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arxiv: 2607.06524 · v1 · pith:MWULPXBH · submitted 2026-07-07 · math.AT · cs.CG· math.CO

Lower Bounds for Approximating the Vietoris-Rips Filtration

Kenneth McCabe This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-07-08 02:48 UTCglm-5.2pith:MWULPXBHrecord.jsonopen to challenge →

classification math.AT cs.CGmath.CO MSC 55N3168U05
keywords Vietoris-Rips filtrationapproximation lower boundshomotopy interleavingstopological data analysisflag complexesgirthTuran graphs
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The pith

VR approximations must blow up without geometry

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

The paper proves that for any fixed approximation factor c, there exist infinite families of finite metric spaces where any c-approximation to the Vietoris-Rips filtration has at least superlinear size. For c in [1, sqrt(2)), the bound is exponential. The key mechanism is Lemma 3.1, which lower-bounds the size of any c-approximation by the rank of the structure map H_i(G)_r -> H_i(G)_{c^2 r}. The author then constructs metric spaces (from Turan graphs, generalized polygons, and Lazebnik-Ustimenko-Woldar graphs) where this rank is large, and extends the results to the intrinsic Cech filtration and to any bifiltration containing VR(-) as a slice.

Core claim

The central observation is that any finitely presented c-approximation F to a filtration G must have size at least the rank of the structure map H_i(G)_r -> H_i(G)_{c^2 r}. By constructing metric spaces from graphs with high girth and large first Betti number (Turan graphs T(3n,n) for exponential bounds, generalized polygon incidence graphs and LUW graphs CD(k,p) for superlinear bounds), the author shows this rank can be made exponentially or superlinearly large, establishing that linear-size approximations are impossible for arbitrary metric spaces at any fixed approximation factor.

What carries the argument

homotopy interleavings

Load-bearing premise

The superlinear lower bounds (Theorems 3.6 and 3.9) rely on Lemma 3.5, which uses a result from Adamaszek (2013) stating that if the 1-skeleton of VR(X)_1 has girth at least 3j+1, then the inclusion VR(X)_1 -> VR(X)_j is a homotopy equivalence. If that girth-to-stability result had hidden conditions, the superlinear bounds would fail. The exponential bound (Theorem 3.2) does not depend on this lemma.

What would settle it

Construct a finitely presented c-approximation to VR(X) for the specified metric spaces that achieves linear or sub-superlinear size, which would contradict the rank lower bound of Lemma 3.1.

Figures

Figures reproduced from arXiv: 2607.06524 by Kenneth McCabe.

Figure 1
Figure 1. Figure 1: A simplicial complex (left) and its Alexander dual (right). say M is finitely presented if F0 and F1 can be chosen to be finitely generated. A presentation is minimal if the ranks of F0 and F1 are as small as possible; for P = T1 × · · · × Tn with each Ti totally ordered, a minimal presentation exists and is unique up to isomorphism [33]. We write β0(M) and β1(M) for the ranks of F0 and F1 in a minimal pre… view at source ↗
Figure 2
Figure 2. Figure 2: The graph G3. 3.1. Lower Bounds for the Vietoris-Rips Filtration. We are now ready to present our lower bounds for approximations to the Vietoris-Rips filtration. Theorem 3.2. For every c ∈ [1, √ 2), there exists an infinite family of finite metric spaces {Xn} such that any finitely presented c-approximation to VR(Xn) has size at least 2 |Xn|/3 . Proof. For n ≥ 2, let Gn be the complete n-partite graph wit… view at source ↗
Figure 3
Figure 3. Figure 3: The incidence graph of the generalized quadrangle of order (2, 2). The blue vertices correspond to points, while the red vertices correspond to lines. By [1, Proposition 2.2], the assumption g(H) ≥ 3j + 1 implies that the inclusion Cl(H) ,→ Cl(Hj ) is a homotopy equivalence. □ We now use Lemma 3.5 to give superlinear lower bounds on approximations to the Vietoris-Rips filtration. The metric spaces we will … view at source ↗
read the original abstract

The Vietoris-Rips filtration $\mathcal{VR}(-)$ is a standard tool for analyzing the shape of data within topological data analysis. Beginning with seminal work of Sheehy, a substantial amount of research has centered on constructing linear-size sparse approximations to $\mathcal{VR}(-)$ and related filtrations for metric spaces of bounded doubling dimension. We show that this geometric assumption is necessary in a precise sense. Working in the framework of homotopy interleavings, we show that for any fixed $c \in [1, \sqrt{2})$, there exists a family of finite metric spaces for which any finitely presented $c$-approximation to $\mathcal{VR}(-)$ has exponential size. We also show that for any fixed $c \geq 1$, there exists a family of finite metric spaces for which any finitely presented $c$-approximation to $\mathcal{VR}(-)$ has superlinear size, yielding an obstruction to linear-size approximations for any fixed approximation factor. Both results extend to the intrinsic \v{C}ech filtration and to any bifiltration containing $\mathcal{VR}(-)$ as a $1$-parameter slice, including the function-Rips, degree-Rips, and subdivision-Rips bifiltrations.

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 / 7 minor

Summary. The paper proves lower bounds on the size of approximations to the Vietoris-Rips filtration VR(−) for arbitrary finite metric spaces, in the framework of homotopy interleavings. The central tool is Lemma 3.1, which lower-bounds the size of any finitely presented c-approximation by the rank of a structure map H_i(G)_r → H_i(G)_{c²r}. Two main results are established: (1) for c ∈ [1, √2), exponential lower bounds via Turán graphs T(3n, n) (Theorem 3.2), and (2) for any fixed c ≥ 1, superlinear lower bounds via incidence graphs of generalized polygons (Theorem 3.6, for c < √6) and Lazebnik–Ustimenko–Woldar graphs (Theorem 3.9, for all c ≥ 1). Both results extend to the intrinsic Čech filtration (Corollaries 3.10, 3.11) and to any bifiltration containing VR(−) as a 1-parameter slice (Corollaries 3.13, 3.14).

Significance. This paper makes a substantial contribution by providing the first explicit lower bounds on the size of c-approximations to VR(−) for arbitrary metric spaces. The exponential bound (Theorem 3.2) cleanly complements Sheehy's linear-size (1+ε)-approximations for bounded doubling dimension, showing that the geometric assumption is necessary. The superlinear bound (Theorem 3.9) is particularly notable: it shows that no fixed approximation factor yields linear-size approximations for arbitrary metric spaces, closing a natural question in the area. The extensions to the intrinsic Čech filtration and to bifiltrations (function-Rips, degree-Rips, subdivision-Rips) broaden the impact considerably. The proofs are clean and rely on standard, well-established tools (Künneth for joins, Alexander duality, Adamaszek's girth-to-stability result). The reliance on [1, Prop 2.2] for Lemma 3.5 is well-grounded: the girth conditions are verified correctly in all cases, and the exponential bound (Theorem 3.2) is independent of this lemma. The Turán graph construction for the exponential bound is well-motivated by the Beers–Botnan extremal result.

minor comments (7)
  1. §3.1, proof of Theorem 3.2: The claim that VR(X_n)_r = Cl(G_n) for r ∈ [1,2) should specify that this holds because G_n is the 1-skeleton at scale 1 and the metric is twice the shortest path metric, so no new edges appear until scale 2. This is implicit but making it explicit would aid the reader.
  2. §3.1, proof of Theorem 3.6: The table listing the three cases uses q^4, q^6, q^12 for dim H_1(G_q), but the text states |X_q| = Θ(q^3), Θ(q^5), Θ(q^11). The exponents for ϵ(c) follow from these, but a brief sentence explaining the computation (e.g., 'since |X_q| = Θ(q^n) and dim H_1 = Θ(q^{2n-2}), we get ϵ = (2n-2)/n - 1 = (n-2)/n') would make the derivation more transparent.
  3. §3.1, equation (1): The formula d(c) := k(c) − ⌊(k(c)+2)/4⌋ + 1 could use a brief explanation of its origin, or a forward reference to where it appears in [30], as its role is not immediately clear.
  4. §3.2, proof of Corollary 3.10: The ball computation B(x,2) = {x} ∪ (X_n ∖ P_j) is correct but the subsequent argument that ⋂_{x∈σ} B(x,2) ≠ ∅ iff |σ ∩ P_j| ≤ 1 for some j could be stated more carefully; the current phrasing conflates 'there exists j' with 'for a specific j.'
  5. §2.4: The definition of c-interleaving via the category I_c is standard but slightly non-standard in that it uses [0,∞) × {0,1} rather than the more common R × {0,1}. A remark noting that this is equivalent for filtrations indexed by [0,∞) would help readers familiar with the [8] formulation.
  6. Figure 2: The graph G_3 is labeled but the caption could note that this is T(9,3), connecting it to the Turán graph terminology used in Remark 3.3.
  7. References [25] and [31]: Both are dated 2026, which appears to be a forward-dating issue; the authors should verify these are correct.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for a careful reading and for the positive assessment of the paper's contributions. The referee's report recommends minor revision but does not raise any specific major or minor comments requiring changes to the manuscript. We are grateful for the referee's thorough summary of the results and the accurate characterization of the paper's significance. We have reviewed the manuscript in light of the referee's remarks and confirm that the girth conditions, the reliance on [1, Proposition 2.2], and the independence of Theorem 3.2 from Lemma 3.5 are all correctly verified as the referee notes. No revisions to the mathematical content are needed. We will conduct a final proofreading pass to address any typographical issues before the final version.

Circularity Check

0 steps flagged

No circularity found; derivation is self-contained with appropriate external citations

full rationale

The paper's derivation chain is clean and self-contained. Lemma 3.1 (the central tool) is proved in full via a straightforward diagram chase through the interleaving category I_c; the proof does not depend on any unverified self-citation. The size definition is attributed to [32] (Lesnick-McCabe) but is a standard notion recovering simplex counts for filtrations, and the paper re-derives what it needs. Theorem 3.2 (exponential bound) uses only Lemma 3.1 plus the observation that VR(X_n)_1 = VR(X_n)_{c²} = A_n when c² ∈ [1,2) and distances are even integers—a genuine mathematical fact, not a definitional identity. Lemma 3.5, the other load-bearing ingredient for the superlinear bounds (Theorems 3.6 and 3.9), cites [1, Proposition 2.2] by Adamaszek (Israel J. Math, 2013), who is not an author of this paper; this is an external, peer-reviewed result. The graph-theoretic ingredients (Turán graphs from [5], generalized polygons from [35], LUW graphs from [30]) are all externally sourced. The Čech extensions (Corollaries 3.10–3.11) use standard Alexander duality and the classical VR/Čech interleaving. The bifiltration extension (Lemma 3.12) is a direct restriction argument proved in full. No step reduces to its inputs by construction, and no self-citation is load-bearing in a circular sense.

Axiom & Free-Parameter Ledger

0 free parameters · 7 axioms · 0 invented entities

No new entities, particles, forces, or postulated objects are introduced. The paper works entirely with standard mathematical objects (graphs, simplicial complexes, persistence modules) and externally defined graph families.

axioms (7)
  • standard math Künneth formula for reduced homology of simplicial joins (Lemma 2.1, cited from [36])
    Used in Theorem 3.2 to compute H̃_{n-1}(D_3^{*n}) ≅ k^{2^n} by induction.
  • standard math Alexander duality for simplicial complexes (Lemma 2.2, cited from [7])
    Used in Corollary 3.10 to relate H̃_i(I(X_n)_2) to H̃_{|X_n|-i-3}(A_n).
  • domain assumption Girth ≥ 3j+1 implies VR(X)_1 ↪ VR(X)_j is a homotopy equivalence (Lemma 3.5, cited from [1, Prop 2.2])
    Load-bearing for Theorems 3.6 and 3.9; ensures the structure map H_1(VR(X))_1 → H_1(VR(X))_{c²} is an isomorphism.
  • domain assumption Existence and properties of generalized polygons (Proposition 3.4, cited from [35])
    Provides the graph families with required girth and vertex counts for Theorem 3.6.
  • domain assumption Properties of LUW graphs CD(k,p): connected, bipartite, p-regular, girth ≥ k+5, |V| ≤ 2p^{d(c)} (Lemma 3.8, cited from [30, Theorem 3.2])
    Provides the graph family with superlinear edge density and sufficient girth for Theorem 3.9.
  • standard math Feit–Higman theorem: finite generalized n-gons exist only for n ∈ {2,3,4,6,8} (Theorem 3.7, cited from [26])
    Explains why generalized polygons cannot handle c ≥ √6, motivating the switch to LUW graphs.
  • standard math Existence and uniqueness of minimal presentations for persistence modules over products of totally ordered sets (Section 2.2, cited from [33])
    Underpins the definition of size via Betti numbers of chain complexes.

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