the loss differential can be computed by the classical chain rule.Source: §3.4 · p. 7

4.1 OPTIMIZING HOW LONG A COVERAGE DEFECT LASTS

• For coverage optimization, three robots move for 30 frames, and the lifetime objective penalizes only defects exceeding a tolerance of six frames.The total objective also penalizes leaving the lattice and deviating from unit speed.Source: §4.1 · p. 8
• The coverage objective decreases and stabilizes across seeds during 100,000 stochastic subgradient iterations, with Gaussian perturbations decreasing from 0.3 to 0.0008 lattice units.The experiments do not discuss solution optimality because the stated convergence result guarantees critical points rather than global optima.Source: §4.1 · pp. 7–8
• The lifetime strategy reaches a topological loss of 33.8 and median longest defect of 9.4 frames; without the topological term, these rise to 530.9 and 21.3 frames.The count strategy reaches total bar length 211.0 frames, versus 246.7 without the topological term.Source: §4.1 · p. 8

4.2 CONTROLLED DEMONSTRATION OF A LEARNED ZIGZAG FEATURE MAP

• The classification task uses two switching figure-eights: the signal’s alternation rate determines the label, while a decoy alternates at the opposite rate.The model applies a graph convolution, converts the time-varying vertex function to a zigzag barcode and persistence image, then predicts the label.Source: §4.2 · p. 9, Figure 3 · p. 9
• End-to-end training reaches accuracy 0.995 ± 0.016, while the reader with its uninformative filtering function frozen stays at 0.540 ± 0.086.With a filter keeping only the signal figure-eight, the task is solvable through zigzag persistence but not per-frame persistence.Source: §4.2 · p. 9

5 CONCLUSIONS

• Threshold-crossing times from linearly interpolated filtering values yield smooth local lifts and almost-everywhere differentials for zigzag persistence.The results assume definable parametrizations and establish local Lipschitzness only outside an explicit exclusion set.Source: §5 · p. 9
• Local Lipschitzness outside the exclusion set falls short of a hypothesis required for stochastic subgradient convergence guarantees.The paper does not close this gap.Source: §5 · p. 9
• Experiments use the gradients in a topological loss and a learnable feature map, consistently driving the loss to low values in two controlled settings.The authors suggest the framework can support zigzag-based optimization in different machine-learning scenarios.Source: §5 · p. 9

CODE AVAILABILITY

• The methods are implemented in the open-source Python package zz-top, which includes a zigzag persistence engine and differentiable read-out.The authors state that experiment code and configurations for regenerating tables and figures will be added to the repository.Source: CODE AVAILABILITY · p. 10

AI USE STATEMENT

• The authors report using generative AI for writing, literature and dataset discovery, experiment planning, code, and synthetic-data generation, with outputs reviewed and verified.Source: AI USE STATEMENT · p. 10

A RELATED WORK

• Differentiable persistent-homology layers and learned filtrations have been developed for fixed complexes with monotone filtrations, while zigzag applications commonly use hand-designed filtrations.The paper identifies derivatives of barcodes with respect to the values generating a sequence as the missing capability.Source: A RELATED WORK · pp. 14–15
• Zigzag persistence tracks features through sequences with insertions and deletions, but prior algorithms compute or maintain barcodes rather than differentiating their endpoints with respect to filtering values.The construction reads gradients through returned endpoints without changing the pairing.Source: A RELATED WORK · pp. 14–15
• Existing zigzag stability results bound distances between already-given modules, not barcode changes under filtering-value perturbations that can create or destroy positive-length bars.The paper describes the resulting exclusion set and discontinuities in Section 3.3 and Appendix C.6.Source: A RELATED WORK · p. 15

B.1 WHITNEY STRATIFICATIONS AND O-MINIMAL GEOMETRY

• O-minimal structures include sets closed under finite unions, intersections, and complements, and definable functions are Whitney-stratifiable; structures with exponentials and restricted analytic functions cover examples such as sigmoids and Gaussian kernels.Source: §B.1 · p. 16

B.2 THE STOCHASTIC SUBGRADIENT METHOD

• Stochastic subgradient descent updates parameters opposite noisy Clarke subgradients using positive step sizes and random noise.A locally Lipschitz Whitney-stratifiable objective, standard step-size conditions, bounded iterates, and unbiased noise with bounded variance underpin the stated convergence result.Source: §B.2 · pp. 16–17
• Under the stated assumptions, every limit point is critical and the objective values converge to the objective at a limit point.Source: §B.2 · p. 17

B.3 BOTTLENECK DISTANCE

• Bottleneck distance is the smallest matching radius for diagram points, allowing unmatched finite points to match the diagonal while essential points must match each other.Source: §B.3 · p. 17

C.1 SEQUENCE REPRESENTATIONS

• A sequence representation encodes a valid simplex-wise event sequence with nondecreasing time stamps; a fixed linear extension gives each sequence a unique ordered representation.Validity requires insertions to have their faces present and deletions to have no cofaces present.Source: §C.1 · pp. 17–19
• Swapping simultaneous events on distinct face-unrelated simplices preserves the persistence diagram after diagonal points are removed, so valid representations of one sequence yield the same diagram.Source: §C.1 · pp. 19–20
• Detours can be removed without changing the diagram off the diagonal only for specified simplex dimensions and face-unrelated intervening events; other detours can alter positive-length bars.The persistence diagram in dimension p depends only on events involving p- and (p+1)-simplices.Source: §C.1 · pp. 20–21

C.2 PROOFS OF SECTION 3.1

• The local lift expresses the persistence diagram through a fixed coordinate projection of the interpolated crossing times.The projection is unchanged across filtering functions in the same cell.Source: §C.2 · p. 21
• If the parametrized filtering values are C^r, the local lift is C^r and the diagram map is r-differentiable.For r ≥ 1, its differential follows by applying the classical chain rule to the lift.Source: §C.2 · p. 22

C.3 PROOFS OF SECTION 3.2

• Phantom crossings are paired exit-and-entry crossings at the same integer time, which cancel and are excluded from the event sequence.Genuine crossings of the same simplex never share a time, so the sequence has no detours.Source: §C.3 · pp. 22–23
• The sequence S(v) consists of the subcomplexes present above threshold, changing only at genuine crossings and remaining right-continuous and piecewise constant.The sequence begins and ends empty; phantom crossings do not produce events.Source: §C.3 · pp. 22–23
• The finite partition into semialgebraic cells preserves crossing types, phantom status, and the global order and ties of genuine events.Consequently, each cell admits a fixed combinatorial event sequence and pairing, while crossing times vary with the filtering values.Source: §C.3 · pp. 23–24
• Within a cell, the resulting local lift is C^r when the parametrized filtering values are C^r, and its differential follows by the chain rule.The construction also extends to ordered, non-integer time indices by changing the interpolation formula.Source: §C.3 · p. 24

C.4 PROOFS OF SECTION 3.3

• For definable parametrizations, the map is C^r on strata, and the complement of the open dense top-dimensional strata is a definable measure-zero set.The result applies for finite r, or for r = ω when the o-minimal structure has analytic decomposition.Source: §C.4 · p. 25
• The proof obtains this regularity by partitioning the parameter domain into cells compatible with the finite semialgebraic partition of filtering-function space.On each top-dimensional stratum, the parametrization stays in one cell, allowing the local-lift differentiability result to apply.Source: §C.4 · p. 25

C.5 SUBGRADIENT COMPUTATION

• Subgradient evaluation first computes crossings and pairings without gradient tracking, then recomputes crossing times differentiably and backpropagates through that phase.Differentiability is used for the time map and loss composition, not for the discrete pairing step.Source: §C.5 · p. 26
• Small Gaussian perturbations make iterates land outside the measure-zero non-differentiability set almost surely, where the computed gradient is a Clarke subgradient.This is the strategy used in the experiments alongside the two-phase evaluation.Source: §C.5 · p. 26

C.6 THEOREM 3.9

• Ordinary and merged crossing-time functions are smooth with bounded derivatives near the reference filtering function, while transient-pair branches agree on their shared boundary.These cases provide a common local Lipschitz bound for all tracked crossing times.Source: §C.6 · pp. 27–28
• The proof tracks a finite set of ordinary, merged, and transient crossings; the active event set may vary, but distinct reference times remain separated locally.Simultaneous crossings near the reference point must already be simultaneous there.Source: §C.6 · p. 28
• Along a segment, finitely many changes in active crossings and event order divide the path into intervals with fixed admissible ordering data.At interval boundaries, transient pairs collapse to equal-time events and can be removed without changing the persistence diagram.Source: §C.6 · pp. 29–30
• The intervalwise bounds sum to show that PHp composed with S is locally Lipschitz outside the exclusion set Np.The reduction extends the estimate from the relevant p- and (p+1)-dimensional simplices to the full filtering function.Source: §C.6 · pp. 27, 30

C.7 REGION OF POTENTIAL NON-LOCAL LIPSCHITZNESS

• An arbitrarily brief disappearance can split a class’s interval into two bars of non-vanishing length, producing a discontinuity.In the example, an arbitrarily small perturbation turns a phantom exit–re-entry pair into genuine crossings and breaks the bar in two.Source: §C.7 · p. 30
• When consecutive filtering values equal the threshold, an arbitrarily small perturbation can create a bar whose bottleneck distance remains above 1/2.The unperturbed diagram is empty, while the perturbation creates a bar whose length tends to 1.Source: §C.7 · p. 30
• The potential non-Lipschitz set has measure zero, and its complement is open and connected, although membership in the set does not guarantee failure.The examples show discontinuities can occur when an arbitrarily short disappearance splits a bar or a small perturbation creates a bar of non-vanishing length.Source: §C.7 · pp. 30–31

D.1 RELATED WORK

• The coverage criterion uses a fixed per-cell cycle and an external evaluator, unlike the coordinate-free relative-homology construction of de Silva & Ghrist (2007; 2006).The fixed cycle uses known sensor positions; coverage gaps are checked separately by the evaluator.Source: §D.1 · p. 32
• Gamble et al. (2015) track coverage holes with zigzag barcodes; this work differentiates through the barcode to optimize trajectories, unlike an instantaneous spectral objective.The comparison with Yadokoro & Bhattacharya (2023) distinguishes scheduling defect lifetimes from static redistribution of a mobile team.Source: §D.1 · p. 32

D.2 EXPERIMENTAL SETUP

• Each of the nine cells is a nonbounding H1 class, and three robots can close at most a third of the arena at once, making hole-time distribution the objective.The arena uses sixteen fixed sensors on a 4-by-4 lattice.Source: §D.2 · p. 32, Figure 7 · p. 33
• Coverage is checked through pointwise Rips–Čech inclusion, which certifies one cycle at one instant but gives no bar-by-bar lifetime correspondence; the barcode remains the optimization objective.The physical sensing radius is used by the evaluator, not to define the barcode objective.Source: §D.2 · p. 32

D.3 EXPERIMENT EVALUATION AND CONTROLS

• The primary physical metric is the maximum over 14,641 probes of the longest coverage gap, independently evaluated at the final iterate.Barcode quantities are read from the diagram computed by the loss on 88 time stamps.Source: §D.3 · p. 34
• In every run, the longest barcode bar exceeded the maximum coverage gap by a median 0.38 frames (lifetime) and 0.21 frames (count).The respective ranges were 0.04–1.87 and 0.00–1.14 frames.Source: §D.3 · p. 34
• Reported results use held-out seeds 10–19, with settings selected on development seeds 0–3 across 30 configurations.Medians are reported across seeds; 95% bootstrap intervals estimate paired within-seed differences.Source: §D.3 · p. 34
• Step sizes at or above 3 · 10^-2 diverged, while steps at or below 3 · 10^-3 left the longest coverage gap above 12 frames.The penalty weight gives curvature 80 and a stability limit near 0.025.Source: §D.3 · p. 34
• At γ0 = 10^-2 and 30,000 steps, the development median improved with noise σ through 30, then worsened at σ = 60 and 100.The reported medians were 11.6, 11.5, 9.6, and 8.8 frames at σ = 0.5, 2, 5, and 30, then 9.3 and 11.4 at σ = 60 and 100.Source: §D.3 · p. 34
• Allowing robots to slow down reduced travel from 87 to 68 units without improving loss, so the speed bound was rejected.The change would undermine the equal-effort comparison between objectives.Source: §D.3 · p. 34
• Extending the selected budget from 30,000 to 100,000 steps changed the held-out metric by −0.20 frames [−0.71, +0.41], lower in 8 of 10 seeds.The reported interval includes zero.Source: §D.3 · p. 34

D.4 EXTENDED RESULTS

• As ∆ rises from 6 to 12, lifetime-strategy longest bars rise from 9.4 to 13.8 frames and remain 1.8–3.4 frames above target.Table 1 reports the three-robot experiment’s count and lifetime strategies, with four tolerances for the latter.Source: Table 1 · p. 35, §D.4 · p. 35
• With more robots, the lifetime-objective result moves closer to its target, as covering the holes becomes easier.Table 2 varies robot count while holding ∆=6.Source: Table 2 · p. 35, §D.4 · p. 35
• Removing noise shifts the median longest bar from 9.4 to 10.6 frames, with a paired interval spanning zero, but sharply widens seed spread; removing topology yields no topological descent.Against motion-only optimization, lifetime shortens the longest bar by a median 12.2 frames, while count changes it by −1.3 frames.Source: §D.4 · p. 35, Table 3 · p. 35

D.5 THE LOSS ALONG THE ITERATE SEQUENCE

• The count term falls 14%, from 244.8 to 211.0, but a hard floor remains because at least six of nine cells are open at every instant.The total loss ends 1.1 above the topological term after penalties decay.Source: §D.5 · p. 36
• The lifetime term falls by factors of 15, 18, 27, and 18 at ∆=6, 8, 10, and 12, respectively.The corresponding changes are 507.9→33.8, 310.7→17.2, 180.1→6.7, and 96.9→5.4; final longest bars remain above their targets.Source: §D.5 · p. 36

D.6 COMPUTATIONAL COST.

• At ∆=6, three robots, and 100,000 steps, a run takes a median 1.5 hours on one CPU core and peaks at 350 MB of memory.The zigzag layer’s forward pass takes 10 ms per step; rebuilding the ambient complex takes 23 ms and is described as an implementation artifact.Source: §D.6 · p. 36

• Each example pairs a signal figure-eight with a decoy running the opposite switching pattern; the label identifies whether the signal alternates every 16 or every 2 frames.The two classes have identical per-frame Betti curves, while their zigzag barcodes differ under the generator-known filtering function.Source: §E.1 · p. 37, Figure 11 · p. 38
• The zigzag read-out converts a thresholded barcode into four persistence-image channels using separate H0 and H1 weightings.The field uses the upper-star rule, with threshold ϵ = 0.5; the reader maps the images to two logits.Source: §E.1 · p. 38
• The learned zigzag trains a frame-shared graph convolution and reader, while fixed-field and signal-only controls train only the reader.Per-frame variants read separate persistence images at each frame, without matching loops across frames; all variants use the same reader.Source: §E.1 · p. 38
• Training uses cross-entropy with λ = 10^-4; stage 1 fits the reader on the frozen initial field, and stage 2 runs stochastic subgradient descent.The graph-convolution weights are rescaled during training so their steps are larger despite its smaller parameter count.Source: §E.1 · p. 39
• Stage-2 settings were fixed using four development draws, separate from the draws used for reported results.The caption identifies Table 4 as the settings and rationale record.Source: Table 4 · p. 39
• Evaluation uses twenty held-out splits, with four seeds per learned variant; the run with the lowest final validation loss is kept for each split.A split is solved when test accuracy exceeds 0.9, and reported results use final iterates.Source: §E.1 · p. 39

E.2 EXTENDED RESULTS

• At a shared filtering function, signal-only zigzag separates the classes completely without variance, while per-frame persistence scores 0.500 against 1.000.The signal-only pair differs only in whether features are matched across time; the task is designed so that this pairing separates the classes.Source: §E.2 · p. 39, Table 5 · p. 40
• The kept runs’ mean validation loss falls from 0.55 to 0.024, reaching 0.078 at iteration 1 000, while final training loss is 0.023.The training loss follows validation loss closely; the reported curves are logarithmic and average the selected runs across twenty splits.Source: §E.2 · p. 39, Figure 12 · p. 40
• 22 of 80 runs fail to converge; faster decay preserves found solutions but leaves more runs at the uninformative start, so no fixed admissible schedule removes both failure modes.The authors state that an adapting step size would be needed, outside the schedules covered by Proposition B.7.Source: §E.2 · p. 40

E.3 COMPUTATIONAL COST

• A learned zigzag run takes a median of 3.1 minutes, compared with 71 minutes for per-frame persistence with a learned field.The per-frame variant’s 32 separate persistence computations account for its higher per-network cost.Source: §E.3 · p. 40