{"id":"0f5320be-da73-4c1a-8eaf-20ad30578471","arxiv_id":"2412.07805","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"A distilled Vietoris-Rips filtration is shown to have persistent homology isomorphic to the standard Vietoris-Rips filtration, enabling a memory-efficient parallel degree-1 algorithm.","lead":"This paper introduces a distilled Vietoris-Rips filtration, a smaller complex that stores only a fraction of the simplices needed by the standard construction. The authors prove its persistent homology is isomorphic to the standard one, and propose a parallel algorithm whose memory savings rest on an unproven conjecture.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The distilled complex omits 1-cycles whose edges are not faces of an included 2-simplex; for the four-point unit square, D^1_1 is empty while standard Rips has H_1=Z_2, so Theorem 3.9 is false as stated.","rationale":"The reader identified Lemma 3.8 as the weak point, and the square example confirms that Lemma 3.8 is not merely unproved but false. The distilled complex is defined by taking only faces of reachable critical (q+1)-simplices of scale at most r. This means a q-simplex (or a graph cycle, for q=1) is absent from D_r unless it is a face of such a (q+1)-simplex. In the unit square, the 1-cycle born at r=1 consists of four edges that are not faces of any 2-simplex of diameter at most 1, so the distilled filtration misses the entire class [1, sqrt(2)). This is a direct counterexample to the central isomorphism theorem, not just a gap in the proof. Consequently the main claim that the distilled filtration computes the same degree-q persistent homology as standard Vietoris-Rips cannot stand as stated. The paper could potentially be repaired by including all simplices up to dimension q with their own filtration values, but that would change the construction and the memory-efficiency claims. As written, the central theorem is false.","tokens_in":16682,"tokens_out":24174,"duration_ms":257584,"concrete_test":"Implement Definition 3.5 for X = {(0,0),(1,0),(1,1),(0,1)} with q=1 and compute the PH_1 barcode of D^1 and of the standard Vietoris-Rips filtration. The check is whether the interval [1, sqrt(2)) appears in the D^1 barcode: the construction gives D^1_1 = empty, so it cannot, while standard Rips has that interval. This would settle that Theorem 3.9 fails for this example.","verdict_should_be":"REJECT","load_bearing_attack":"Definition 3.5 makes D^q_r depend only on (q+1)-simplices in A with diameter at most r, plus their faces. For q=1, take X to be the four corners of the unit square in R^2. In the reduced Rips complex R^1, the only 2-simplices are the two triangles, both of diameter sqrt(2). The matching mu_1 pairs the diagonal with one triangle; the other triangle is critical. Hence A, the union of reach of critical 2-simplices, has every element of diameter sqrt(2), so D^1_r is empty for every r < sqrt(2). In particular D^1_1 is empty, giving H_1(D^1_1)=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. Lemma 3.8 would require Z_1(Crit_*(D_1)) = Z_1(Crit_*(R_1)), yet the square cycle lies in the latter and not in the former. Since Theorem 2.42 identifies H_1(R^1_1) with H_1(V_1(X)), Theorem 3.9 is false as stated. The failure is the unproved step in Lemma 3.8: a homology class can be born from edges that are not faces of any A-simplex at its birth scale.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":16993,"tokens_out":10385,"duration_ms":107243,"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":[{"comment":"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":"Section 3, Lemma 3.8 and Definition 3.5"},{"comment":"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":"Section 3, proof of Lemma 3.8"},{"comment":"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.","section":"Section 4, Algorithm 1"}],"minor_comments":[{"comment":"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.","section":"Throughout"},{"comment":"Definition 2.3 has a formatting glitch: it states 'dim( σ) = q' with an incomplete phrasing; please fix.","section":"Definition 2.3"},{"comment":"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":"Theorem 3.9"},{"comment":"The phrase 'not necessarily in R2(µ1)' appears immediately after Algorithm 1 and seems to be a leftover fragment of the main text.","section":"Section 4.1"},{"comment":"Figure 3 uses the label 'DVRC' while the text uses 'distilled Vietoris-Rips complex'; please use one consistent abbreviation.","section":"Figure 3"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is false as stated, and I am not confident that a simple edit to the proof will fix it; the square counterexample suggests that a redefinition of D^q_r is needed, for example by including all q-simplices that are faces of A-simplices with respect to their own diameter. If the authors can provide a corrected definition and a valid proof, the paper could be a useful contribution. In its current form I would not accept it. I also note the heavy reliance on the companion paper [5]; the editor may wish to ensure that manuscript is available before final acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nFor 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.","headline":"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.","tokens_in":17511,"tokens_out":11280,"would_cite":false,"duration_ms":101258,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55N31","55U10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The distilled Vietoris–Rips filtration carries the same degree-q persistent homology as the standard Vietoris–Rips filtration while using far fewer simplices.","keywords":["distilled Vietoris–Rips filtration","persistent homology","discrete Morse theory","reduced Vietoris–Rips complex","lune","apparent pairs","parallel algorithm","memory efficiency"],"falsifier":"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.","tokens_in":16478,"feed_emoji":"🧮","tokens_out":10488,"duration_ms":100313,"temperature":0.7,"pith_summary":"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.","feed_headline":"Distilled Rips filtration preserves the persistence barcode","feed_subtitle":"Same degree-q homology, far fewer simplices: a Morse-theoretic shortcut for Vietoris–Rips computations.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Defines apparent pairs and the simplex-wise Vietoris–Rips ordering; supplies Lemma 3.1 that apparent pairs form an acyclic partial matching.","marker":"[1]"},{"why":"Introduces the reduced Vietoris–Rips complex and lune functions; supplies Theorem 2.42 identifying its persistent homology with the standard Vietoris–Rips filtration.","marker":"[5]"},{"why":"Provides the discrete Morse theory used throughout: closure maps, the critical subcomplex, and Theorem 2.64 transferring persistent homology from critical subcomplexes to the filtration.","marker":"[11]"}],"fun_headline_variants":["Distilled Rips cuts memory, keeps same persistent homology","A leaner Vietoris–Rips: distilled filtration preserves barcodes","Memory-efficient persistent homology via distilled Rips filtration","Distilled Vietoris–Rips: same homology, less memory","New distilled Rips filtration shrinks memory, not homology"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Distilled Rips cuts memory, keeps same persistent homology","A leaner Vietoris–Rips: distilled filtration preserves barcodes","Memory-efficient persistent homology via distilled Rips filtration","Distilled Vietoris–Rips: same homology, less memory","New distilled Rips filtration shrinks memory, not homology"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000125,"raw_usage":{"total_tokens":1102,"prompt_tokens":938,"completion_tokens":164,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":554,"completion_tokens_details":{"reasoning_tokens":79}},"tokens_in":554,"tokens_out":164,"duration_ms":2504,"temperature":1.0,"reasoning_tokens":79,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T19:21:21.430448+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Ripser: efficient computation of Vietoris–Rips persistence barcodes,","cited_arxiv_id":null,"evidence_quote":"Defines apparent pairs and the simplex-wise Vietoris–Rips ordering; supplies Lemma 3.1 that apparent pairs form an acyclic partial matching."},{"cited_title":"Faster computation of degree-1 persistent homology using the reduced vietoris-rips filtration,","cited_arxiv_id":null,"evidence_quote":"Introduces the reduced Vietoris–Rips complex and lune functions; supplies Theorem 2.42 identifying its persistent homology with the standard Vietoris–Rips filtration."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the discrete Morse theory used throughout: closure maps, the critical subcomplex, and Theorem 2.64 transferring persistent homology from critical subcomplexes to the filtration."}],"review_version":1}