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arxiv: 2606.06310 · v1 · pith:5UXWWS3Rnew · submitted 2026-06-04 · 💻 cs.CG · cs.MS· math.AT

RedZeD: Computing persistent homology by Reduction to Zero Differentials

Chris Kapulkin , Nathan Kershaw This is my paper

Pith reviewed 2026-06-27 22:34 UTC · model grok-4.3

classification 💻 cs.CG cs.MSmath.AT
keywords persistent homologyVietoris-Rips filtrationcomputational topologyreduction algorithmactive enumerationtopological data analysisalgorithm speedup
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The pith

RedZeD reformulates persistent homology as reduction to zero differentials to enable faster active enumeration on Vietoris-Rips filtrations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a new theoretical framework called Reduction to Zero Differentials that recasts the computation of persistent homology. Within this view the authors define an active enumeration procedure that identifies the same persistence pairs as the standard algorithm. The framework is applied specifically to Vietoris-Rips filtrations and is shown to produce identical output while running faster in many cases. A reader would care because persistent homology is a standard tool for extracting topological features from data sets, and any reliable reduction in running time expands the size of point clouds that can be analyzed.

Core claim

Persistent homology of a Vietoris-Rips filtration can be computed by successively reducing differentials to zero; the resulting RedZeD framework makes an active enumeration algorithm possible that returns exactly the same birth-death pairs as the classical persistence pairing algorithm yet runs faster on many inputs.

What carries the argument

The Reduction to Zero Differentials (RedZeD) framework, which recasts the boundary matrix operations of persistent homology so that active enumeration can locate the necessary reductions without exhaustive search.

If this is right

  • The computed persistence diagrams are identical to those produced by the classical algorithm.
  • Running time improves over the existing implementation for many Vietoris-Rips inputs.
  • Active enumeration becomes a viable replacement for the standard pairing step once the RedZeD view is adopted.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same reduction perspective could be tested on filtrations other than Vietoris-Rips to check whether the speedup generalizes.
  • An implementation of active enumeration might be combined with existing matrix-reduction libraries to measure concrete wall-clock gains on benchmark data sets.
  • If the framework extends cleanly, it could reduce the memory footprint required for large-scale topological data analysis pipelines.

Load-bearing premise

The RedZeD reformulation produces exactly the same persistence pairs as the standard algorithm without omitting cases or introducing algebraic errors.

What would settle it

A concrete Vietoris-Rips filtration on which the active enumeration algorithm reports a different set of birth-death pairs than the standard persistence pairing algorithm.

Figures

Figures reproduced from arXiv: 2606.06310 by Chris Kapulkin, Nathan Kershaw.

Figure 1
Figure 1. Figure 1: Ripser vs RedZeD VR runtimes for random distance(n) Not only is RedZeD VR faster in this case, but it is also scaling better than Ripser. For n = 200, the average runtime for Ripser is 1.93 seconds, whereas the average RedZeD VR runtime is 0.0335 seconds, which is roughly 58 times faster than Ripser. For n = 400, the largest we test on, the average runtime for Ripser is 227 seconds, and the average runtime… view at source ↗
Figure 2
Figure 2. Figure 2: Ripser vs RedZeD VR runtimes for noisy circle(n, 0.1) While RedZeD VR is not scaling any better in this case, it remains faster roughly by a constant factor just under 2. For n = 2500, the Ripser runtime was 46.7 seconds, and the RedZeD runtime was 20.8 seconds. Next, we look at varying n for stacked circles(n, 20, 0.1), once again computing in degree one. Note that here our datasets will be of size 20n, s… view at source ↗
Figure 3
Figure 3. Figure 3: Ripser vs RedZeD VR runtimes for stacked circles(n, 20, 0.1) 23 [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Ripser vs RedZeD VR runtimes for random euclidean(n, 2) This is the first case in our tests where Ripser outperforms RedZeD VR. As stated at the beginning of this section, this is likely due to the lack of topological signal in this case. Ripser is faster by a constant factor just under 2 in this case. For n = 5000 the runtime for Ripser is 8.67 seconds and the runtime for RedZeD VR is 12.9 seconds. The fi… view at source ↗
Figure 5
Figure 5. Figure 5: Ripser vs RedZeD VR runtimes for random euclidean(n, 10) Interestingly, when the dimension of the ambient space increases, RedZeD VR is faster again, and also is once again scaling better. For n = 5000, the Ripser runtime is 33.0 seconds, and the RedZeD VR runtime is 14.8 seconds. We see that in most cases in homology degree 1, RedZeD VR is faster than Ripser. The degree to which RedZeD VR is faster appear… view at source ↗
Figure 6
Figure 6. Figure 6: Ripser vs RedZeD VR runtimes for random euclidean(n, 10) in degree 2 We see that RedZeD VR is slower than Ripser by a constant factor of roughly 2. For n = 600, the Ripser time was 11.4 seconds and the RedZeD VR time was 22.4 seconds. We next test random distance(n) in degree two, which can be seen in [PITH_FULL_IMAGE:figures/full_fig_p026_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Ripser vs RedZeD VR runtimes for random distance(n) in degree 2 In this case, RedZeD VR is faster and scales better than Ripser. For n = 180, the Ripser time was 382 seconds and the RedZeD VR time was 4.39 seconds. We find that for most other cases where the 26 [PITH_FULL_IMAGE:figures/full_fig_p026_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Ripser vs RedZeD VR runtimes for random distance(n) in degree 3 Again, we find RedZeD VR scaling better than Ripser. While slower for smaller datasets, it is still faster as n gets larger. For n = 100, the average Ripser runtime was 65.2 seconds, and the average RedZeD VR runtime was 4.04 seconds. We do not test in any higher homology degrees, since neither method is able to compute on any sizable datasets… view at source ↗
Figure 9
Figure 9. Figure 9: Computation times on standard benchmark datasets [PITH_FULL_IMAGE:figures/full_fig_p027_9.png] view at source ↗
read the original abstract

We introduce a new algorithm for computing persistent homology of Vietoris--Rips filtrations, which in many cases offers a considerable speedup over the existing implementation of the persistence pairing algorithm. The key innovation, called active enumeration, is made possible by a new theoretical framework of Reduction to Zero Differentials (hence RedZeD) in which to view persistent homology.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The manuscript introduces RedZeD, a new theoretical framework for persistent homology based on reduction to zero differentials, which enables an active enumeration algorithm for computing the persistence pairing on Vietoris-Rips filtrations. The central claim is that this approach yields identical results to the standard persistence pairing algorithm while providing considerable speedup in many cases.

Significance. If the reformulation is correct and the speedup holds under standard validation, the work could offer a practical improvement for persistent homology computations, which are widely used in topological data analysis. The new framework provides an alternative perspective on the problem that directly supports the algorithmic innovation.

minor comments (1)
  1. The abstract and introduction would benefit from explicit cross-references to the sections containing the formal definition of the RedZeD framework and the pseudocode for active enumeration to improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The provided abstract and context describe the introduction of a new theoretical framework (RedZeD) and algorithm (active enumeration) for persistent homology computation. No equations, derivations, self-citations, or reformulations are quoted that reduce a claimed result to its own inputs by construction, fitted parameters renamed as predictions, or load-bearing self-citations. The central claim is presented as a novel reformulation enabling speedup, with no evidence of circular steps in the available text; the derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no free parameters, axioms, or invented entities are explicitly detailed or required for the central claim.

pith-pipeline@v0.9.1-grok · 5576 in / 969 out tokens · 20914 ms · 2026-06-27T22:34:18.242474+00:00 · methodology

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