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REVIEW 3 major objections 6 minor 55 references

HOPSE: Scalable Higher-Order Positional and Structural Encoder for Combinatorial Representations

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper argues that higher-order message passing is unnecessary: decomposing a combinatorial complex into Hasse graphs and applying graph positional and structural encodings matches or beats existing topological models while training up…

desk verdict A genuinely useful empirical recipe for message-passing-free TDL, but the paper's headline expressivity claim is not supported by the math and should be softened or proven. read the letter →

arxiv 2505.15405 v3 pith:3ZPYLAL3 submitted 2025-05-21 cs.LG

Guillermo Bernárdez , Marco Montagna , Louis Van Langendonck , Martin Carrasco , Amirreza Akbari , Louisa Cornelis , Mathilde Papillon , Pere Barlet-Ros
show 2 more authors
Nina Miolane Lev Telyatnikov
This is my paper
classification cs.LG
keywords topologicaldeeplearninghigher-ordermessagepassingcombinatorialcomplexesHassegraphspositionalandstructuralencodingsgraphneuralnetworksscalabilityMANTRAtriangulations
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

This paper argues that higher-order message passing (HOMP) is not required to learn from combinatorial complexes: a complex can instead be broken into its Hasse graphs and characterized by standard graph positional and structural encodings (PSEs). The proposed HOPSE framework precomputes these encodings, aggregates them by cell rank, and feeds the result into small MLPs, so training cost scales linearly with the number of cells. On molecular graph benchmarks and topological triangulation tasks, HOPSE matches or beats existing topological models while training up to 7 times faster, with the strongest gains on predicting Betti numbers and orientability of 2-manifold triangulations. The paper reads this as evidence that the expensive propagation routes of HOMP architectures can often be replaced by cheap, structure-aware encodings.

What carries the argument

The central object is the strictly augmented Hasse graph decomposition of a combinatorial complex: for each chosen neighborhood function, the complex becomes a directed graph whose nodes are cells and whose edges record the neighborhood relation. The machinery is then any graph positional or structural encoder applied to these Hasse graphs, aggregated per target rank by concatenation and processed by MLPs. The paper's theoretical motivation is that a combinatorial complex is determined by its Hasse decomposition, so encodings of the decomposition act as piecewise invariants of the complex; the fixed PSEs used in practice are permutation-equivariant but not fully injective, which is why the paper presents them as a practical instantiation rather than a complete characterization.

What would settle it

Find two triangulations with different MANTRA labels that produce identical HOPSE encodings under the tested neighborhood set and PSE choices; if HOPSE's final embedding is identical for both, the fixed encoding is not injective and the characterization claim fails on that class.

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Extended reading notes

Core claim

HOPSE establishes that structural characterization of an arbitrary combinatorial complex can be decoupled from attribute learning. Given a set of neighborhood functions, the complex is decomposed into strictly augmented Hasse graphs; each graph is encoded by one or more PSEs (Laplacian eigenvectors, random-walk, heat-kernel, electrostatic potentials, or the pretrained GPSE), and the encodings are aggregated by target rank. These rank-wise structural features are projected and combined with initial cell features by MLPs, and a readout produces the complex-level prediction. Because all structural computation happens before training, the learning step is linear in the number of cells and independent of connectivity, and the paper reports that this pipeline matches or surpasses HOMP-based models on standard benchmarks while achieving up to 7x training-time speedups.

Load-bearing premise

The method assumes the fixed, lossy encodings of the chosen Hasse graphs preserve every distinction the label requires; the paper proves only that some pair of neighborhoods distinguishes non-isomorphic complexes, not that the tested set does.

Editorial extensions

If this is right

  • If correct, HOMP is not a hard requirement for topological tasks; PSE-augmented models can serve as cheaper baselines for higher-order learning.
  • Training time for HOPSE scales linearly with the number of cells, allowing higher-order learning on complexes too large for message-passing topological networks.
  • Choice of neighborhood functions and PSEs becomes the main design decision, with ablations showing adjacency-only sets often enough on graph tasks while mixed incidence sets matter on topological tasks.
  • The speedups (up to 7x) hold across training, and preprocessing can be amortized once per dataset.
  • A pretrained graph encoder (GPSE) can transfer to higher-order domains, giving a path to domain-specific pretraining for topological learning.

Reading between the lines

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

  • Editorial inference: if the Hasse-decomposition characterization holds broadly, HOPSE could be used as a generic front-end for any architecture, replacing message-passing layers with precomputed structural features and making the framework a drop-in alternative to GNN+PSE pipelines on graph data.
  • Editorial inference: the linear-complexity claim depends on how many neighborhoods and PSEs are selected; the paper's complexity analysis fixes these as constants, so a testable extension is to measure wall-clock scaling as neighborhood sets and PSE dimensions grow.
  • Editorial inference: the strong results on MANTRA Betti-number prediction suggest that spectral and walk-based signatures of Hasse graphs may carry homological information; one could test whether HOPSE embeddings correlate with persistent homology features on the same triangulations.
  • Editorial inference: the paper does not prove injectivity for the tested PSEs, so a natural extension is to learn injective encoders on the Hasse graphs, trading some speed for guaranteed distinguishability.
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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

3 major / 6 minor

Summary. The paper proposes HOPSE, a message-passing-free framework for learning over combinatorial complexes. HOPSE decomposes a combinatorial complex into strictly augmented Hasse graphs for a chosen set of neighborhood functions, applies graph positional/structural encoders (PSEs) to these graphs, aggregates the resulting encodings by target rank, and then learns rank-wise MLP projections that are combined by a readout. The claimed contributions are (i) linear training-time scaling with the number of cells, (ii) preservation of the expressive power and permutation equivariance of higher-order message passing (HOMP) approaches, and (iii) empirical competitive or superior accuracy with large speedups relative to HOMP baselines. Experiments are run on TUDataset graph benchmarks (MUTAG, PROTEINS, NCI1, NCI109, ZINC-12K) and the MANTRA topological benchmarks under the TopoBench protocol, with two instantiations: HOPSE-M (handcrafted PSEs) and HOPSE-G (pretrained GPSE).

Significance. If the expressivity and scaling claims were rigorously established, the paper would make a useful contribution by questioning the necessity of HOMP and providing a scalable preprocessing-based alternative for topological deep learning. The empirical study is a strength: it uses a standardized protocol, reports runtimes and ablations over neighborhood sets, and provides first TopoBench-protocol baselines on MANTRA. The observed results, e.g., HOPSE-M achieving the best F1 on MANTRA beta1 and beta2 and consistent speedups, are valuable regardless of the theoretical framing. However, the central theoretical claim in the abstract is not supported by the stated results, and the paper's significance as a 'preserving expressive power' method is therefore reduced to a conjecture with empirical support.

major comments (3)
  1. [Abstract and Section 4.1] The claim that HOPSE 'preserves the expressive power ... of the HOMP approaches' is not established. Section 4.1 states that 'a CC is determined by its Hasse decomposition,' but Proposition B.2 only shows that for any non-isomorphic pair of CCs there exists some pair of neighborhood functions whose strictly augmented Hasse graphs are non-isomorphic. This does not imply that the fixed neighborhood sets NS used in Section 5 (e.g., Mix-2, Inc-3) provide Hasse graphs that distinguish all CCs that HOMP can distinguish. The abstract's expressivity claim needs either a proof for the specific NS choices or a substantially weakened formulation, e.g., 'can preserve expressive power for suitable neighborhood sets'.
  2. [Section D.1 and Proposition B.2] Even if the Hasse decomposition were injective for a fixed NS, the concrete PSEs used in HOPSE-M are explicitly admitted in Section D.1 to be 'not fully injective'. LapPE cannot distinguish cospectral graphs, RWSE and HKdiag diagonal features collide for certain graphs, and EStat is a summary statistic. Consequently, non-isomorphic Hasse graphs can map to identical HOPSE-M encodings, so the composition 'Hasse decomposition + PSE aggregation' is not injective. The claim of preserving HOMP expressive power therefore fails for the actual HOPSE-M instantiation. The paper should either use injective graph characterizations (e.g., homomorphism counts with sufficient width, as mentioned in the Limitations) or reposition the theoretical statement as a property of the framework in the limit of injective encoders rather than a property of the evaluated variants.
  3. [Section E and Table 12/13] The 'scales linearly with the size of combinatorial representations' claim applies only to training time, not to preprocessing. The complexity analysis in Section E computes O(N_r D^2 (K+1)L) for the trainable components, but preprocessing includes LapPE eigendecomposition, which is O(n^3) per Hasse graph, as the same section acknowledges. Tables 12 and 13 show that preprocessing can be large: e.g., HOPSE-G with Mix-1 on MANTRA requires 7329 seconds of preprocessing. The abstract should specify that the linear scaling is for training and inference, not end-to-end, and the total cost (preprocessing + training) should be reported in the main runtime comparison.
minor comments (6)
  1. [Throughout] There are frequent typos and grammatical errors, including 'it's' for 'its' (Section 4.1), 'intercheangeably' (Section 1), 'scenarions' (Section 5.2), 'usefull' (Section 5.2), and 'does not included' (Section 5.2). The manuscript would benefit from a careful proofreading pass.
  2. [Figure 2 and Section 4.1] The caption labels in Figure 2 (A-F) are not explicitly referenced in the text in the same order; the text should state where each subfigure is discussed in the pipeline description.
  3. [Table 2] The heading 'β 1 β2' is missing a separator and is ambiguous; it should be 'β1 / β2' or two separate columns.
  4. [Section F.2] The discussion of ZINC discrepancies is useful but should appear earlier, since Table 1 shows HOPSE-M underperforming GCCN on ZINC; a reader may otherwise misinterpret the result as a failure rather than a protocol choice.
  5. [Section D.2] The statement that GPSE 'produces latent descriptors using standard message passing' contradicts the main text's claim that HOPSE is 'free of message passing layers'; HOPSE-G should be described as 'uses a pretrained message-passing encoder in preprocessing, but has no message passing during downstream training.'
  6. [Section B.2 (Proposition B.2 proof)] The proof of Proposition B.2 contains a vague argument when it says 'this maps encode rank information' and 'this implies that there is a CC-Isomorphism'; the proof should be expanded to make the construction of the function f and the preservation of set-inclusion explicit.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: HOPSE's structural encodings are precomputed from the input complex and its downstream predictions are trained on fixed splits, so the central derivation is not reduced to its own outputs.

full rationale

The derivation chain is self-contained in the relevant sense. HOPSE computes PSEs from Hasse graphs as a preprocessing step (Eq. 1 and Eq. 2) and then trains MLPs on the precomputed matrices; no target label or test statistic is used to fit the structural constants. The claim that a CC is determined by its Hasse decomposition (Sec. 4.1) is supported by an in-house argument in Sec. B, and although Proposition B.2 only establishes existence of some distinguishing neighborhoods rather than injectivity for the fixed NS sets used in Sec. 5, this is an expressivity gap, not a circular reduction. Section D.1 explicitly concedes that the chosen PSEs are 'not fully injective' while being permutation equivariant, which is an honest limitation statement rather than a hidden reuse of the target result. Citations to TopoBench (Telyatnikov et al., 2024) and Papillon et al. (2024) overlap with the authors, but they supply benchmark infrastructure and definitions; under the review rules, parameter-free definitions with stated assumptions count as independent support, and they are not used to prove the central claim by assumption. No fitted parameter is renamed as a prediction, and no equation reduces to its own input. The expressivity criticism raised in the skeptical reading is a correctness and evaluation concern, not a circularity of the derivation chain.

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

The central claim rests on the premise that Hasse graphs, after PSE encoding, preserve the information needed for each task. This is not established: Proposition B.2 gives only an existence result over all possible neighborhoods, and the PSEs used are heuristic graph descriptors. The method also inherits the choice of neighborhood set and GPSE checkpoint as per-dataset hyperparameters. No new physical or mathematical entities are introduced.

free parameters (3)
  • Neighborhood set NS = Per-dataset best among 8 configurations (e.g., Mix-2 for MANTRA beta2, Adj-2 for ZINC simplicial)
    Performance depends on which Hasse graphs are included; the paper searches over 8 configurations and selects per dataset by validation.
  • GPSE pretraining dataset = One of ZINC, GEOM, MOLPCBA, PCQM4MV2, treated as hyperparameter
    HOPSE-G's encoding quality depends on which pretrained GPSE checkpoint is chosen; the best is selected per dataset.
  • PSE hyperparameters = K=60 steps for RWSE/HK; eigenvector count and EStat dimension not fixed in text
    These choices control the richness of the handcrafted structural descriptors in HOPSE-M.
assumptions (3)
  • ad hoc to paper A combinatorial complex is determined by its Hasse decomposition
    Section 4.1 states this without proof. Prop B.2 only shows existence of some distinguishing neighborhood pair, not that the fixed NS or lossy PSEs are injective.
  • domain assumption Graph PSEs provide sufficient characterization of Hasse graphs for the downstream tasks
    LapPE, RWSE, HK, EStat, and GPSE are approximate, non-injective graph descriptors; no expressivity bound is given for the pipeline.
  • domain assumption Baselines, including the re-implemented SaNN, are configured fairly under the TopoBench protocol
    The 'matches or surpasses' claim relies on published baseline results and a new re-implementation of SaNN without validation against the original.

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Cite this review

Pith. "Pith review of HOPSE: Scalable Higher-Order Positional and Structural Encoder for Combinatorial Representations." pith.science (2026). https://pith.science/paper/3ZPYLAL3

@misc{pith2026250515405,
  author       = {Pith},
  title        = {Pith review of: HOPSE: Scalable Higher-Order Positional and Structural Encoder for Combinatorial Representations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3ZPYLAL3}},
  note         = {Machine review of arXiv:2505.15405}
}
read the original abstract

While Graph Neural Networks (GNNs) have proven highly effective at modeling relational data, pairwise connections cannot fully capture multi-way relationships naturally present in complex real-world systems. In response to this, Topological Deep Learning (TDL) leverages more general combinatorial representations--such as simplicial or cellular complexes--to accommodate higher-order interactions. Existing TDL methods often extend GNNs through Higher-Order Message Passing (HOMP), but face critical scalability challenges due to the steep complexity overhead of propagating messages through combinatorial structures. To overcome this limitation, we propose HOPSE (Higher-Order Positional and Structural Encoder), a framework free of message passing layers that uses Hasse graph decompositions to derive efficient and expressive encodings over arbitrary higher-order domains. Notably, HOPSE scales linearly with the size of combinatorial representations while preserving the expressive power and permutation equivariance of the HOMP approaches. Experiments on molecular and topological benchmarks show that it matches or surpasses state-of-the-art performance while consistently achieving speedups over HOMP-based models, opening a new path for scalable TDL. The code is available at https://github.com/geometric-intelligence/topobench.git.

Figures

Figures reproduced from arXiv: 2505.15405 by the authors.

Figure 1
Figure 1. Neighborhood expansions of a complex. Given a complex T (left), three examples of strictly augmented Hasse graphs GN are illustrated corresponding to 4 neighborhood functions: adjacency of nodes w.r.t. edges (A0,1), incidence from nodes to edges (I0→1), incidence from faces to nodes (I2→1), and incidence from nodes to faces (I0→2). 2 Background This section introduces key concepts underpinning TDL. For brevity and g… view at source ↗
Figure 2
Figure 2. HOPSE pipeline. (A) Considering the collection of neighborhood functions NC = {A0,1, I0→1, I2→1, I0→2}, the input combinatorial complex T is decomposed into the corresponding Hasse graphs expansions {GN }NC . (B) PSEs are computed for each Hasse graph, G(GNi ), as defined in Equation (1). (C) Higher-order PSEs are derived via the target rank-aware aggregation ψk (see Equation (2)). (D) The PSEs Xr,k are transformed … view at source ↗
Figure 3
Figure 3. Illustration of a combinatorial complex (top) and different neighborhood types, grouped by the [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

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    \@ifxundefined[1] #1\@undefined \@firstoftwo \@secondoftwo \@ifnum[1] #1 \@firstoftwo \@secondoftwo \@ifx[1] #1 \@firstoftwo \@secondoftwo [2] @ #1 \@temptokena #2 #1 @ \@temptokena \@ifclassloaded agu2001 natbib The agu2001 class already includes natbib coding, so you should ...

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    \@lbibitem[] @bibitem@first@sw\@secondoftwo \@lbibitem[#1]#2 \@extra@b@citeb \@ifundefined br@#2\@extra@b@citeb \@namedef br@#2 \@nameuse br@#2\@extra@b@citeb \@ifundefined b@#2\@extra@b@citeb @num @parse #2 @tmp #1 NAT@b@open@#2 NAT@b@shut@#2 \@ifnum @merge>\@ne @bibitem@firs...

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    , " * write output.state after.block = add.period write newline

    @open @close @open @close and [1] URL: #1 \@ifundefined chapter * \@mkboth \@ifxundefined @sectionbib * \@mkboth * \@mkboth\@gobbletwo \@ifclassloaded amsart * \@ifclassloaded amsbook * \@ifxundefined @heading @heading NAT@ctr thebibliography [1] @ \@biblabel @NAT@ctr \@bibset...

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    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 gl...

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.