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REVIEW 4 major objections 4 minor 26 references

Positional Encoding meets Persistent Homology on Graphs

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper establishes that positional encodings and persistent homology are incomparable on graphs, and that its PiPE method, which feeds persistence summaries of positional encodings into message passing, is strictly more expressive than…

desk verdict The PiPE architecture is a reasonable engineering contribution, but the central proofs that it strictly beats LSPE and PH rest on false Laplacian spectra claims. read the letter →

arxiv 2506.05814 v1 pith:PJVVNUZX submitted 2025-06-06 cs.LG cs.AIcs.ETcs.SI

Yogesh Verma , Amauri H. Souza , Vikas Garg This is my paper
classification cs.LGcs.AIcs.ETcs.SI MSC 05C6055N3168T07
keywords graphneuralnetworkspositionalencodingpersistenthomologyexpressivityWeisfeiler-LehmantestLaplacianeigenmapstopologicaldataanalysismessagepassing
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

Positional encodings (PE) and persistent homology (PH) are two standard ways to give message-passing graph neural networks structural information, but the paper argues that neither subsumes the other: there are non-isomorphic graph pairs that each approach distinguishes while the other cannot. It then shows the two can be combined: when positional encodings supply the vertex colors for a persistent-homology filtration, the resulting 0-dimensional persistence diagrams are at least as expressive as the encodings themselves. On that basis it introduces PiPE, a learnable encoding that recomputes persistence diagrams from the current positional embeddings at every layer and feeds their vectorized summaries back into message passing. The paper's central positive claims are that PiPE with Laplacian PE is strictly more expressive than LSPE, a prominent learnable PE scheme, and strictly more expressive than PH combined with Laplacian PE, and that these gains show up on molecular-property and graph-classification benchmarks.

What carries the argument

The central object is the persistence diagram of a vertex-color filtration whose colors are the current positional embeddings, computed inside the message-passing loop. At layer $\ell$, the encodings are pushed through a filtration function $f_\ell$; the 0- and 1-dimensional diagrams $D^0_\ell$ and $D^1_\ell$ are vectorized by learned maps $\Psi^0_\ell$ and $\Psi^1_\ell$ into node-level topological embeddings, which then participate in the update of the positional embeddings and of the node features (the feature vector concatenates the node, positional, and topological embeddings). The expressive-power arguments turn on two facts: the multiset of birth times in $D_0$ encodes the multiset of vertex colors, and differing numbers of connected components force different 0-dimensional diagrams. Those facts let the same machinery establish both incomparability of PE and PH and strict gains when the two are combined.

What would settle it

Compute the normalized Laplacian spectra of a 4-cycle ($\{0,1,1,2\}$) and a 6-cycle ($\{0,0.5,0.5,1.5,1.5,2\}$); the multisets differ, so the disjoint unions used in Proposition 4.1 cannot have identical $k$ smallest eigenpairs for $k \ge 2$, and checking the stated eigenpairs directly would invalidate the construction.

Watch

Extended reading notes

Core claim

On unattributed graphs, Laplacian, random-walk, and distance positional encodings are incomparable with persistent homology built from vertex-color filtrations: statement S1 gives graph pairs with the same Betti numbers but different encodings $\Phi_k$, while S2 and S3 give pairs with different $\beta_0$ or $\beta_1$ but identical encodings (Propositions 3.1 and 3.2). The paper then proves that if the positional encodings are used as the vertex colors of a filtration, the 0-dimensional persistence diagram $D_0$ preserves all the distinguishing power of the encodings (Lemma 3.3) and can separate graph pairs the encodings alone cannot (Propositions 3.4, 3.5, 3.6). PiPE operationalizes this by computing, at each message-passing layer, the 0- and 1-dimensional persistence diagrams of the current positional embeddings, vectorizing them with learned maps into topological node features, and concatenating those features into the next layer's updates. The claimed consequences are that PiPE based on Laplacian PE strictly dominates both LSPE and PH+LPE in distinguishing non-isomorphic graphs (Propositions 4.1 and 4.2), that random-walk-based PiPE still misses some 3-WL-separable pairs (Proposition 4.3), and that any k-FWL-separable pair can be separated by 0-dimensional diagrams under an explicitly constructed coloring (Proposition 4.4).

Load-bearing premise

The proof that PiPE strictly beats LSPE assumes that graph complexes built from disjoint 4-cycles and disjoint 6-cycles share the same lowest Laplacian eigenvectors; their spectra differ, so that construction needs a different witness pair or a revised argument.

Editorial extensions

If this is right

  • Appending PiPE's persistence branch to Laplacian, random-walk, distance, SignNet, or SPE embeddings should strictly increase distinguishing power over the base PE alone, per Propositions 3.4, 3.5, and 4.2.
  • PH methods that filter on Laplacian positional colors do not match PiPE: the cycle-versus-path construction in Proposition 4.2 gives graph pairs that PH+LPE cannot separate but a message-passing GNN can.
  • Random-walk-based PiPE is not a 3-WL surrogate: the cospectral 4-regular pair of Proposition 4.3 is 2-WL-equivalent, so RW-based PiPE cannot separate it even though 3-WL can.
  • Proposition 4.4 makes persistent homology a principled interface to the WL hierarchy: whenever k-FWL distinguishes two graphs, an explicit node coloring produces different 0-dimensional persistence diagrams.
  • On the reported benchmarks, adding persistent-homology vectorizations to existing positional encodings lowers test MAE on ZINC and Alchemy, raises ROC-AUC on OGBG-MOLHIV, and improves OOD-Test AUC on DrugOOD.

Reading between the lines

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

  • The same recipe—persistence diagrams built from whatever node-level signal a model already has—could be applied to distance encodings, random feature maps, or graph-transformer attention weights, since the only requirement is a filtration over nodes.
  • The empirical gains on molecular benchmarks may not rest on the strict expressivity theorems; ablating the persistence branch while keeping the extra parameters would show whether the topological summaries help as features, as regularization, or through the claimed distinguishing power.
  • Since 0-dimensional diagrams already carry the component-count information behind the PE-failure examples, a natural test is whether the 1-dimensional cycle branch adds measurable value on datasets where cycles matter.
  • The k-FWL result suggests a general principle: any graph invariant that yields stable colorings can be converted into a filtration that separates the same graph pairs, potentially connecting persistent homology to the full WL hierarchy.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The manuscript proposes PiPE, a graph representation method that augments a base positional encoding (Laplacian or random-walk PE) with persistent-homology features computed from filtrations over the PE features, and integrates the result into message-passing GNNs. The theoretical portion claims (i) PE and PH are incomparable (Propositions 3.1 and 3.2); (ii) PH computed on top of PE is strictly more expressive than the base PE (Propositions 3.3–3.6); and (iii) PiPE with Laplacian PE is strictly more expressive than LSPE and than PH+LPE (Propositions 4.1 and 4.2), with additional results connecting RW-based PiPE to 3-WL (Proposition 4.3) and k-FWL to persistent-homology color-separating sets (Proposition 4.4). The empirical evaluation covers BREC expressivity, ZINC, Alchemy, OGBG-MOL*, DrugOOD, and synthetic tree tasks. The central advertised claim is that PiPE is provably more expressive than both PH and PE.

Significance. The idea of feeding PE-based persistence diagrams back into a GNN is natural, and the experimental study is substantial: the paper compares against SignNet, BasisNet, PEG, SPE, DE, and TOGL/RePHINE across several benchmarks and releases code. If the strict-expressivity theorems were correct, the paper would make a meaningful contribution to the theory of graph positional encodings. However, the definitive claims rest on proofs that contain false spectral statements (Propositions 4.1 and 4.2) and an invalid inference about k-FWL colorings (Proposition 4.4). Because the advertised contribution is precisely the provable separation, these errors are load-bearing; the empirical gains alone do not establish the paper's central claim.

major comments (4)
  1. [Appendix C.7, Proposition 4.1] The witness pair G=∪_{i=1}^{3n}C4 and G′=∪_{i=1}^n(C6∪C6) is claimed to satisfy Φ_k(G)=Φ_k(G′) 'having identical constant eigenvector' for all k≤12n. This is false for the normalized Laplacian Δ=I−D^{-1/2}AD^{-1/2} defined in Section 2. The multiset of eigenvalues of C4 is {0,1,1,2} and that of C6 is {0,1/2,1/2,3/2,3/2,2}. Hence G has 3n zero eigenvalues and G′ has 2n, and the first nonzero eigenvalues of the two graphs are 1 and 1/2, respectively. The k-lowest eigenpairs cannot coincide for any k>2n, and even for k≤2n the zero eigenspace dimensions differ, so no canonical choice of eigenvectors makes the encodings equal. Since this witness is the only evidence in the paper that LSPE fails on the pair, the strict-superiority conclusion of Proposition 4.1 is unsupported.
  2. [Appendix C.8, Proposition 4.2] For G=∪_{i=1}^n C4 and G′=∪_{i=1}^n P4, the proof asserts that 'the k smallest laplacian eigenvalues ... are all equal to 0 with the identical constant eigenvector, where k<4n'. This is false: the normalized Laplacian has exactly n zero eigenvalues (one per connected component), so for any k>n the k-lowest eigenvalues cannot all be zero, and the nonzero eigenvalues of C4 and P4 are different. The subsequent inference that PH+LPE cannot separate the pair is therefore not established. The construction may be salvageable by restricting to k≤n, but the proof as written needs to be corrected before the proposition can be accepted.
  3. [Appendix C.10, Proposition 4.4] The proof assumes that if k-FWL distinguishes two graphs, then there is a tuple v_w with color C∞(v_w) that does not appear in the other graph. k-FWL distinguishes graphs when the histograms of stable tuple colors differ, which can occur even when the sets of colors are identical and only the multiplicities differ. The proof also assumes 'hash functions are injective in nature', which is not guaranteed by any standard hash and is not an axiom in the paper. These two assumptions are needed to conclude that Q=∅ is a color-separating set, so the theorem is not proven.
  4. [Appendices C.1, C.4, C.7, C.8] Several proofs rely on the phrase 'identical constant eigenvector' for graphs with multiple connected components. For the normalized Laplacian, the zero eigenvalue has multiplicity equal to the number of connected components, and its eigenvectors are degree-normalized component indicators, not a single constant vector. The paper never specifies a canonical choice of basis for repeated eigenvalues, so equality statements of the form Φ_k(G)=Φ_k(G′) for graphs with different component counts are not well-defined. This is not a stylistic issue: the witness constructions in C.7 and C.8 choose k inside the zero eigenspace, and their validity depends on a basis convention that the paper neither states nor justifies.
minor comments (4)
  1. [Appendix C.1, S2/S3 and Appendix C.4] The statement that for G consisting of n/2 copies of K1∪K3 the k smallest eigenvalues are all zero for k≤n is only true for k≤n/2 when the encodings are required to come from the isolated-node components, since the triangle components contribute eigenvectors with entries 1/√3. The existence argument still works with k=n/2, but the stated range is incorrect.
  2. [Appendix C.5] The sentence 'have different diagrams D0(G1,Φ_k(G1), f) = D0(G2,Φ_k(G2), f)' appears to contain an equality where the intended relation is inequality, as stated in Proposition 3.6.
  3. [Appendix headers and Figure 1] The theorem numbering is inconsistent: Appendix C.5 is labeled 'Proof of Theorem 3.6' and C.6 is labeled 'Proof of Theorem 3.5', while the main text refers to Propositions 3.5 and 3.6; Figure 1 mixes 'Theorem' and 'Proposition' labels. Please unify the terminology.
  4. [Appendix C.9, Eq. (19)] The two matrices displayed for Φ_k(K) and Φ_k(K′) are not equal as arrays but are row-permutations of each other; please state explicitly that the encodings are compared as multisets of node features up to vertex relabeling.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: PiPE's derivations are self-contained special-case constructions; the flawed spectral witnesses are correctness errors, not circular reductions.

full rationale

The paper's derivation chain does not reduce to its inputs. PiPE's expressivity subsumption arguments (Proposition 4.1, Appendix C.7; Proposition 4.2, Appendix C.8) proceed by exhibiting choices of aggregation/update functions that recover LSPE or PH+LPE, which is a standard special-case argument rather than a circular definition. Strict expressivity is then attempted through witness graph pairs, relying on the elementary fact that differing numbers of connected components yield different 0-dimensional persistence diagrams; the paper cites Immonen et al. (2023) for this fact, and although that prior work shares authors (Souza, Garg), the cited theorem concerns VC filtrations and does not assume PiPE's conclusion, so it is independent support rather than a self-citation chain. Lemma 3.3 is likewise a direct corollary of prior PH theory, not a restatement of the paper's target result. No fitted parameter is relabeled as a prediction, and the empirical benchmarks are external. The most serious problems in the paper are mathematical errors, not circularity: Appendix C.7 asserts that union-of-C4 and union-of-C6 graphs have identical Laplacian PEs for k up to 12n, and Appendix C.8 makes the analogous claim for C4 versus P4, both of which are false for the normalized Laplacian because the component spectra and nullspace row norms differ. A false premise invalidates a proof without making it circular. Accordingly, the circularity score is 0.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central expressivity claims rest on standard persistent homology facts (different β0 implies different 0-dimensional persistence diagrams) and on the message-passing GNN framework. Three of the four cited PH facts come from Immonen et al. (2023), which shares two authors with this paper; these are background theorems rather than fitted results. The proofs of Props 4.1 and 4.2 do not depend on the fitted hyperparameters but contain false spectral claims. No new physical or mathematical entities are postulated.

free parameters (3)
  • PH embed dim = 64
    Latent dimension of PH features, chosen manually for experiments (Table 3).
  • Num Filt = 8
    Number of filtrations for PH layers, chosen manually (Table 3).
  • Hidden Filtration = 16 or 128
    Hidden dimension of the learnable filtration function, chosen per task (Tables 3 and 4).
assumptions (4)
  • standard math Persistent homology of graphs captures only 0- and 1-dimensional topological features (β0, β1) for vertex-color filtrations.
    Used throughout Section 3 to bound PH expressivity; standard in TDA.
  • domain assumption Message-passing GNNs with order-invariant aggregation cannot distinguish graphs with identical 1-WL colorings.
    Standard result (Xu et al. 2019), used in proofs of Props 4.1.
  • domain assumption Theorem 1 and Lemma 5 from Immonen et al. (2023): different β0 implies different 0-dim persistence diagrams for any injective filtration; birth times encode vertex colors.
    Cited as prior work by same group; used in Props 3.4, 3.6, 4.2.
  • standard math Injective hash functions exist and preserve tuple colors in Prop 4.4.
    Used in Prop 4.4 proof; the implication is logically flawed.

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Pith. "Pith review of Positional Encoding meets Persistent Homology on Graphs." pith.science (2026). https://pith.science/paper/PJVVNUZX

@misc{pith2026250605814,
  author       = {Pith},
  title        = {Pith review of: Positional Encoding meets Persistent Homology on Graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PJVVNUZX}},
  note         = {Machine review of arXiv:2506.05814}
}
read the original abstract

The local inductive bias of message-passing graph neural networks (GNNs) hampers their ability to exploit key structural information (e.g., connectivity and cycles). Positional encoding (PE) and Persistent Homology (PH) have emerged as two promising approaches to mitigate this issue. PE schemes endow GNNs with location-aware features, while PH methods enhance GNNs with multiresolution topological features. However, a rigorous theoretical characterization of the relative merits and shortcomings of PE and PH has remained elusive. We bridge this gap by establishing that neither paradigm is more expressive than the other, providing novel constructions where one approach fails but the other succeeds. Our insights inform the design of a novel learnable method, PiPE (Persistence-informed Positional Encoding), which is provably more expressive than both PH and PE. PiPE demonstrates strong performance across a variety of tasks (e.g., molecule property prediction, graph classification, and out-of-distribution generalization), thereby advancing the frontiers of graph representation learning. Code is available at https://github.com/Aalto-QuML/PIPE.

Figures

Figures reproduced from arXiv: 2506.05814 by the authors.

Figure 1
Figure 1. Overview of our key contributions the cornerstone of topological data analysis and offers a powerful framework to capture multi-scale topological in￾formation from data. In the context of graphs, PH has been recently used, e.g., to boost the expressive and representa￾tional power of GNNs (Horn et al., 2022; Immonen et al., 2023; Carriere et al., 2020; Verma et al., 2024). However, while both PE and PH schemes enhanc… view at source ↗
Figure 2
Figure 2. Overview of PiPE and integration with backbone GNN. At each layer ℓ, the node embeddings {x ℓ−1 u }u are updated using the positional embeddings {p ℓ−1 u }u and the topological embeddings {r ℓ−1,0 u , rℓ−1,1 u }u. The position embeddings {p ℓ−1 u }u are updated and then leading to the computation of persistence diagrams D0 ℓ , D1 ℓ leading to topological embeddings {r ℓ,0 u , rℓ,1 u }u. In readout phase, the final l… view at source ↗
Figure 3
Figure 3. Incomparability of PH and PE. The graph G , resembling an anthracene molecule, consists of three conjoined 3-cycles sharing a ring. Using the k = β0(G) + 1 lowest Laplacian PE eigenmaps, PE fails to capture the number of basis cycles. The k th (Fiedler’s) eigenvector partitions G into two components, as shown in G′ , missing the cyclic structure. The graphs K consist of two 4-cycles sharing consecutive nodes, and K′… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Construction for Theorems 3.4 and 4.3. The graph G with two copies of C5 is indistinguishable from graph G′ with C10 by RW-based PE , despite having disconnected components. This follows since every node in G and G′ has the same degree and same RW-based PE Φk. In contr…
Figure 5
Figure 5. Figure 5: Test MAE for drug molecule property prediction and graph classification results. Integration of PiPE into backbone message passing schemes leads to better downstream performance across diverse datasets. For more details, see [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: AUC-ROC (↑) results for DrugOOD Benchmark. PiPE outperforms the competing baselines in achieving better scores for OOD-Test. For more details, see [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: Identity filtrations and Runtime Comparisons. [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: Exemplar graphs for Theorems 3.1, 3.4 and 4.2. Similarly to statement S1, here we proceed with a proof by example. To prove this statement, we will provide a pair of graphs G1 and G2 with β0(G1) ̸= β0(G2) and β1(G1) ̸= β1(G2) for which Φk(G1) = Φk(G2) for some k. Let C…

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Reference graph

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