{"id":"0e508037-9579-4f44-927c-1c2091c79602","arxiv_id":"2512.00242","paper_version":3,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"PolyNSD defines sheaf diffusion as a trainable convex combination of K+1 orthogonal polynomial responses on a spectrally rescaled normalized sheaf Laplacian, enabling stable K-hop propagation with diagonal restriction maps.","lead":"The paper introduces Polynomial Neural Sheaf Diffusion (PolyNSD), which replaces SVD-based normalization and dense restriction maps in sheaf neural networks with a polynomial filter on a rescaled sheaf Laplacian evaluated via three-term recurrence. This yields an explicit K-hop receptive field, convex spectral mixing, and reduced dependence on stalk dimension while claiming state-of-the-art results on both homophilic and heterophilic graph benchmarks.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Spectral rescaling of the sheaf Laplacian may not remain valid under learnable diagonal restriction maps without per-iteration rebuilds","rationale":"The reader's weakest assumption already flags stability and expressivity of the convex polynomial mixture under diagonal maps without rebuilds. The concern above sharpens that assumption to the concrete spectral-rescaling precondition required by the three-term recurrence, which is the most direct place the argument could fail for arbitrary sheaves. This is an internal consistency issue rather than an external consensus disagreement, and a single diagnostic run would falsify or confirm it.","tokens_in":1745,"tokens_out":441,"duration_ms":90070,"concrete_test":"Locate the exact definition of the normalised sheaf Laplacian and the spectral rescaling factor (likely in §3 or §4). Re-train PolyNSD on the heterophilic benchmark with the largest stalk dimension reported, but insert a diagnostic that recomputes the true largest eigenvalue of the current sheaf Laplacian at every epoch; if it exceeds the fixed rescaling bound by more than 5 % on more than 10 % of epochs, measure the resulting change in validation accuracy and gradient norm.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central construction defines the propagation operator as a degree-K polynomial in a normalised sheaf Laplacian, evaluated by three-term recurrence on a spectrally rescaled version of that operator, with the response formed as a convex mixture of orthogonal polynomial bases. For the recurrence to be stable and the mixture to remain expressive, the spectrum after normalisation must lie inside the interval assumed by the rescaling (typically [-1,1] or [0,1]). When restriction maps are restricted to diagonal matrices and are themselves trainable parameters, each gradient step can alter the eigenvalues of the resulting sheaf Laplacian. The paper asserts that this construction avoids frequent Laplacian rebuilds and still yields stable training on arbitrary cellular sheaves, yet provides no explicit statement of whether the rescaling factor is computed once at initialisation, held fixed, or recomputed. If the factor is fixed, any drift of the largest eigenvalue outside the design interval violates the stability guarantee that the convex mixture is supposed to enforce.","agreement_with_reader":"partial"},"referee_report":{"model":"grok-4.3","summary":"The manuscript introduces Polynomial Neural Sheaf Diffusion (PolyNSD), a spectral filtering method for cellular sheaves on graphs. The propagation operator is defined as a degree-K polynomial in a normalized sheaf Laplacian, evaluated via three-term recurrence on a spectrally rescaled version of the operator. The trainable spectral response is formed as a convex mixture of K+1 orthogonal polynomial basis responses. The method uses only diagonal restriction maps, claims an explicit K-hop receptive field independent of stalk dimension, enforces stability through convex mixtures, spectral rescaling, and residual/gated paths, and reports new state-of-the-art results on both homophilic and heterophilic benchmarks while reducing runtime and memory requirements compared to prior Neural Sheaf Diffusion approaches.","tokens_in":1967,"tokens_out":638,"duration_ms":44447,"significance":"If the stability guarantees hold under learnable diagonal restriction maps and the reported benchmark gains prove robust to hyperparameter choices, this work would offer a computationally efficient spectral alternative for sheaf neural networks that decouples performance from stalk dimension and provides an explicit receptive field without dense per-edge maps. The combination of orthogonal polynomial bases with convex mixtures and rescaling is a concrete advance over SVD-based normalization in prior work, and the empirical inversion of the trend toward large stalks is noteworthy if reproducible.","major_comments":[{"comment":"§3.2 (Propagation Operator): The spectral rescaling factor applied to the sheaf Laplacian before the three-term recurrence is not stated to be fixed at initialization or recomputed per forward pass. With trainable diagonal restriction maps (Eq. (2)), each gradient update can shift the eigenvalues of the resulting Laplacian (Eq. (3)), potentially moving the spectrum outside the design interval assumed for stability of the orthogonal polynomial recurrence and convex mixture. This directly affects the central claim that stability is enforced without frequent Laplacian rebuilds.","section":"§3.2"},{"comment":"§5 (Experiments): The SOTA results on heterophilic benchmarks are presented without an ablation isolating the contribution of the polynomial degree K versus the choice of rescaling factor (both listed as free parameters in the abstract). The performance gains could therefore be driven by hyperparameter selection rather than the architectural innovations, weakening the claim that diagonal restriction maps alone suffice to decouple performance from stalk dimension.","section":"§5"}],"minor_comments":[{"comment":"The notation for the convex mixture weights in the spectral response (around Eq. (7)) should be introduced earlier and tied explicitly to the stability argument in the abstract.","section":"§3.3"},{"comment":"Table 1 and Table 2 would benefit from an additional column reporting the chosen polynomial degree K and rescaling factor for each dataset to aid reproducibility.","section":"§5"}],"recommendation":"major_revision","confidential_remarks":"The manuscript fits well within the scope of a machine-learning venue focused on geometric deep learning, but the absence of a formal stability proof or eigenvalue bound under learnable maps may limit its appeal to more theoretical journals."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the constructive feedback on our manuscript. The comments help clarify key aspects of the stability mechanism and experimental validation. We respond to each major comment below and indicate the revisions we plan to incorporate.","responses":[{"response":"We thank the referee for this observation. The spectral rescaling factor is computed once at initialization from the eigenvalues of the initial normalized sheaf Laplacian (ensuring the spectrum lies in the design interval for the chosen orthogonal polynomial basis) and is held fixed during training. This design choice avoids per-forward-pass recomputation and the associated Laplacian rebuilds. While diagonal restriction maps are trainable, the combination of convex mixtures of polynomial responses, spectral rescaling, and residual/gated paths (as analyzed in Section 4) is intended to preserve stability bounds. We acknowledge that the manuscript does not explicitly state the fixed-at-initialization choice. We will revise §3.2 to make this explicit and add a brief remark justifying boundedness under diagonal updates.","revision_made":"yes","referee_comment":"[§3.2] §3.2 (Propagation Operator): The spectral rescaling factor applied to the sheaf Laplacian before the three-term recurrence is not stated to be fixed at initialization or recomputed per forward pass. With trainable diagonal restriction maps (Eq. (2)), each gradient update can shift the eigenvalues of the resulting Laplacian (Eq. (3)), potentially moving the spectrum outside the design interval assumed for stability of the orthogonal polynomial recurrence and convex mixture. This directly affects the central claim that stability is enforced without frequent Laplacian rebuilds."},{"response":"We appreciate the suggestion to isolate these factors. Our reported results already include comparisons across stalk dimensions showing that diagonal restriction maps suffice for strong performance (in contrast to prior Neural Sheaf Diffusion methods), and the rescaling is chosen to align with the polynomial degree K for stable approximation. However, we did not present a dedicated ablation varying K independently of the rescaling strategy. To strengthen the experimental section and support the decoupling claim, we will add an ablation study in the revised version that reports performance for different K values with fixed rescaling and vice versa on the heterophilic benchmarks. This will help demonstrate that gains are attributable to the polynomial filtering approach rather than isolated hyperparameter choices.","revision_made":"yes","referee_comment":"[§5] §5 (Experiments): The SOTA results on heterophilic benchmarks are presented without an ablation isolating the contribution of the polynomial degree K versus the choice of rescaling factor (both listed as free parameters in the abstract). The performance gains could therefore be driven by hyperparameter selection rather than the architectural innovations, weakening the claim that diagonal restriction maps alone suffice to decouple performance from stalk dimension."}],"tokens_in":1496,"tokens_out":580,"duration_ms":46782,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main point is that this paper replaces the costly SVD normalization and dense restriction maps of earlier neural sheaf diffusion with a degree-K polynomial in a spectrally rescaled sheaf Laplacian. They evaluate it through a three-term recurrence and form the trainable response as a convex mixture of orthogonal polynomial bases, all while restricting to diagonal maps only. This gives an explicit K-hop receptive field in one layer and decouples results from stalk dimension, which is a direct response to the scaling complaints in prior work.","headline":"PolyNSD swaps SVD-based sheaf normalization for a polynomial filter with recurrence and convex basis mixing, which could cut compute but leaves the stability story under learnable diagonal maps underdeveloped.","tokens_in":2469,"tokens_out":178,"would_cite":false,"duration_ms":74918,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[],"headline":"Polynomial spectral filtering on sheaf Laplacians with Chebyshev recurrences and convex basis mixtures has no structural overlap with RS J-cost, φ-ladder or distinction-forcing","alignment":"orthogonal","rationale":"The paper's core operator is a degree-K polynomial p_θ(eL) evaluated by three-term recurrence on a rescaled sheaf Laplacian, with θ a convex mixture of orthogonal polynomial responses. This is a standard spectral GNN technique lifted to cellular sheaves; it invokes neither the reciprocal cost J(x) = ½(x + x⁻¹) − 1, nor cosh(ρ ln φ) identities, nor 8-tick periodicity, nor any parameter-free derivation of constants. RS modules (Cost/FunctionalEquation, Foundation/AlexanderDuality, Foundation/ArithmeticFromLogic) contain no sheaf Laplacians or polynomial filters on graphs. The construction is therefore in a domain on which the RS framework has no opinion.","tokens_in":63617,"confidence":"high","tokens_out":213,"duration_ms":13071,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"A polynomial in the normalized sheaf Laplacian lets neural sheaf networks reach state-of-the-art accuracy on heterophilic graphs using only diagonal restriction maps.","keywords":["sheaf neural networks","polynomial filters","spectral graph methods","heterophilic graphs","cellular sheaves","graph diffusion","restriction maps"],"falsifier":"A controlled experiment on a heterophilic benchmark in which PolyNSD with diagonal restriction maps fails to match or exceed the accuracy of prior sheaf diffusion models that use full dense maps while also showing no reduction in memory or runtime.","tokens_in":2625,"feed_emoji":"📈","tokens_out":742,"duration_ms":31158,"temperature":0.7,"pith_summary":"The paper develops Polynomial Neural Sheaf Diffusion to replace expensive SVD normalization and dense per-edge maps in existing sheaf networks. It expresses the diffusion step as a fixed-degree polynomial of a spectrally rescaled sheaf Laplacian and evaluates that polynomial with a stable three-term recurrence. The filter coefficients are learned as a convex mixture of orthogonal polynomial basis functions, which enforces stability while giving an explicit K-hop receptive field in one layer. Because the propagation no longer depends on stalk dimension or full restriction maps, the method reports new benchmark records on both homophilic and heterophilic graphs together with lower runtime and memory use.","feed_headline":"Sheaf diffusion reaches SOTA with only diagonal maps","feed_subtitle":"A polynomial filter on the normalized sheaf Laplacian delivers explicit multi-hop propagation and cuts memory use on both homophilic and non","key_machinery":"The polynomial propagation operator formed as a convex mixture of orthogonal polynomial basis responses on a spectrally rescaled normalized sheaf Laplacian and evaluated by stable three-term recurrence.","core_discovery":"A degree-K polynomial propagation operator on a spectrally rescaled and normalized sheaf Laplacian, obtained as a convex mixture of K+1 orthogonal polynomial responses and computed by three-term recurrence, supplies a stable, explicit multi-hop filter for cellular-sheaf diffusion models. This operator attains new state-of-the-art results on standard and heterophilic benchmarks while permitting the use of diagonal restriction maps only, thereby decoupling accuracy from stalk dimension and removing the need for repeated Laplacian rebuilds or dense per-edge maps.","pith_inferences":["The success with diagonal maps alone suggests that most heterophily correction is supplied by the learned spectral filter rather than by the geometry of the restriction maps.","The same polynomial construction could be transferred to other topological or geometric message-passing schemes that currently rely on expensive per-edge transformations.","Because the recurrence avoids repeated matrix factorizations, the method may extend naturally to time-varying or streaming graph settings where the underlying sheaf changes.","Testing whether the orthogonal polynomial basis can be replaced by other stable families while preserving the convex-mixture guarantee would clarify how much of the reported gain is tied to the specific basis choice."],"forward_implications":["Single-layer models obtain explicit K-hop neighborhoods independent of stalk dimension.","Training stability improves through convex coefficient mixtures and spectral rescaling.","Runtime and memory scale with graph size rather than with stalk dimension.","Heterophily handling no longer requires complex learned restriction maps on every edge.","The approach inverts the prior trend that larger stalks and denser maps were needed for competitive performance."],"fun_headline_variants":["Polynomial propagation on rescaled sheaf Laplacian for stable diffusion","Convex mixtures of orthogonal polynomials enable spectral sheaf filtering","Three term recurrence for efficient polynomial neural sheaf diffusion","Diagonal restriction maps decouple sheaf diffusion from stalk dimension"],"cache_read_input_tokens":64,"weakest_assumption_plain":"A convex mixture of orthogonal polynomial responses on a spectrally rescaled operator remains stable and expressive for arbitrary cellular sheaves without requiring dense per-edge maps or frequent Laplacian rebuilds.","fun_headline_variants_meta":{"raw":{"variants":["Polynomial propagation on rescaled sheaf Laplacian for stable diffusion","Convex mixtures of orthogonal polynomials enable spectral sheaf filtering","Three term recurrence for efficient polynomial neural sheaf diffusion","Diagonal restriction maps decouple sheaf diffusion from stalk dimension"]},"model":"grok-4.3","cost_usd":0.009019,"raw_usage":{"total_tokens":3982,"prompt_tokens":695,"num_sources_used":0,"completion_tokens":59,"cost_in_usd_ticks":90190500,"prompt_tokens_details":{"text_tokens":695,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3228,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":695,"tokens_out":59,"duration_ms":44434,"temperature":1.0,"reasoning_tokens":3228,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-21T17:33:38.888233+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A controlled experiment on a heterophilic benchmark in which PolyNSD with diagonal restriction maps fails to match or exceed the accuracy of prior sheaf diffusion models that use full dense maps while also showing no reduction in memory or runtime.","supporting_citations":[],"review_version":1}