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arxiv: 2606.02846 · v1 · pith:2NH67ID2new · submitted 2026-06-01 · 🧮 math.AT

Cheeger Inequalities for the Persistent Laplacian

Magnus Bakke Botnan , Rui Dong This is my paper

Pith reviewed 2026-06-28 11:21 UTC · model grok-4.3

classification 🧮 math.AT
keywords persistent LaplacianCheeger inequalitysimplicial complex inclusionpersistent homologyisoperimetric inequalitypseudomanifoldgraph Laplacian reduction
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The pith

A Cheeger inequality relates the persistent Cheeger constant to the smallest nonzero eigenvalue of the persistent up-Laplacian.

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

The paper proves a Cheeger-type inequality for persistent Laplacians arising from inclusions of simplicial complexes K into L. It defines the persistent up p-Laplacian and shows that its p=1 case produces a nonzero persistent Cheeger constant phi_q^{K,L} that is controlled by the smallest positive eigenvalue of the ordinary persistent up-Laplacian. The inequality extends the classical Cheeger relation between expansion and spectral gap into the persistent setting. Additional results give an extension of an isoperimetric inequality under a local-completeness assumption and a reduction of the Laplacian to a weighted graph Laplacian for pseudomanifolds.

Core claim

For an inclusion of simplicial complexes K hookrightarrow L we introduce the persistent up p-Laplacian Delta_{q,p,up}^{K,L}. When p equals 1 this yields the nonzero persistent Cheeger constant phi_q^{K,L}, and we prove a Cheeger-type inequality relating phi_q^{K,L} to the smallest nonzero eigenvalue of Delta_{q,up}^{K,L}. The same construction recovers the usual persistent up-Laplacian when p equals 2 and extends the non-persistent Cheeger inequality of Jost and Zhang.

What carries the argument

The persistent up p-Laplacian Delta_{q,p,up}^{K,L} associated to the inclusion, which for p=1 defines the persistent Cheeger constant phi_q^{K,L}.

If this is right

  • The persistent Cheeger constant supplies a combinatorial lower bound on the spectral gap of the persistent up-Laplacian.
  • Under the locally complete q-skeleton assumption the Parzanchevski-Rosenthal-Tessler isoperimetric inequality extends directly to the persistent setting.
  • For orientable (q+1)-dimensional pseudomanifolds the persistent up-Laplacian reduces to a vertex- and edge-weighted graph Laplacian, possibly with Dirichlet terms.
  • Two-sided Cheeger inequalities hold in the pseudomanifold case.
  • In the non-branching pseudomanifold case the nonzero persistent Cheeger constant admits an explicit description in terms of the dual graph.

Where Pith is reading between the lines

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

  • The reduction to weighted graph Laplacians may allow standard graph-partitioning algorithms to approximate higher-dimensional persistent spectral quantities.
  • The explicit dual-graph formula suggests that persistent expansion can be read off from a single auxiliary graph even when the original complexes are higher-dimensional.
  • Direct comparison of the persistent Cheeger constants with the Kron-reduction constants of Memoli et al. may identify settings where the two notions coincide.

Load-bearing premise

The locally complete q-skeleton assumption on K is required to extend the Parzanchevski-Rosenthal-Tessler isoperimetric inequality to the persistent setting.

What would settle it

An explicit pair of finite simplicial complexes K and L together with a computed value of phi_q^{K,L} that lies strictly above the bound implied by the smallest nonzero eigenvalue of Delta_{q,up}^{K,L} would falsify the inequality.

Figures

Figures reproduced from arXiv: 2606.02846 by Magnus Bakke Botnan, Rui Dong.

Figure 1
Figure 1. Figure 1: The relevant morphisms in the definition of the persistent Laplacian ∆ [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Example of locally q-complete K ⊂ L, with K = K1 ` K2 . Before defining the Cheeger constant, we introduce some additional notation. Since each component Kα has a complete q-skeleton, any set of q + 1 vertices in V Kα spans a q-simplex of Kα. Thus, if a (q + 1)-simplex τ ∈ S L q+1 has exactly q + 1 vertices in V Kα , then these vertices span a unique q-face of τ contained in Kα. We define the (q + 1)-front… view at source ↗
Figure 3
Figure 3. Figure 3: The octahedron from Example 5.2. Bringing it all together. Combining Cases 3 and 4 over all choices of i and α, we get Xq+1 i=0 X N α=1   X τ∈GP (Ft(Kα),i) [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A filtration of a planar simplicial complex; see Example [PITH_FULL_IMAGE:figures/full_fig_p031_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Eigenvector entries on the 2-simplices for the smallest non-zero eigenvalue of [PITH_FULL_IMAGE:figures/full_fig_p032_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Eigenvector entries on the 2-simplices for the second-smallest non-zero eigenvalue of [PITH_FULL_IMAGE:figures/full_fig_p033_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Evolution of the smallest nonzero eigenvalue of the persistent Laplacian for [PITH_FULL_IMAGE:figures/full_fig_p034_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: A numerical solution of the eigenvalue problem [PITH_FULL_IMAGE:figures/full_fig_p035_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: A triangulated Klein bottle. Define xk ∈ C2(Uk) by (xk)σ = ( +1, σ ∈ S c , −1, otherwise. Here the second case includes all 2-simplices in the Lk summand. By the choice of orientations, the contributions along Cut B cancel, whereas the contributions along Cut A do not cancel. Moreover, there are no contributions along the connected-sum seam. Hence ∂ Uk 2 xk is supported on the finitely many edges of Cut A.… view at source ↗
Figure 10
Figure 10. Figure 10: Figures in Example 6.3 . The boundary matrix BL 2 is B L 2 =         [123] [124] [134] [234] [12] 1 1 0 0 [13] −1 0 1 0 [23] 1 0 0 1 [14] 0 −1 −1 0 [24] 0 1 0 −1 [34] 0 0 1 1         , and B =   1 1 0 0 −1 0 1 0 1 0 0 1   , Z =     1 0 0 −1 0 1 0 −1     , and thus (BZ) T =  1 −1 1 −1 1 −1  . Consider (BZ) T as the incidence matrix of a dual graph G containing two vertices and… view at source ↗
Figure 11
Figure 11. Figure 11: The pair of simplicial complexes and the associated dual graph in Example [PITH_FULL_IMAGE:figures/full_fig_p040_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The simplicial complex L and the associated dual graph in Example 6.5. 7 Persistent Cheeger constants for graph inclusions In this section, we focus on persistent Cheeger constants for graphs. Throughout, let G = (V G, EG) be a connected graph and let H = (V H, EH) be a subgraph of G. Note that ∆H,G 0 depends only on the vertex set V H, and not on the edge set EH. As observed in Proposition 3.2, the persi… view at source ↗
Figure 13
Figure 13. Figure 13: 1 2 3 n − 1 n [PITH_FULL_IMAGE:figures/full_fig_p044_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The graphs from Example 7.1. Since ∆H,Gc 0 is the graph Laplacian of the corresponding Kron reduction, Eq. (21) gives ∥PcχS∥1 = 2χ T S PcχS. Therefore φ H,G 0 ≤ min ∅̸=S⊆V H |S|≤|V H|/2 ∥BT HχS∥1 + ∥PcχS∥1 |S| ≤ min ∅̸=S⊆V H |S|≤|V H|/2 2χ T S (BHBT H + Pc)χS |S| = 2φ H,G Kron. Example 7.1. Consider the graphs H ⊂ G from [MWW22], where H consists of the three outermost vertices, and He is the associated K… view at source ↗
read the original abstract

We study Cheeger-type inequalities for persistent Laplacians associated with inclusions of simplicial complexes $\mathcal{K}\hookrightarrow \mathcal{L}$. We introduce a persistent up $p$-Laplacian $\Delta_{q,p,\mathrm{up}}^{\mathcal{K},\mathcal{L}}$ for $p\geq 1$. For $p=2$, this recovers the usual persistent up Laplacian, while for $p=1$ it yields a nonzero persistent Cheeger constant $\varphi_q^{\mathcal{K},\mathcal{L}}$. We prove a Cheeger-type inequality relating $\varphi_q^{\mathcal{K},\mathcal{L}}$ to the smallest nonzero eigenvalue of $\Delta_{q,\mathrm{up}}^{\mathcal{K},\mathcal{L}}$. This gives a persistent extension of recent work by Jost and Zhang (Ann. Sc. Norm. Super. Pisa Cl. Sci., 2024; arXiv:2302.01069). We then study two more structured settings. Under a locally complete $q$-skeleton assumption on $\mathcal{K}$, we extend the complete-skeleton isoperimetric inequality of Parzanchevski--Rosenthal--Tessler (Combinatorica, 2016; arXiv:1207.0638) to the persistent setting. For orientable $(q+1)$-dimensional pseudomanifolds, we prove a Kron-type reduction of the persistent up Laplacian to a vertex- and edge-weighted graph Laplacian, possibly with Dirichlet boundary terms, and obtain two-sided Cheeger inequalities; this is related to the dual-graph perspective in the work of Steenbergen--Klivans--Mukherjee (Adv. Appl. Math., 2014; arXiv:1209.5091). We also describe the nonzero persistent Cheeger constant $\varphi_q^{\mathcal{K},\mathcal{L}}$ explicitly in terms of the dual graph in the non-branching pseudomanifold case. Finally, for graph inclusions $H\hookrightarrow G$, we compare the persistent Cheeger constants introduced here with the Kron-reduction Cheeger constants of M\'emoli et al. (SIAM J. Math. Data Sci., 2022; arXiv:2012.02808).

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 / 4 minor

Summary. The manuscript introduces the persistent up p-Laplacian Δ_{q,p,up}^{K,L} for an inclusion of simplicial complexes K ↪ L. For p=2 it recovers the standard persistent up Laplacian; for p=1 it yields a nonzero persistent Cheeger constant ϕ_q^{K,L}. The central result is a Cheeger-type inequality relating ϕ_q^{K,L} to the smallest nonzero eigenvalue of Δ_{q,up}^{K,L}, extending Jost-Zhang. Under a locally complete q-skeleton assumption on K the authors extend the Parzanchevski-Rosenthal-Tessler isoperimetric inequality to the persistent setting. For orientable (q+1)-pseudomanifolds they establish a Kron-type reduction of the persistent up Laplacian to a weighted graph Laplacian (possibly with Dirichlet terms) and obtain two-sided Cheeger inequalities; they also give an explicit description of ϕ_q^{K,L} via the dual graph in the non-branching case. Finally they compare the persistent Cheeger constants with the Kron-reduction constants of Mémoli et al. for graph inclusions H ↪ G.

Significance. If the proofs hold, the work supplies a direct persistent extension of the Jost-Zhang Cheeger inequality together with concrete reductions and comparisons that are likely to be useful in topological data analysis. The explicit dual-graph description and the Kron reduction on pseudomanifolds are particularly valuable because they convert spectral questions on higher-dimensional complexes into graph-Laplacian computations. The scoping of the locally complete q-skeleton assumption solely to the isoperimetric extension (and not to the main inequality) is a strength.

minor comments (4)
  1. [§3] §3 (definition of Δ_{q,p,up}^{K,L}): the construction is stated via the usual coboundary and boundary operators on the pair (K,L), but the precise normalization constants that ensure the p=1 case recovers exactly ϕ_q^{K,L} are not written out; adding the explicit formula for the 1-Laplacian would remove any ambiguity.
  2. [Theorem 4.2] Theorem 4.2 (persistent Cheeger inequality): the statement is given for the smallest nonzero eigenvalue, but the proof sketch does not record the dependence on the dimension q or on the number of simplices; a short remark on whether the constants are dimension-independent would clarify the result.
  3. [§6] §6 (pseudomanifold reduction): the claim that the reduced operator is a vertex- and edge-weighted graph Laplacian is stated without an explicit matrix representation; including the 2×2 block form for the lowest-dimensional case would make the Dirichlet boundary terms immediately visible.
  4. [Figure 2] Figure 2 (dual-graph example): the caption does not indicate whether the weights are induced by the original simplicial volumes or by the inclusion map; a one-sentence clarification would prevent misreading.

Circularity Check

0 steps flagged

No significant circularity; derivation is a self-contained proof extending external results

full rationale

The central claim is a mathematical proof of a Cheeger-type inequality for persistent Laplacians, explicitly positioned as an extension of Jost-Zhang (external, 2024). Other results extend Parzanchevski-Rosenthal-Tessler, Steenbergen-Klivans-Mukherjee, and Mémoli et al. (all external citations). No self-citations are load-bearing for the main inequality. The locally complete q-skeleton assumption is scoped only to a secondary extension and not invoked for the core result. No equations reduce claims to fitted parameters, self-definitions, or renamed known results by construction. The work is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Abstract only; no explicit free parameters or ad-hoc axioms are visible. The constructions rest on standard domain assumptions of simplicial homology.

axioms (1)
  • domain assumption Standard algebraic properties of simplicial complexes and their inclusions hold
    Invoked to define the persistent up Laplacian and the Cheeger constant.
invented entities (1)
  • persistent up p-Laplacian Δ_{q,p,up}^{K,L} no independent evidence
    purpose: Generalizes the persistent Laplacian to p ≥ 1 so that p=1 yields a nonzero Cheeger constant
    Defined in the paper to obtain the p=1 case of the inequality.

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Reference graph

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