The distilled Vietoris Rips filtration for persistent homology and a new memory efficient algorithm

Katharine Turner; Musashi Ayrton Koyama; Vanessa Robins

REVIEW 3 major objections 5 minor 2 cited by

The distilled Vietoris Rips filtration for persistent homology and a new memory efficient algorithm

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The distilled Vietoris–Rips filtration carries the same degree-q persistent homology as the standard Vietoris–Rips filtration while using far fewer simplices.

desk verdict The central theorem is false as stated: a four-point square gives H_1(D^1_1)=0 while H_1(R^1_1)=Z_2, though the distilled filtration idea is novel and worth a second look. read the letter →

arxiv 2412.07805 v1 pith:QKEJDAHK submitted 2024-12-09 math.AT

Musashi Ayrton Koyama , Vanessa Robins , Katharine Turner This is my paper · ORCID

classification math.AT MSC 55N3155U10

keywords distilledVietoris–RipsfiltrationpersistenthomologydiscreteMorsetheoryreducedcomplexluneapparentpairsparallelalgorithmmemoryefficiency

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Persistent homology of a point cloud is usually read off the Vietoris–Rips filtration, whose memory cost grows so fast that degree-1 computations on 100,000-point clouds are not feasible on ordinary machines. This paper aims to establish that a much smaller filtration, the distilled Vietoris–Rips filtration, has the same degree-q persistent homology as the standard Vietoris–Rips filtration for every finite metric space and every q>0. The distilled complex is built from the reduced Vietoris–Rips complex by keeping only simplices reachable from critical (q+1)-simplices of a discrete Morse matching, so it discards simplices that cannot affect the degree-q barcode. If the proof is correct, computing on the distilled complex reproduces the same barcode while storing far fewer simplices, and the paper's degree-1 algorithm can build it directly from the pairwise distance matrix with a highly parallel loop.

What carries the argument

The carrying object is the acyclic partial matching $\mu_q$ on the reduced Vietoris–Rips complex. A $q$-simplex $\sigma$ with nonempty lune is paired with the $(q+1)$-simplex obtained by adding the earliest 0-simplex in its lune; these are apparent pairs, meaning the face appears latest among faces of the coface and the coface earliest among cofaces of the face, so the matching is acyclic and respects the filtration. The distilled complex is generated by the reachable sets $\operatorname{reach}(\sigma)$ of the critical $(q+1)$-simplices, where $\operatorname{reach}$ is defined through the directed graph $G_{q+1}(\mu_q)$ of the matching. Discrete Morse theory supplies the closure maps $\varphi_q$ and decomposes the chain complex into a critical subcomplex and an acyclic matched part, so the homology of the critical subcomplex is the homology of the whole complex; Lemma 3.8 transfers this to the distilled complex by identifying its critical $q$-cycle group with that of the reduced complex.

What would settle it

A concrete way to test the claim is to enumerate small finite metric spaces and compare the degree-1 barcode of $D^1_\bullet(X)$ with the standard Vietoris–Rips barcode. If a space is found where the barcodes differ, or where a $1$-cycle in $\operatorname{Crit}_\ast(R^1_r(X))$ is only a boundary at a later scale through a critical $1$-simplex that lies in no $\operatorname{reach}(\sigma)$ for a critical $2$-simplex $\sigma$, Lemma 3.8 fails and the theorem is false.

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Extended reading notes

Core claim

The central claim is Theorem 3.9: for every finite metric space $X$ and every $q>0$, the degree-$q$ persistent homology of the distilled filtration $D^q_\bullet(X)$ is isomorphic to the degree-$q$ persistent homology of the reduced filtration $R^q_\bullet(X)$, and by Theorem 2.42 the latter is the standard Vietoris–Rips persistent homology. The distilled complex is defined from a discrete Morse matching $\mu_q$ on $R^q_\bullet(X)$: let $A$ be the union, over all critical $(q+1)$-simplices $\sigma \in R_{q+1}(\mu_q)$, of $\operatorname{reach}(\sigma)$, the simplices reachable from $\sigma$ in the matching's directed graph; then $D^q_r(X)$ consists of the elements of $A$ of diameter at most $r$ together with all their faces. Lemma 3.8 asserts the equality $Z_q(\operatorname{Crit}_\ast(D^q_r(X),\mu_q)) = Z_q(\operatorname{Crit}_\ast(R^q_r(X),\mu_q))$ of cycle groups, and Theorem 3.9 combines this with discrete Morse theory (Theorem 2.64) to give the persistence isomorphisms. The result is meant to let practitioners compute the same barcode from a complex with far fewer simplices.

Load-bearing premise

The proof leans on the assumption that no degree-q cycle is lost when cutting the reduced complex down to the distilled complex; this works only if every specially selected q-simplex that shows up in the boundary of a later specially selected (q+1)-simplex is included as a face of one of the kept (q+1)-simplices.

Editorial extensions

If this is right

For any $q>0$, the degree-$q$ barcode of $D^q_\bullet(X)$ equals the degree-$q$ barcode of the standard Vietoris–Rips filtration, so downstream analysis can use the smaller complex without changing results.
The distilled filtration can be constructed from pairwise distances alone, so it applies to arbitrary finite (semi-)metric spaces, not just point clouds in Euclidean space.
For degree 1, Algorithm 1 builds $D^1_\bullet(X)$ one edge at a time and parallelizes the main loop over $m$ machines, with total complexity bounded by $O\left(\frac{1}{m}((n^2-b(X))n + b(X)n^4\log n)\right)$, where $b(X)$ counts edges whose lune has more than one component.
Before the final persistence reduction, the algorithm needs to store only the distance matrix and the 2-simplices of $D^1_\bullet(X)$; the 1-simplices are read off as faces.
Under Conjecture 4.1, $D^1_\infty(X)$ for samples from a $k$-dimensional manifold in Euclidean space has $O(kn)$ simplices, which would make memory use linear in the number of points.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An extension the paper leaves implicit: if Conjecture 4.1 holds for higher degrees, the same construction could bring degree-2 and degree-3 Vietoris–Rips computations on large clouds within reach of a single machine; the paper only presents evidence for degree 1.
A general recipe suggested by the proof: any acyclic, filtration-respecting matching whose critical simplices' reachable sets are closed under the Morse boundary should yield a distilled subcomplex with the same persistent homology, so testing other collapse schemes could produce even smaller complexes.
A testable prediction: because lune connectivity controls the size of $D^1$, metric spaces with high doubling dimension or many separated local neighborhoods may need more simplices, so memory savings should degrade exactly where the lune components proliferate.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The headline: the main theorem is false as stated, and it is not just a gap in a proof. Take the four corners of the unit square. The reduced Rips complex R^1_1 contains the four side edges forming a 1-cycle, so H_1(R^1_1) = Z_2. But the distilled complex D^1_1 is empty: the only critical 2-simplices are the two triangles of diameter sqrt(2), so A contains only those and their faces, none of diameter <=1. Hence H_1(D^1_1)=0, contradicting Theorem 3.9. The failure is exactly the unproved step in Lemma 3.8: a 1-cycle can be born from edges that are not faces of any A-simplex at its birth scale.

The paper is not without merit. The distilled filtration is a genuinely new construction, and the idea of using apparent pairs as an acyclic matching to prune the reduced complex is worth taking seriously. The exposition is clear, and the paper does a good job of collecting the Morse-theoretic machinery. The algorithm section is honest about being highly parallel rather than fast on a single core, and the O(n^6 log n) bound is at least explicit.

The soft spots are not minor. Beyond the counterexample, the memory-efficiency claim rests on Conjecture 4.1, which is supported only by small point-cloud experiments and no code or data. The paper also leans on Theorem 2.42 from the authors' companion preprint [5]; that is a dependency, not circularity, but it means the main result stands on unrefereed ground twice.

For a reader working on efficient VR persistence, the construction is interesting enough to want a fixed version. As written, though, the central isomorphism is false, and the algorithmic claims are unsubstantiated. I would send it to a referee if the authors can repair Lemma 3.8; otherwise it is a counterexample, not a theorem. For now, do not cite it as a positive result. Worth a reading-group discussion, because the failure mode is instructive.

Referee Report

3 major / 5 minor

Summary. The paper introduces the 'distilled Vietoris-Rips filtration' D^q_•(X), defined from the reduced Vietoris-Rips complex R^q_•(X) and an acyclic matching induced by apparent pairs. It claims (Theorem 3.9) that H_q(D^q_r(X)) is isomorphic to H_q(R^q_r(X)) compatibly with the filtrations, so that the distilled complex has the same degree-q Vietoris-Rips persistence barcode while using fewer simplices. The paper also presents a parallel algorithm for degree-1 persistence based on this construction, with a complexity analysis and numerical evidence for the size of D^1_∞(X).

Significance. The proposed construction is creative and, if correct, would be a meaningful step toward memory-efficient persistent homology computation for large point clouds. The use of a discrete Morse matching on the reduced Rips complex and the idea of keeping only reachable critical (q+1)-simplices are interesting. However, the central theorem is false as stated, and the proof of the key lemma (Lemma 3.8) contains an invalid inference. The paper therefore cannot currently support its main claim.

major comments (3)

[Section 3, Lemma 3.8 and Definition 3.5] The equality in Eq. (14) is false. Take X to be the four corners of the unit square and q=1. In R^1_∞(X), one of the two triangles is matched to the diagonal edge, so the other triangle is the only critical 2-simplex. Its reach A consists of the two triangles. Since both have diameter sqrt(2), Definition 3.5 yields D^1_1(X)=∅, so H_1(D^1_1(X))=0. But the standard Rips complex V_1(X) contains the four unit edges forming a 1-cycle, so H_1(V_1(X))=Z_2. This directly contradicts Theorem 3.9 and shows that the error is not a minor gap.
[Section 3, proof of Lemma 3.8] The step 'since all degree-q homology classes eventually die' is not sufficient. From γ ∈ Z_q(Crit_*(R^q_r)) and γ ∈ ∂(Crit_{q+1}(R^q_t)) for some t>r it does not follow that γ is a cycle in Crit_*(D^q_r), because the boundary chains at scale t may involve critical (q+1)-simplices whose faces are not faces of A-simplices of diameter at most r. This is exactly what happens in the square example: the 1-cycle is born from edges that are not faces of any triangle in A with diameter at most 1.
[Section 4, Algorithm 1] The correctness of the algorithm relies entirely on Theorem 3.9. In light of the square counterexample, the claim that computing PH1(X) using D^1_•(X) yields the true PH1(X) is not justified; the experiment in Figure 3 may be measuring a different filtration. The paper should state that the algorithm's correctness is contingent on a corrected version of Theorem 3.9.

minor comments (5)

[Throughout] There are numerous typographical errors, such as 'parrallelizable' in Section 4, 'witll' in Section 4.1, and 'Nieghborhood' in the Section 5 title; these should be corrected.
[Definition 2.3] Definition 2.3 has a formatting glitch: it states 'dim( σ) = q' with an incomplete phrasing; please fix.
[Theorem 3.9] In the statement of Theorem 3.9, the second inclusion in the lower row should read D^q_{r_1}(X) ⊂ D^q_{r_2}(X), not D^q_{r_1}(X) ⊂ D^q_{r_1}(X).
[Section 4.1] The phrase 'not necessarily in R2(µ1)' appears immediately after Algorithm 1 and seems to be a leftover fragment of the main text.
[Figure 3] Figure 3 uses the label 'DVRC' while the text uses 'distilled Vietoris-Rips complex'; please use one consistent abbreviation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the distilled construction is new, and the cited reduced-VR theorem is a dependency, not a definitional or self-referential reduction.

full rationale

The distilled filtration D^q_• is defined in Definition 3.5 from reach sets of critical (q+1)-simplices of the reduced Rips complex, an independent construction not equal to any input. Theorem 3.9 is proved through standard discrete Morse theory (Theorem 2.64 from Kozlov [11]) plus Lemma 3.8. Lemma 3.8 does not simply assert the result; it attempts a cycle-group argument using eventual death of degree-q classes. Even if that argument has a scale gap (the step from a boundary at scale t back to a cycle at scale r), a logical gap is not a circularity, and the four-point-square objection would be a counterexample, not a circularity. Theorem 2.42 from the authors' companion paper [5] supplies the equivalence with the standard Vietoris-Rips filtration, but [5] is about the reduced complex, not the distilled complex, so the target result does not reduce to the cited result. No fitted parameter is renamed as a prediction, no known object is merely relabeled, and no uniqueness theorem is imported from the authors. I therefore find no circular step.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters or fitted constants appear. The main unproved input is the same-group reduced-VR theorem from [5], plus the terse Lemma 3.8 step; no invented physical or empirical entities are introduced.

assumptions (4)

domain assumption Theorem 2.42 from [5]: H_q of the degree-q reduced Vietoris-Rips filtration is isomorphic to that of the standard Vietoris-Rips filtration with commuting maps.
Imported from the authors' companion paper; not proved in this text. The distilled theorem composes through this result.
standard math Lemma 3.1 from [1]: the apparent pairs of the simplex-wise Vietoris-Rips filtration form an acyclic partial matching.
Used to conclude that the matching µ_q defined on the reduced complex is acyclic.
standard math Discrete Morse theory results from [11]: Theorems 2.56, 2.59 and 2.64 giving the critical-cell decomposition and persistent homology isomorphism.
Provides the mathematical framework that turns the matching into a homology-preserving reduction; treated as background.
domain assumption All degree-q homology classes in the finite Vietoris-Rips filtration eventually die, and cycles can be represented as boundaries at a sufficiently large scale.
Invoked in Lemma 3.8 to replace a cycle by a boundary at a later scale; the proof does not show the boundary chains are contained in the original distilled complex.

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Cite this review

Pith. "Pith review of The distilled Vietoris Rips filtration for persistent homology and a new memory efficient algorithm." pith.science (2026). https://pith.science/paper/QKEJDAHK

@misc{pith2026241207805,
  author       = {Pith},
  title        = {Pith review of: The distilled Vietoris Rips filtration for persistent homology and a new memory efficient algorithm},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QKEJDAHK}},
  note         = {Machine review of arXiv:2412.07805}
}

read the original abstract

The long computational time and large memory requirements for computing Vietoris Rips persistent homology from point clouds remains a significant deterrent to its application to big data. This paper aims to reduce the memory footprint of these computations. It presents a new construction, the distilled Vietoris Rips filtration, and proves that its persistent homology is isomorphic to that of standard Vietoris Rips. The distilled complex is constructed using a discrete Morse vector field defined on the reduced Vietoris Rips complex. The algorithm for building and reducing the distilled filtration boundary matrix is highly parallelisable and memory efficient. It can be implemented for point clouds in any metric space given the pairwise distance matrix.

Figures

Figures reproduced from arXiv: 2412.07805 by the authors.

**Figure 2.** (a) CRNC1(X) and CRNC2(X) of 50 points uniformly distributed over a cube. CRNC1(X) is depicted in green and CRNC2(X) is depicted in red. (b) CRNC1(X) and CRNC2(X) of 50 points uniformly distributed over a cube. CRNC1(X) is depicted in green and CRNC2(X) is depicted in red. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗

**Figure 3.** (a) Number of 2-simplices used in D1 ∞(X) (DVRC) on point clouds of size 50, 100, 200, 300, 400, 500, 600, 700 generated from a uniform distribution in a 10 × 10 × 10 cube. (b) Number of 2-simplices used in D1 ∞(X) (DVRC) on point clouds of size 50, 100, 200, 300, 400, 500, 600, 700 generated on a unit sphere. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗

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Reference graph

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [5]

Faster computation of degree-1 persistent homology using the reduced vietoris-rips filtration,

M. A. Koyama, F. Memoli, V. Robins, and K. Turner, “Faster computation of degree-1 persistent homology using the reduced vietoris-rips filtration,” 2024

work page 2024

[2] [1]

Ripser: efficient computation of Vietoris–Rips persistence barcodes,

U. Bauer, “Ripser: efficient computation of Vietoris–Rips persistence barcodes,” Journal of Applied and Computational Topology , 2021

work page 2021

[3] [2]

GPU-Accelerated Computation of Vietoris-Rips Persistence Barcodes,

S. Zhang, M. Xiao, and H. Wang, “GPU-Accelerated Computation of Vietoris-Rips Persistence Barcodes,” in 36th International Symposium on Computational Geometry (SoCG 2020) , Schloss Dagstuhl-Leibniz-Zentrum f¨ ur Informatik, 2020

work page 2020

[4] [3]

Dory: Computation of persistence diagrams up to di- mension two for vietoris–rips filtrations of large data sets,

M. Aggarwal and V. Periwal, “Dory: Computation of persistence diagrams up to di- mension two for vietoris–rips filtrations of large data sets,” Journal of Computational Science, vol. 79, p. 102290, 2024

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Swap, Shift and Trim to Edge Collapse a Filtration,

M. Glisse and S. Pritam, “Swap, Shift and Trim to Edge Collapse a Filtration,” in 38th International Symposium on Computational Geometry (SoCG 2022) (X. Goaoc and M. Kerber, eds.), vol. 224 of Leibniz International Proceedings in Informatics (LIPIcs), (Dagstuhl, Germany), pp. 44:1–44:15, Schloss Dagstuhl – Leibniz-Zentrum f¨ ur Infor- matik, 2022

work page 2022

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J. R. Munkres, Elements of algebraic topology . CRC press, 2018

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Edelsbrunner and J

H. Edelsbrunner and J. L. Harer, Computational topology: an introduction . American Mathematical Society, 2022

work page 2022

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Skeletonization and partitioning of digital images using discrete morse theory,

O. Delgado-Friedrichs, V. Robins, and A. Sheppard, “Skeletonization and partitioning of digital images using discrete morse theory,” IEEE Transactions on Pattern Analysis and Machine Intelligence , vol. 37, no. 3, pp. 654–666, 2015

work page 2015