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Learning Efficient Positional Encodings with Graph Neural Networks

T0 review · 5 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read The paper claims that graph positional encodings can be learned by message-passing GNNs with random or basis node initializations, achieving linear complexity, stability, and high expressiveness without computing eigenvectors.

desk verdict A genuinely useful PE method with strong experiments; the headline sample-complexity theorem has an inverted inequality and an unverified norm assumption, but the empirical core holds up and deserves refereeing. read the letter →

arxiv 2502.01122 v1 pith:YDM37ZVD submitted 2025-02-03 cs.LG

Charilaos I. Kanatsoulis , Evelyn Choi , Stephanie Jegelka , Jure Leskovec , Alejandro Ribeiro This is my paper · ORCID
classification cs.LG MSC 68T0705C50
keywords positionalencodingsgraphneuralnetworkseigenvectorspermutationequivariancesamplecomplexityexpressivitystabilitymessagepassing
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

The paper introduces PEARL, a way to build graph positional encodings entirely from message-passing GNNs instead of from eigenvector decompositions. It argues that a GNN is a nonlinear function of the graph shift operator's eigenvectors, so feeding the GNN a batch of random or standard-basis node attributes and averaging the outputs recovers eigenvector-style positional information. The paper claims this construction is permutation equivariant in expectation, stable under graph perturbations independently of eigenvalue gaps, and expressive enough to count cycles up to length seven and to beat the 1-WL test. If true, this would give graph transformers and GNNs a scalable substitute for spectral encodings: linear or quadratic complexity instead of cubic, with comparable accuracy.

What carries the argument

The load-bearing object is the observation that message-passing GNN layers of the form $X^{{(l)}}$ = σ(Σ_{k=0}^{K-1} S^k $X^{{(l-1)}}$ H_k) act as nonlinear mappings applied to the eigenvectors V of the graph shift operator S, with first-layer weights constrained by eigenvalues and node features. PEARL exploits this by breaking node symmetries with M unique random or basis identifiers, processing each independently, and restoring permutation equivariance through statistical pooling (empirical mean for R-PEARL, sum for B-PEARL). The sample complexity theorem and the stability bound both flow from viewing the GNN as a bank of bounded filters H(S) applied to unit-variance inputs.

What would settle it

Train R-PEARL on graphs of increasing size with filter norms larger than 1/F, then measure the empirical variance of the pooled outputs as M grows; if the variance grows with depth or graph size rather than staying bounded, Theorem 4.3's premise fails. A second test is to apply random node permutations to a graph and measure whether the finite-M PEARL output changes by more than the predicted equivariance error, which would show that exact equivariance requires more samples than the theorem assumes.

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Extended reading notes

Core claim

The central claim is that learnable positional encodings can be generated by message-passing GNNs with statistical pooling, and that these encodings approximate equivariant functions of the graph's eigenvectors. For large graphs, R-PEARL initializes each node with M random scalar samples, runs the same GNN on each sample, and averages the outputs; the paper proves the required number of samples is bounded by 1/(δ $ε^{2}$), independent of graph size, and that the encoding inherits GNN stability while counting node-level 3-7 node cycles. For smaller graphs, B-PEARL initializes with the N standard basis vectors and sums the GNN outputs, which the paper shows is equivalent to the SPE architecture when the spectral filters are analytic, at quadratic rather than cubic cost.

Load-bearing premise

The paper's guarantee that a small number of random samples suffices assumes the learned message-passing filters have norm at most 1/F and the nonlinearities do not amplify variance; if trained filters are stronger than that, the sample bound can degrade with network depth and graph size.

Editorial extensions

If this is right

  • R-PEARL scales to large graphs: its feed-forward cost is Θ(M N F² + M |E| F), so with 10–100 samples it is roughly linear and avoids the O(N³) eigendecomposition.
  • B-PEARL matches or beats full eigenvector PEs on molecular and social benchmarks while reducing cost from cubic to quadratic, and it improves out-of-distribution size generalization on DrugOOD.
  • PEARL-augmented GNNs are strictly more expressive than the 1-WL test and can count node-level cycles of length 3 through 7, without pre-specified motif biases.
  • Because the stability bound does not depend on eigenvalue gaps, PEARL avoids the eigengap fragility of truncated eigenvector encodings such as SignNet and BasisNet.

Reading between the lines

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

  • Editorial extension: If the graph-size-independent sample bound transfers to trained models, PEARL-style encodings could make spectral information practical for graph transformers on graphs with billions of edges, where eigendecomposition is impossible.
  • Editorial extension: Finite-sample R-PEARL is only approximately equivariant, so there is a concrete trade-off between sample count and permutation invariance; the paper's own experiments show one random sample degrades performance sharply, but the exact error as a function of M is not characterized.
  • Editorial extension: Because B-PEARL coincides with SPE for analytic spectral filters, any future improvement to SPE's stability or expressivity should carry over to B-PEARL at reduced cost; this gives a cheaper testbed for spectral PE research.
  • Editorial extension: The cycle-counting result suggests a direct probe: measure whether PEARL-derived encodings recover known cycle counts on graphs where closed-form counts are available, and whether structurally aware sampling extends counting to longer motifs, which the paper leaves outside its scope.
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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

5 major / 5 minor

Summary. The paper proposes PEARL, a framework for constructing graph positional encodings from message-passing GNNs. In R-PEARL, each node is initialized with M i.i.d. random signals, the graph is processed M times, and the outputs are averaged; in B-PEARL, standard basis vectors are used and the outputs are summed. The authors argue that message-passing GNNs are nonlinear functions of the graph shift operator's eigenvectors (Prop. 3.1), that PEARL can universally approximate basis-invariant functions (Thm. 3.1), that the sample complexity of the empirical average is independent of graph size (Thm. 4.3), that the encodings count cycles and exceed 1-FWL expressiveness (Cors. 4.4, 4.5), and that they inherit GNN stability guarantees (Cor. 4.6). Experiments on REDDIT, ZINC, DrugOOD, RelBench, CSL, and Peptides-struct show competitive or better performance relative to eigenvector-based baselines, often at lower claimed complexity.

Significance. If the theoretical claims held as stated, PEARL would be a substantial practical contribution: it would offer a scalable, stable, and expressive alternative to full spectral positional encodings. The paper includes useful open-source code, a broad experimental evaluation, and careful connections to existing spectral PE methods. The empirical results are genuinely promising, especially B-PEARL on ZINC and the low-sample behavior of R-PEARL on REDDIT. However, several load-bearing theoretical statements are not currently justified for the architectures actually evaluated, and one central theorem contains a reversed inequality. The paper's conceptual contribution is solid, but the formal backing needs correction before the scalability guarantee can be accepted.

major comments (5)
  1. [Section 4.1, Theorem 4.3 and Appendix F] The inequality in Theorem 4.3/Eq. (10) is reversed. From Chebyshev's inequality as written in Eq. (49), P(|1/M Σ P(m) - E[Φ]| ≥ ε) ≤ Var/(M ε²). To make this probability at most δ, one needs M ≥ Var/(δ ε²), which with Var ≤ 1 gives M ≥ 1/(δ ε²), not M ≤ 1/(δ ε²). The proof therefore supports a lower bound on the number of samples, not the upper bound printed in the theorem. Since this theorem is the basis for the claimed graph-size-independent sample complexity, the statement must be corrected.
  2. [Section 4.1, Assumption 4.2; Section 6.1] Theorem 4.3 depends critically on Assumption 4.2 with β = 1/F and Cσ = 1, but the architecture described in Section 6.1 uses a 9-layer GNN with batch normalization, skip connections, K up to 18, and widths 40/128. Nothing in the paper enforces ∥H(S)∥ ≤ 1/F for the learned filters. Skip connections add an identity path with operator norm 1, and batch normalization changes the effective per-layer Lipschitz constants. If βF > 1, the bound in Eq. (73) grows as (βF)^{2L}, so the sample count needed for a fixed error can grow exponentially in depth. The constant-M scalability claim is therefore not established for the implemented PEARL architecture.
  3. [Section 3.2, Theorem 3.1 and Appendix C] The basis-universality proof uses polynomial filters of order K equal to the number q of distinct eigenvalues (Lemma C.1) and an unrestricted equivariant pooling function ρ, e.g., a high-order tensor IGN. For large graphs q can be as large as N, making the filter order and pooling cost incompatible with the paper's linear-complexity claim and with the fixed, small K values used in experiments (K ≤ 18). The universal-approximation and linear-complexity claims cannot both be supported by the current proof as written.
  4. [Section 5.1, Remark 5.2 and Appendix I] The claimed equivalence between B-PEARL and SPE is established by expanding the spectral functions α_i as Taylor series and truncating them to polynomials. An MLP, the actual choice in the SPE implementation cited by the authors, is not generally a polynomial. Exact equivalence holds only for polynomial α_i; for general analytic α_i the argument gives an approximation without an error bound. The remark overstates the relationship between the two architectures.
  5. [Section 4.3, Corollary 4.6 and Appendix H.2] The statement of Corollary 4.6 defines Lipschitz filters with constant C but the displayed bound is independent of C. In the stability framework of Gama et al. that the paper cites, the Lipschitz constant typically enters the bound, often through a factor such as C^L. Either C is implicitly set to 1, in which case that should be stated explicitly, or the bound must include C. As written, the displayed inequality does not follow from the stated assumptions.
minor comments (5)
  1. [Appendix F] The word 'requqired' appears in the proof of Theorem F.1; it should be 'required'.
  2. [Table 9] The rows for R-PEARL with K=1 and K=12 report the identical Test MAE of 0.0699; please clarify whether this is a duplicate row or an actual coincidence of results.
  3. [Tables 1 and 2] Typography: 'RREDDIT-M' in the Table 1 caption and 'ouperforms' in the Table 2 caption should be corrected.
  4. [Appendix A] The phrase 'to captural structural information' should read 'to capture structural information'.
  5. [Section 4.3, Eq. (11)] The notation Φ(G, ·)[:, f] is used without a definition; please define it when the stability bound is first stated.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; the paper's central derivations are either proved in-paper or rest on independent published theorems, though Theorem 4.3 contains a correctness issue in its sample-complexity bound.

full rationale

Walking the derivation chain, the paper's core claims do not reduce to their own inputs by construction. Proposition 3.1 is a direct algebraic re-expression of a message-passing layer as a nonlinear function of eigenvectors, not a circular claim. Theorem 3.1 (Basis Universality) is proved by explicitly constructing a GNN that outputs the basis projection matrices V_mu V_mu^T and then invoking the published BasisNet universality result (Lim et al.) as an external building block; this is a reduction, not an assumption of the conclusion. Corollary 4.4 (cycle counting) cites Theorem K.1 from Kanatsoulis & Ribeiro, and Corollary 4.6 (stability) cites Gama et al.; both are peer-reviewed, stated-assumption theorems from prior work, so under the review rules they count as independent support rather than circular self-citation. Remark 5.2's equivalence between B-PEARL and SPE is derived in Appendix I by Taylor-expanding analytic alpha functions, again a constructive reduction. The main theoretical concern is not circularity but correctness: Theorem 4.3 assumes beta = 1/F and C_sigma = 1, which are not verified for the implemented architectures with batch norm, skip connections, and K up to 18, and the Chebyshev argument yields M >= 1/(delta epsilon^2), not M <= 1/(delta epsilon^2) as stated. These issues undermine the strength of the scalability claim but do not make the claim equivalent to its inputs. The empirical comparisons against external baselines (SignNet, BasisNet, SPE, structure-based PEs) further show the work is not self-confirmatory. Accordingly, the circularity score is low.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The central claims rest on several assumptions that are either imported from prior work or chosen to make the bounds work. The most significant is the β=1/F bound on filter norms, which is not verified empirically and is needed for the sample complexity result. The universality claims further require filter orders up to the number of distinct eigenvalues and an unrestricted pooling function ρ, neither of which matches the practical architecture.

free parameters (2)
  • M (number of random samples) = 10-200 per dataset
    Selected on validation data; R-PEARL's practical equivariance and variance control depend on M, though the theorem says the required M is O(1/(δ ε²)).
  • K (graph filter order) = 2-18 per dataset
    Selected on validation data; universality requires K equal to the number of distinct eigenvalues, but experiments use small K.
assumptions (7)
  • domain assumption Assumption 4.1: pointwise nonlinearity σ is Lipschitz continuous with constant Cσ; Cσ=1 for ReLU, tanh, sigmoid.
    Used in the proof of Theorem 4.3 to bound variance propagation through activations; holds for ReLU, tanh, sigmoid.
  • ad hoc to paper Assumption 4.2: the linear operators H(S)=Σ h_k S^k satisfy ||H(S)|| ≤ β, with β=1/F in the central theorem.
    The specific value β=1/F is chosen so that var(X^(L)) ≤ 1 (Eqs. 73-74). This is not guaranteed by training and is not verified in experiments.
  • standard math The graph shift operator S has a complete eigendecomposition S = V Λ V^T with orthonormal eigenvectors (or α_n = V^{-1}[n,:] when S is non-symmetric).
    Used in Proposition 3.1 to express GNN layers as functions of eigenvectors.
  • standard math Polynomial filters of order K=q (number of distinct eigenvalues) can isolate individual eigenspaces (Vandermonde system, Lemma C.1).
    Used in the universality proofs (Appendix C, D); requires K up to the number of distinct eigenvalues, which can be as large as N.
  • domain assumption For Corollary 4.4, the nonlinearity σ is assumed to be a power function σ(x)=x^p for p≥2, or an analytic function analyzed via Taylor expansion.
    The cycle-counting result is existential for power activations; the practical architecture uses ReLU, so the guarantee does not directly transfer to the trained model.
  • domain assumption Expressiveness result relies on Theorem K.1 from Kanatsoulis & Ribeiro (ICLR 2024).
    The proof of Corollary 4.4 imports a prior result by the same first author and senior author.
  • domain assumption Stability bounds from Gama et al. (2020) apply to the GNN encoder Φ.
    Corollary 4.6 uses the stability framework of Gama et al.; Ribeiro is a co-author of both papers.

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Cite this review

Pith. "Pith review of Learning Efficient Positional Encodings with Graph Neural Networks." pith.science (2026). https://pith.science/paper/YDM37ZVD

@misc{pith2026250201122,
  author       = {Pith},
  title        = {Pith review of: Learning Efficient Positional Encodings with Graph Neural Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YDM37ZVD}},
  note         = {Machine review of arXiv:2502.01122}
}
read the original abstract

Positional encodings (PEs) are essential for effective graph representation learning because they provide position awareness in inherently position-agnostic transformer architectures and increase the expressive capacity of Graph Neural Networks (GNNs). However, designing powerful and efficient PEs for graphs poses significant challenges due to the absence of canonical node ordering and the scale of the graph. {In this work, we identify four key properties that graph PEs should satisfy}: stability, expressive power, scalability, and genericness. We find that existing eigenvector-based PE methods often fall short of jointly satisfying these criteria. To address this gap, we introduce PEARL, a novel framework of learnable PEs for graphs. Our primary insight is that message-passing GNNs function as nonlinear mappings of eigenvectors, enabling the design of GNN architectures for generating powerful and efficient PEs. A crucial challenge lies in initializing node attributes in a manner that is both expressive and permutation equivariant. We tackle this by initializing GNNs with random node inputs or standard basis vectors, thereby unlocking the expressive power of message-passing operations, while employing statistical pooling functions to maintain permutation equivariance. Our analysis demonstrates that PEARL approximates equivariant functions of eigenvectors with linear complexity, while rigorously establishing its stability and high expressive power. Experimental evaluations show that PEARL outperforms lightweight versions of eigenvector-based PEs and achieves comparable performance to full eigenvector-based PEs, but with one or two orders of magnitude lower complexity. Our code is available at https://github.com/ehejin/Pearl-PE.

Figures

Figures reproduced from arXiv: 2502.01122 by the authors.

Figure 1
Figure 1. PEARL framework: The input graph undergoes anonymization by removing its node and edge attributes. For each node, a set of M random or basis attributes is generated. Each sample is then independently processed by a message-passing GNN, and a pooling function ρ is applied to produce equivariant PEs. The graph structure, together with the generated PEs and any node or graph attributes, is subsequently processed using … view at source ↗
Figure 2
Figure 2. Ablation studies on the sample size for R-PEARL; It converges with only a few samples. [PITH_FULL_IMAGE:figures/full_fig_p028_2.png] view at source ↗

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.