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REVIEW 3 major objections 5 minor 93 references

Differentiable Lifting for Topological Neural Networks

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper introduces ∂lift, a trainable graph-lifting method that selects higher-order cells for topological neural networks end-to-end, outperforming static liftings by up to 45% on benchmarks.

desk verdict A genuinely useful framework for task-adaptive lifting, but the 'end-to-end differentiable' claim overreaches because the candidate proposal step (kNN, cycle basis) is non-differentiable; the empirical wins are real though the attribution to learning the topology is not fully isolated. read the letter →

arxiv 2608.01160 v1 pith:NMCUJRF2 submitted 2026-08-02 cs.LG cs.SI

Jorge Luiz Franco , Gabriel Duarte , Alexander Nikitin , Moacir Ponti , Diego Mesquita , Amauri H. Souza This is my paper · ORCID
classification cs.LGcs.SI
keywords topologicalneuralnetworksgraphliftingdifferentiablestructurelearninghypergraphcellcomplexesstraight-throughestimationend-to-endhigher-ordermessagepassing
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 proposes ∂lift (DiffLift), a method that learns, end-to-end with the task loss, which higher-order cells (hyperedges, edges, cycles, or simplices) should be added to an input graph before feeding it to a topological neural network. Instead of fixing a lifting rule a priori, ∂lift uses a backbone graph model to embed nodes, proposes candidate cells from those embeddings (nearest neighbors for hyperedges, cycle bases for higher cells), and accepts or rejects each candidate by sampling a Bernoulli variable whose probability is a learned function of the cell's node embeddings. The paper reports that this trainable lifting outperforms static liftings in 22 of 24 graph-classification settings, with gains of up to 45%, and beats a previous differentiable lifting method on node classification. If correct, the method turns graph lifting from a preprocessing choice into part of the model, letting the task itself decide which multi-node interactions matter.

What carries the argument

The load-bearing mechanism is the differentiable Bernoulli acceptance of candidate cells. A backbone graph model computes node embeddings; candidate hyperedges or edges are defined by $k$-nearest neighbors in embedding space, with $k$ itself sampled from a learnable distribution; higher-dimensional cells are proposed from the cycle basis of the current complex. Each candidate receives an acceptance probability from a permutation-invariant network over its node embeddings, and the discrete samples are trained with the straight-through estimator. This converts structure selection into a differentiable subproblem inside any topological neural network pipeline.

What would settle it

Construct a graph classification task where the label depends on a specific cycle that is not included in the computed cycle basis (e.g., a cycle that is a modulo-2 sum of basis cycles but not itself a basis element), and ensure no embedding-based $k$-NN hyperedge covers its vertices. If ∂lift then fails to beat a static cycle lifting on this graph despite the structure being essential to the label, the fixed candidate set is confirmed as the limiting factor.

Watch

Extended reading notes

Core claim

The central claim is that the discrete choice of a graph lifting can be optimized with gradient descent. ∂lift parameterizes a distribution over candidate higher-order cells: node embeddings $z_v$ produced by any graph neural network define neighborhoods (via $k$-nearest neighbors in embedding space), a sampled size $k_v$ comes from a learnable categorical distribution, and each candidate cell $C$ is included with probability $\Psi(\{\{z_v : v \in C\}\})$, where $\Psi$ is a learned permutation-invariant function. Training uses the straight-through estimator to propagate gradients through the Bernoulli decisions, so the entire pipeline—backbone graph model, proposal distributions, acceptance

Load-bearing premise

The optimal higher-order structure must be present among the candidate cells generated from embedding-space nearest neighbors and from the current cycle basis; if the task requires a cell that is not proposed, the learned acceptance probabilities cannot recover it and the method's gains are capped.

Editorial extensions

If this is right

  • Lifting becomes part of the objective: the task loss directly shapes which topological structures are retained.
  • The same recipe covers hypergraphs, simplicial complexes, cell complexes, and combinatorial complexes, so one learnable lifting replaces several hand-crafted ones.
  • Gains of up to 45% over connectivity- and feature-based static liftings suggest that fixed rules often leave performance on the table.
  • The method is architecture-agnostic: reported improvements hold across CW networks, cellular message passing, and hypergraph networks.
  • Adaptive hyperedge sizes emerge from a learned distribution over $k_v$, avoiding manual tuning of the neighborhood scale.

Reading between the lines

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

  • Because candidate generation is unsupervised and fixed per domain, the ceiling of the method is set by whether the optimal cells are expressible as $k$-NN neighborhoods or cycle-basis elements; a richer proposal mechanism (e.g., attention-based cell proposals) could further raise this ceiling.
  • The learned acceptance probabilities double as a per-cell importance score, offering a way to inspect which higher-order interactions the model relies on—something static liftings cannot provide.
  • The paper's own argument that 2-cells can shortcut bottleneck bridges suggests ∂lift could serve as a testbed for oversquashing: one could measure whether learned long-range cells actually reduce effective graph distances during training.
  • The framework's independence of the downstream topological model suggests it could be plugged into persistent-homology or equivariant layers, not just message-passing ones.
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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 / 5 minor

Summary. The paper introduces ∂lift (DiffLift), a framework for learning graph liftings to hypergraphs and cell/simplicial complexes in a task-driven way. The method first computes node embeddings with a GNN, then defines candidate higher-order cells (via k-nearest-neighbor sets for hypergraphs/1-cells and a cycle basis for D-cells), computes acceptance probabilities with MLP/DeepSet blocks, and samples accepted cells with a straight-through Bernoulli estimator. The authors report experiments on six graph-classification and four node-classification datasets across several TNN architectures, claiming consistent gains over static lifting baselines, with improvements up to 45%.

Significance. If the empirical gains are attributable to the learned lifting mechanism rather than to confounds, ∂lift would be a useful and broadly applicable tool for task-adaptive topological lifting. The paper's strengths include its generality across several topological domains, the breadth of the benchmark evaluation, the inclusion of a deterministic variant, a complexity/runtime appendix, and an anonymous code release. However, the central 'end-to-end differentiable lifting' claim is weakened by the fact that candidate generation itself is not differentiable, and the headline experimental claim is compromised by a capacity confound: ∂lift adds a GNN encoder and acceptance networks that static liftings do not include. These issues are load-bearing for the paper's main contribution and conclusions.

major comments (3)
  1. [§5.1, Tables 1–2] The comparison does not hold model capacity fixed. ∂lift augments the TNN with a GNN backbone (GPS/GIN) and acceptance MLPs/DeepSets, while the static-lifting baselines use the TNN alone. The reported gains could therefore be due to the additional parameters rather than to learning the lifting. Please add ablations that insert the same GNN encoder and acceptance network before a static lifting, or otherwise match parameter counts, so the effect of the learned acceptance mechanism can be isolated.
  2. [§3.1–§3.3, Eq. (3)] Candidate generation is not differentiable. The kNN selection in Eq. (3) and the cycle-basis computation in §3.3 are hard, discrete operations that receive no gradient; the straight-through estimator is applied only to the Bernoulli samples at Step 3. Thus the proposal set is not directly learned—only acceptance probabilities over a heuristic candidate family are. This weakens the abstract's 'end-to-end' and 'learned vertex-level latent representations to identify ... candidate cells' claims. Either make the proposal step differentiable (e.g., via a soft kNN relaxation) or explicitly recharacterize the contribution as learning acceptance over a fixed candidate family, and provide evidence that the learned embeddings improve the candidate sets themselves.
  3. [§5.1–§5.2, Tables 1–4] Reported results are based on three runs with no significance tests, and several standard deviations are large or overlapping. For example, Table 3 Texas CWN∂Lift is 80.18±3.37 versus Cycle 63.06±7.75, and Table 1 MUTAG UniGCNII∂lift is 89.47±4.30 versus k-hop 61.40±2.48. Without confidence intervals or paired tests, the claim that ∂lift 'consistently outperforms' static liftings is not statistically supported. Please add significance testing or at least report per-seed results and effect sizes.
minor comments (5)
  1. [§3.2, Eq. (3)] The set-builder notation defines a family of sets rather than a unique candidate hyperedge. Specify the intended k-nearest-neighbor set and state tie-breaking behavior.
  2. [Abstract / §5.1] The 'gains of up to 45%' claim is ambiguous: it is unclear whether this is relative improvement or absolute percentage points, and which table cell/cells it refers to. Define the metric explicitly.
  3. [§6, Limitations] The abstract calls ∂lift 'scalable', but the Limitations section states that cycle-basis computation for cell complexes can be cubic in the number of nodes. Qualify the scalability claim accordingly.
  4. [Table 1 caption] The caption says ∂lift is better in '90% of cases' without giving the exact numerator/denominator. Since Table 1 contains one or more counterexamples, provide the precise count.
  5. [Figure 2] The caption text is grammatically awkward ('sampled rank-1 cell sampled rank-2 cell'); consider rewording for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the learned acceptance probabilities are trained end-to-end on task losses and evaluated on held-out splits; candidate generation is heuristic but not a fitted prediction.

full rationale

We find no circular step in the claimed derivation. The paper's central mechanism is a learning algorithm: node embeddings z_v from a GNN define candidate cells (Eq. 3 for hypergraphs; cycle-basis selection for D≥2 in Section 3.3), and an MLP/DeepSet maps cell embeddings to acceptance probabilities (Eq. 4), with Bernoulli samples trained via a straight-through estimator. The reported gains come from training these parameters, including the backbone GNN, on train/validation splits and evaluating on held-out test splits; no fitted constant or learned parameter is reused to define the evaluation metric or to construct the comparison. The kNN and cycle-basis proposal steps are discrete and non-differentiable, but the paper does not claim gradients flow through the neighbor/cycle selection itself, and the only flagged limitation is computational (cubic cycle-basis cost in the Limitations section). Self-citations (e.g., Nikitin et al. 2022 in the kernel-lifting baseline and Verma et al. 2024 in related work) are not load-bearing: the central claim of task-adaptive lifting does not reduce to those citations, and the baselines are standard static methods. Thus the derivation is self-contained rather than circular.

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

The method introduces no new physical or mathematical entities. The central claim rests on the expressiveness of the candidate sets (k-NN balls and cycle bases), the reliability of straight-through gradient estimates, and the transferability of TNN hyperparameters chosen for static liftings. The trained weights and the hyperparameters kmax, kmin, and gamma are the fitted or hand-chosen quantities that affect performance.

free parameters (4)
  • kmax (max neighborhood size) = searched in {3,5,7,9,11}
    Upper bound on the cardinality of sampled hyperedges and edges; chosen on validation in Appendix C; directly controls the candidate search space.
  • kmin (min neighborhood size) = not reported
    Lower bound in the Categorical distribution over k_v; hand-set, not reported in the paper.
  • gamma (deterministic variant threshold) = 0.5
    Acceptance threshold for the deterministic version in Appendix E.2; fixed without tuning.
  • GNN backbone and acceptance network weights = learned during training
    Weights of the GNN, the MLP parameterizing pi_v, the acceptance network Psi, and the DeepSet are fit on training data. The empirical claim depends on these standard fitted parameters, but they are not free constants in a derivation.
assumptions (4)
  • domain assumption Candidate hyperedges are k-NN balls in the Euclidean space of learned node embeddings (Section 3.2, Eq. 3).
    Assumes task-relevant higher-order interactions are localized in the GNN embedding space; if not, the proposal distribution never generates the useful cells.
  • domain assumption Candidate D-cells for D>=2 are drawn from a cycle basis of the current complex K_{D-1} (Section 3.3, Case D>=2, Step 2).
    Assumes the optimal 2-cells are cycles of the augmented graph; non-cycle cells that could help the task are never proposed.
  • standard math Straight-through estimator provides usable gradients through Bernoulli samples (Section 3.1, Step 4, citing Bengio et al., 2013).
    The estimator is known to be biased; the paper relies on it for end-to-end training without analyzing bias or variance in this setting.
  • domain assumption TNN hyperparameters from TopoBench transfer to ∂lift-augmented pipelines (Appendix C.1).
    The paper reuses optimal TNN hyperparameters tuned for static liftings; ∂lift changes the input distribution, so these may be suboptimal.

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Cite this review

Pith. "Pith review of Differentiable Lifting for Topological Neural Networks." pith.science (2026). https://pith.science/paper/NMCUJRF2

@misc{pith2026260801160,
  author       = {Pith},
  title        = {Pith review of: Differentiable Lifting for Topological Neural Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NMCUJRF2}},
  note         = {Machine review of arXiv:2608.01160}
}
abstract

Topological neural networks (TNNs) enable leveraging high-order structures on graphs (e.g., cycles and cliques) to boost the expressive power of message-passing neural networks. In turn, however, these structures are typically identified a priori through an unsupervised graph lifting operation. Notwithstanding, this choice is crucial and may have a drastic impact on a TNN's performance on downstream tasks. To circumvent this issue, we propose $\partial$lift (DiffLift), a general framework for learning graph liftings to hypergraphs and cellular- and simplicial complexes in an end-to-end fashion. In particular, our approach leverages learned vertex-level latent representations to identify and parameterize distributions over candidate higher-order cells for inclusion. This results in a scalable model which can be readily integrated into any TNN. Our experiments show that $\partial$lift outperforms existing lifting methods on multiple benchmarks for graph and node classification across different TNN architectures. Notably, our approach leads to gains of up to 45% over static liftings, including both connectivity- and feature-based ones.

Figures

Figures reproduced from arXiv: 2608.01160 by the authors.

Figure 1
Figure 1. [Left] Lifting to different domains can lead to disparate performances. Accuracies taken from the best TNNs in (Telyat￾nikov et al., 2025). [Right] Performances of liftings to the same domain (hypergraph) vary greatly. Values taken from [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Overview of ∂lift. For a given input graph, we first compute node embeddings using GNNs. Then, we use these embeddings to select cells/hyperedges. Cell-level embeddings run through MLPs responsible for returning acceptance probabilities. For hierarchical domains (e.g., cell complexes), cells are generated in increasing dimensionality. From the accepted cells, we form a relational object that is sent to an off-the-sh… view at source ↗
Figure 3
Figure 3. Examples of topological domains. set V is a set K of subsets of V (the simplices) such that, for every σ ∈ K and every non-empty τ ⊂ σ, we have that τ ∈ K. Thus, we can define ASCs as a family of subsets K ⊆ 2 V of V that is closed under taking subsets. The dimension of a simplex is equal to its cardinality minus 1, and the dimension of an ASC is the maximal dimension of its simplices. We say τ is on the boundary of… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Two iterations of ∂lift for cell complexes. At the first iteration, we leverage each node u’s GNN embedding zu to delineate candidate 1-dim cells. Specifically, we consider cell-equivalent of edges linking u to each of its ku NNs in embedding space, where ku is a rando…

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.