{"id":"f0d1f4f8-4a6b-446a-a8fd-a4f8cfdbdb27","arxiv_id":"2607.21885","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"ACE adds a heterophily-aware auxiliary loss to coarsening-based GNN training, recovering discarded node-level information and improving accuracy on heterophilic graphs by up to ~15 points.","lead":"The paper shows that training GNNs on coarsened graphs loses significantly more accuracy on heterophilic graphs than on homophilic ones, and proposes ACE, a plug-in module that adds an auxiliary loss to recover the discarded information. The method consistently improves several coarsening pipelines on large heterophilic benchmarks with modest overhead.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Unspecified label masking in the auxiliary loss (Eqs. 5/11) is the key risk: if Y includes test labels, ACE's gains are an artifact; theory issues are secondary.","rationale":"The single most load-bearing condition for the central claim ('ACE consistently improves coarsening-based training under heterophily') is the validity of the reported test accuracies. That validity depends on whether the auxiliary loss uses test labels. The reader's weakest_assumption targets Proposition 2.2's proof, but those theory defects do not threaten the empirical numbers; a label-masking defect would. The reader did flag 'label masking' as a missing implementation detail and asked for code verification, so there is partial agreement. I keep the verdict at CONDITIONAL (hence UNCHANGED), but the binding condition should be: code inspection must confirm that no test labels enter Eq. (5) or Eq. (11), and ideally that Y' in baselines is uncontaminated. If the code shows leakage, the verdict should be REJECT, not merely conditional. The paper's provision of a code link is positive evidence and makes this test feasible, but it does not replace a stated masking protocol.","tokens_in":27416,"tokens_out":11220,"duration_ms":111778,"concrete_test":"Inspect the released code at https://github.com/vasile-paskardlgm/ACE and identify the index set used for Y in Eq. (5)/(11). If the auxiliary loss is computed over all n nodes, rerun the GCN/SCAL row of Table 3 on Genius with the loss restricted to the training mask and all other settings unchanged. If the +10.16 gain is not reproduced, the paper's central claim is unsupported. Also check that Y' in the baseline is computed without test labels.","verdict_should_be":"UNCHANGED","load_bearing_attack":"ACE's auxiliary loss (Eq. 5; Eq. 11) is written as a loss against Y, defined in Sec. 2.1 as the full n x c one-hot label matrix. The paper never states that a train/validation/test mask is applied. In transductive node classification, Y contains test labels; using them in the auxiliary loss would invalidate the reported test accuracies (Tables 3, 8-11) and make the comparison to coarsening baselines unfair. The concern extends to the toy loss in Eq. 5 and to Y' = C^{-1} P Y in Sec. 2.1, which inherits any test-label contamination. This is not a stylistic omission: the magnitude of the claimed gains (~15 pts) is exactly what one would expect from adding full-label supervision. If the released code applies a training mask to Y, the concern is resolved; if not, the central empirical claim collapses. I am not asserting the authors did this; I am asserting the manuscript as written does not rule it out. The Proposition 2.2/3.1 theory problems are real but secondary, since the empirical contribution could stand without them.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the performance degradation of coarsening-based GNN training on heterophilic graphs. It first documents empirically that existing coarsening pipelines (SCAL, FGC, UGC, SGBGC) lose substantially more accuracy on heterophilic than homophilic graphs. It then proposes ACE, a plug-and-play auxiliary-loss framework: a learnable projector P(ϕ) is optimized with an anisotropic structural regularization term, and the resulting projector is used to lift coarsened-graph predictions to the original graph for an auxiliary cross-entropy loss against the full label matrix; the primary and auxiliary losses are combined via homoscedastic uncertainty weighting. Experiments on 5 heterophilic and 2 homophilic datasets with up to 6 backbones report consistent gains, up to ~15 points, with modest computational overhead. The paper also offers two theoretical propositions (Prop. 2.2 on a mutual-information gap, Prop. 3.1 on the projector's spectral behavior).","tokens_in":27775,"tokens_out":7196,"duration_ms":73768,"significance":"If the empirical results are valid, ACE is a practical and broadly applicable enhancement for coarsening-based training: it is model-agnostic, works across four coarsening pipelines, and the experiments are extensive (10 runs with standard deviations, multiple backbones, ablation studies, noise robustness, and overhead measurements). The paper also releases code and includes careful efficiency analyses. However, the two theoretical propositions are not rigorously established; they are best viewed as post-hoc intuition rather than proofs. The central unresolved issue is whether the auxiliary loss uses only training labels: the manuscript defines Y as the full n×c label matrix and never specifies a mask, which could invalidate the reported test accuracy gains.","major_comments":[{"comment":"The label matrix Y is defined as the full n×c one-hot matrix, and both auxiliary losses (Eq. (5) and Eq. (11)) are written against Y with no train/validation/test mask. In transductive node classification, using test labels in the loss would inflate the reported test accuracies (Tables 3, 8–11) and make the comparison to coarsening baselines unfair. Please state explicitly how the label mask is applied in the auxiliary loss and in Y' = C^{-1} P Y. If test labels are used, the experiments must be rerun with a proper training mask; if a mask is used in the code but omitted from the text, it must be described precisely.","section":"Sec. 2.1, Eq. (5), Eq. (11)"},{"comment":"The lifting operator is written as P^T C (or P(ϕ*)^T C). Given the coarsened labels are defined as Y' = C^{-1} P Y, the natural inverse mapping from coarse predictions to fine nodes is P^T C^{-1}, or simply P^T if f outputs a probability vector. The factor C appears to be the wrong inverse. This makes the definition of the auxiliary loss mathematically inconsistent with the stated coarsening convention. Please correct the equations or clarify the intended operator; if the implementation actually uses C^{-1}, that should be stated explicitly.","section":"Eq. (5), Eq. (11), Sec. 2.1"},{"comment":"The proof is not valid. Lemma A.2 assumes conditional independence A⊥B|Y for the outputs of two models trained on the same labels, which is not justified and generally false for two trained GNNs. Moreover, Eq. (20) applies the data-processing inequality to the chain Y→f(A,X;Θ*)→(f(A,X;Θ**),g(G\\G')), but this Markov chain does not hold: f(A,X;Θ**) depends on Y through the coarsened labels Y' and is not a function of f(A,X;Θ*). Consequently, the inequality in Eq. (4) is not established. The theoretical explanation should either be rigorously proven or explicitly presented as a heuristic.","section":"Appendix A, Prop. 2.2"},{"comment":"The proof analyzes a local objective Li((P(ϕ)µ)_i) and differentiates with respect to the lifted signal (P(ϕ)µ)_i while treating neighboring lifted signals as constants. The resulting update rule (Eq. (29)) is not the fixed-point condition for the minimizer, and the optimization is over ϕ, not directly over the lifted signal. Thus the claimed high-frequency/low-frequency interpolation behavior of P(ϕ*) is not proven. Please revise the proof or weaken the proposition to a stated intuition.","section":"Appendix B, Prop. 3.1"}],"minor_comments":[{"comment":"The homoscedastic uncertainty weighting uses 1/(2σ^2) for both losses, which is the regression form from Kendall et al. For classification, the standard form is (1/σ^2) L + log σ. Please verify and correct the formula.","section":"Eq. (11)"},{"comment":"Typo: 'features anisotropic structural regularization' should likely be 'features an anisotropic structural regularization' or 'incorporates anisotropic structural regularization'.","section":"Sec. 3.2.1"},{"comment":"Inconsistent naming: 'GLOGNN' vs 'GloGNN', 'Ogbn-product' vs 'Ogbn-products', and 'arXiv' vs 'arXiv-year' in a few places. Please standardize.","section":"Throughout"},{"comment":"The description states UGC 'explicitly accounts for the heterophily present in such graphs,' which seems to contradict the introduction's claim that prior coarsening work has been developed almost exclusively for homophilic graphs. Please reconcile.","section":"Appendix D.2"},{"comment":"The text says the degradation is 'often nearly four times larger' on heterophilic graphs. This ratio is not computed or shown; please provide the exact computation or substantiate the claim.","section":"Sec. 2.2, Table 1"}],"recommendation":"major_revision","confidential_remarks":"The most critical issue is the label-masking ambiguity in Eqs. (5) and (11). I strongly recommend that the editor request the authors to clarify and, if necessary, re-run experiments with a proper training mask before considering acceptance. The theoretical propositions are secondary and can be revised or de-emphasized. If the released code confirms that only training labels are used, the empirical contribution could be solid and the paper may be suitable after a rigorous revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take on 2607.21885. The paper is the first systematic look at coarsening-based GNN training under heterophily, and its empirical work is genuinely useful. Across four coarsening pipelines, six backbones, five heterophilic and two homophilic datasets, ACE gives consistent gains up to ~15 points, with small overhead. The toy experiment—a simple auxiliary loss recovering much of the gap—is a nice observation. This is a real gap in the literature and the authors have done the heavy lifting of benchmarking it.\n\nThe soft spots, in order of seriousness:\n\n1. The auxiliary loss in Eq. (5) and Eq. (11) uses Y as defined in Sec. 2.1: the full n x c one-hot label matrix. The paper never states that a train/validation/test mask is applied. In transductive node classification, Y includes test labels. If those labels are used to compute the auxiliary loss, the reported test accuracies are meaningless and the comparison to baselines is unfair. The magnitude of the gains—about what you'd expect from full-label supervision—makes this a load-bearing question, not a formatting nit. The authors must state explicitly that only training labels are used, and the released code must confirm it. The stress-test note is right to make this the primary issue.\n\n2. The theory is not rigorous. Proposition 2.2's proof leans on a conditional-independence assumption (Lemma A.2) that is not justified, and the data-processing inequality in Eq. (20) is applied to a Markov chain that is at best a stretch. Proposition 3.1 characterizes the lifted signal, not the learned projector. Since the empirical contribution stands without the theory, this is fixable by reframing as intuition or tightening the proofs.\n\n3. Minor: implementation details T, N(i), and label masking are not specified. The code repository is referenced but not verified.\n\nProportionately, the paper is a good empirical contribution with an underspecified crucial detail and a weak theoretical overlay. It deserves a serious referee, not a desk reject. If the masking issue checks out, this is a solid paper. If not, it collapses.\n\nRecommendation: send to peer review; make sure the reviewers check the code and ask for an explicit statement about label masking. I'd hold off citing it until that's resolved.","headline":"Solid empirical study of heterophily in coarsening-based GNN training, but the auxiliary loss's label masking is unspecified and could invalidate the headline gains; worth reviewing, not yet citable.","tokens_in":28202,"tokens_out":2712,"would_cite":false,"duration_ms":26244,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Coarsening-based GNN training degrades sharply under heterophily; ACE, an auxiliary-loss wrapper with a learned projector, recovers up to ~15 accuracy points at ~5% overhead.","keywords":["graph coarsening","heterophily","graph neural networks","auxiliary loss","learned projector","mutual information","anisotropic diffusion","homoscedastic uncertainty"],"falsifier":"Estimate the quantities in Proposition 2.2 on a real heterophilic dataset (e.g., Genius): I(f(A,X;Θ*); f(A,X;Θ**)), I(f(A,X;Θ*);Y), and I(g(G\\G');Y | f(A,X;Θ**)) using trained models. If the inequality fails, or if the conditional mutual information Ω is not larger for heterophilic than for comparable homophilic graphs, the paper's information-theoretic explanation of the coarsening gap would be refuted, even if ACE still improves accuracy empirically.","tokens_in":27313,"feed_emoji":"🧩","tokens_out":6266,"duration_ms":58764,"temperature":0.7,"pith_summary":"The paper argues that training a GNN on a coarsened graph — a common trick for scaling to massive graphs — fails on heterophilic graphs, where connected nodes tend to have different labels, because coarsening discards exactly the fine-grained, label-relevant structure inside each supernode. Existing coarsening pipelines lose roughly four times more accuracy on heterophilic benchmarks than on homophilic ones, and even heterophily-specialized GNN backbones cannot escape the drop. The authors propose ACE, which learns a projector that reconstructs original node features from supernode features, regularized by an anisotropic diffusion term that respects local heterophily. Training then combines the usual coarsened-graph loss with an auxiliary full-graph loss, automatically weighted by homoscedastic uncertainty. Across four coarsening pipelines and six GNN backbones, ACE consistently recovers a large fraction of the gap — up to about 15 accuracy points on heterophilic graphs — while adding only a few percent to runtime and memory.","feed_headline":"ACE recovers up to 15 points in coarsened-GNN training","feed_subtitle":"Heterophilic graphs lose 4x more accuracy to coarsening; ACE closes most of the gap for ~5% extra cost.","key_machinery":"The load-bearing object is the learned heterophily-aware projector P(ϕ), defined by a softmax over per-node affinities that combine a structural affinity S = A C⁻¹ P and a feature affinity F (Gaussian kernel to supernode centroids), with the logits produced by a small MLP. It is trained by minimizing J_ASR: an anisotropic smoothness term whose edge weights exp(−||Xi−Xj||²) suppress smoothing across dissimilar neighbors, plus a feature reconstruction cost. The projector bridges the coarsened graph back to the full graph: lifting coarsened logits through P(ϕ) defines an auxiliary cross-entropy loss against full-graph labels, and learnable homoscedastic uncertainty parameters σ₁, σ₂ automatical","core_discovery":"The central claim is that performance degradation in coarsening-based GNN training under heterophily is not a model artifact but a consequence of information loss in the coarsening step itself. Treating the discarded intra-supernode structure G\\G' as a source of label information, the paper proves a mutual-information bound (Proposition 2.2): the similarity between full-graph-trained and coarsened-graph-trained models is capped by I(f(A,X;Θ*);Y) − I(g(G\\G');Y | f(A,X;Θ**)), so the larger the conditional mutual information between discarded structure and labels, the larger the gap. ACE attacks this gap directly: it learns a heterophily-aware projector P(ϕ) that maps supernode centroids back t","pith_inferences":["A direct test of the proposed mechanism would estimate the three mutual-information terms in Proposition 2.2 on real heterophilic and homophilic graphs; if the conditional mutual information Ω is not larger under heterophily, or if the inequality fails, the theory would be refuted even though ACE's empirical gains could still stand on the auxiliary-loss supervision.","Because the projector is learned purely from features and structure, ACE could plausibly transfer to other settings where a reduced graph substitutes for the full one, such as graph distillation, continual graph learning, or scalable inference on evolving graphs.","The homoscedastic weighting likely acts as a form of curriculum: early training may emphasize the cheap coarsened loss and later shift to the fine-grained auxiliary loss. Inspecting the learned σ₁/σ₂ trajectories would reveal whether the benefit comes from a stable early phase or from late fine-grained correction.","The paper's own limitation section concedes that ACE recovers discarded information implicitly rather than through explicit intra-supernode structure, which it says caps the achievable gain; adding a lightweight explicit decoder of intra-supernode edges is a natural next step toward closing the remaining gap to full-graph training."],"forward_implications":["Any existing coarsening-based GNN pipeline can be upgraded by adding ACE's auxiliary loss with the learned projector, without retraining or altering the coarsening algorithm.","Coarsening-based training becomes practical on large heterophilic graphs, where the gap to full-graph training shrinks substantially — for instance, GCN on Genius improves from 67.47 to 77.63 against a full-graph score of 87.42.","The same recipe extends to other graph-reduction methods such as graph condensation and to extreme 1% coarsening ratios, where the relative gains are even larger.","Inference cost stays identical to the base coarsening pipeline because the auxiliary branch is discarded after training; only training time and memory rise by roughly 4–7%.","ACE-enhanced conventional coarsening methods become competitive with state-of-the-art scalable training approaches such as sampling and condensation, while preserving the simpler model-agnostic coarsening paradigm."],"fun_headline_variants":["ACE rescues coarsened GNNs on heterophilic graphs","Coarsening hurts heterophily; ACE closes the gap","ACE: Plug-and-play fix for coarsening-induced GNN loss","Heterophily-aware ACE lifts coarsened GNN accuracy"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing assumption is that the full-graph-trained and coarsened-graph-trained model outputs are conditionally independent given the true labels, which is what makes the mutual-information bound in Proposition 2.2 go through; the empirical ACE recipe does not depend on this assumption.","fun_headline_variants_meta":{"raw":{"variants":["ACE rescues coarsened GNNs on heterophilic graphs","Coarsening hurts heterophily; ACE closes the gap","ACE: Plug-and-play fix for coarsening-induced GNN loss","Heterophily-aware ACE lifts coarsened GNN accuracy"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000604,"raw_usage":{"total_tokens":2697,"prompt_tokens":827,"completion_tokens":1870,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":571,"completion_tokens_details":{"reasoning_tokens":1795}},"tokens_in":571,"tokens_out":1870,"duration_ms":14229,"temperature":1.0,"reasoning_tokens":1795,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T06:24:48.926486+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Estimate the quantities in Proposition 2.2 on a real heterophilic dataset (e.g., Genius): I(f(A,X;Θ*); f(A,X;Θ**)), I(f(A,X;Θ*);Y), and I(g(G\\G');Y | f(A,X;Θ**)) using trained models. If the inequality fails, or if the conditional mutual information Ω is not larger for heterophilic than for comparable homophilic graphs, the paper's information-theoretic explanation of the coarsening gap would be refuted, even if ACE still improves accuracy empirically.","supporting_citations":[],"review_version":1}