{"id":"dcc6ba5d-909b-4a65-8044-8dd38ec08fa3","arxiv_id":"2506.05120","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A grouped additive noise model is identifiable under a technical condition, and the GroupRESIT-MURGS algorithm learns causal graphs among variable groups.","lead":"This paper extends nonlinear additive noise models for causal discovery to data where the causal units are groups of variables, such as brain regions or manufacturing process cells. It introduces a technical identifiability condition, a two-stage algorithm called GroupRESIT, and a new sparse regression method called MURGS, with promising results on synthetic and real manufacturing data.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Condition 1, the premise of the identifiability theorem, is uncharacterized: no examples, no genericity proof, and group-size analysis is explicitly deferred, so the theorem's coverage of the paper's own experimental models is unverified.","rationale":"The reader's weakest assumption is also my primary concern: Theorem 2.1 holds only under Condition 1, a condition that is neither characterized nor shown to be generic. I found no internal inconsistency in the proof of Theorem 2.1; the bivariate argument correctly derives Eq. (3) from the existence of a backward GANM, and the dimensional bookkeeping is coherent when the conventions in Appendix A are followed. The issue is scope: the paper's headline identifiability claim is conditional on an unverified premise, and the authors themselves flag the missing group-size analysis. The proposed numerical test would settle whether Condition 1 is satisfied by the synthetic models, which is the main place the assumption needs to hold for the experiments to support the paper's claims. I also note the self-admitted HSIC comparability caveat after Algorithm 1, but it is secondary to the identifiability premise and would mainly affect heterogeneous group sizes. The paper deserves credit for the publicly available implementation and for the MURGS derivation, which is a genuine contribution even if the identifiability assumption needs further verification. A conditional accept remains the right verdict.","tokens_in":28278,"tokens_out":14137,"duration_ms":164105,"concrete_test":"Evaluate Eq. (3) numerically for the paper's own data-generating process. For bivariate submodels taken from the Section 4 generator (Gaussian-process f_g, log-normal noise), compute the left and right sides of Eq. (3) at, say, 100 random points per model, using automatic differentiation of f_g and density estimates for P_Xj and P_Ng. Repeat across at least 50 random GANMs with equal group sizes d_j=d_g for d=1,2,5 and with unequal sizes such as d_j=2,d_g=5. If the residual is nonzero for essentially all random models, Condition 1 is generic within the experimental family, so the theorem covers the simulations. If a non-negligible fraction (e.g., more than 5%) has residual near numerical tolerance, the identifiability theorem does not cover the evaluated models, and the experiments would be testing the algorithm outside its proven regime.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theoretical claim (Theorem 2.1, Corollary 1) is that a grouped additive noise model is identifiable whenever Condition 1 holds, i.e., whenever Eq. (3) is not satisfied. This is a negative condition about a tensor differential equation involving f_g, P_Xj, and P_Xg. The paper gives no example satisfying or violating Condition 1, no genericity argument, and no characterization of the models to which it applies; the Remark after Eq. (3) explicitly defers the role of group sizes to future work. The gap is load-bearing because in the scalar ANM the analogous condition is known to be satisfied by Gaussian-linear models, and in the vector case the dimensions d_j and d_g change the number of equations in (3), so the non-identifiable set may be larger than the scalar case suggests. Corollary 1 then requires Condition 1 for conditional triples given every admissible conditioning set S, a condition that is not verified in the simulations. As written, Theorem 2.1 is a correct conditional statement, but it does not establish that the models used in the Section 4 experiments or the manufacturing data satisfy its premise; the paper's claim that causal directions are identifiable in the group setting is therefore not yet supported for the settings it targets.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper extends nonlinear additive noise model (ANM) causal discovery from scalar random variables to groups of random vectors. It defines the group additive noise model (GANM), states a condition (Condition 1) under which causal directions in a bivariate GANM are identifiable (Theorem 2.1), and sketches an extension to full graphs (Corollary 1). On the algorithmic side, the paper proposes GroupRESIT, a two-phase procedure: the causal order is estimated by multi-output neural network regression followed by HSIC-based sink identification, and the DAG is pruned by MURGS, a newly proposed multi-response group sparse additive model with a block-coordinate descent/backfitting algorithm and a closed-form soft-thresholding update. The method is evaluated on synthetic GANM data against several baselines and on assembly-line manufacturing data with a partially known cell order.","tokens_in":28589,"tokens_out":7623,"duration_ms":86813,"significance":"If the identifiability theory is made fully rigorous, the paper would be a useful extension of ANM identifiability to multivariate groups and would provide a practical algorithm for grouped causal discovery. The paper has clear strengths: a publicly available implementation, a closed-form block update for a genuinely useful multi-response sparse additive model, evaluation with order- and intervention-oriented metrics (AAID/OAID) in addition to standard edge-based metrics, and a real-data application with a partial ground-truth ordering that checks edge orientation. The main caveat is that the central identifiability theorem is conditional on an uncharacterized premise, and the empirical comparison omits the most directly relevant group-level baselines; these issues need to be addressed before the headline claims are fully supported.","major_comments":[{"comment":"Theorem 2.1 is logically conditional: it establishes identifiability only when the triple (f_g, P_Xj, P_Xg) does not satisfy Eq. (3). The paper does not characterize the set of triples satisfying Condition 1, does not give a single example of a model that satisfies or violates it, and the Remark after Eq. (3) explicitly defers the effect of group sizes to future work. Since the simulation models in Section 4 use Gaussian-process link functions and multivariate log-normal noise, and the manufacturing data is not checked against Eq. (3), the theorem's premise is not verified for any model used in the paper. In particular, the dimensions d_j and d_g determine the number of tensor equations in (3), so one cannot simply infer from the scalar ANM that Condition 1 is generic. Please provide a genericity argument, nontrivial examples (including cases where Eq. (3) holds), or an explicit verification that the simulation settings satisfy Condition 1; otherwise the claim that causal directions are identifiable in the group setting is not supported for the settings studied.","section":"§2.2, Condition 1 and Theorem 2.1"},{"comment":"The proof of Corollary 1 is too sketchy and appears internally inconsistent. It invokes 'Proposition 29 in Peters et al. [2014]' without stating it, and it invokes 'Lemma 2 in Peters et al. [2011]' without stating or verifying its hypotheses. More seriously, the proof says 'For any s=(q,r)' and 'we can choose any s', whereas the corollary's hypothesis only asserts existence of some realization x_S with positive density. The third component of the conditional triple is written as P_Xj rather than P_{X_j | X_S = x_S}, and the phrase 'the conditional distribution of X_S has strictly positive density' appears to refer to the wrong distribution. The extension from the bivariate theorem to the multivariate graph therefore needs to be reworked with precise conditioning notation and a complete proof.","section":"Corollary 1 and Appendix A.2"},{"comment":"The closed-form update in Theorem 3.1 is derived under the assumption E[f_{g,h'}^{(k)} | X_h^{(g)}] = 0 for h' ≠ h, which the paper calls crucial. The paper does not discuss whether this assumption holds for the function classes used in Algorithm 3, and Algorithm 4 applies the same update regardless. Consequently, there is no theorem guaranteeing that the algorithm converges to the MURGS minimizer for the data-generating processes in Section 4. The authors should either state conditions on the smoother and function space under which the assumption is satisfied, provide convergence guarantees for the general case, or explicitly frame Theorem 3.1 as an approximate update and justify its use.","section":"§3.3, Theorem 3.1"},{"comment":"The experimental comparison does not include the most relevant existing group-level causal discovery baselines, such as the groupwise methods of Wahl et al. [2023] or the grouped LiNGAM of Entner and Hoyer [2012] (the latter appears only in the real-data section, not in the synthetic benchmarks). Given the paper's claim of 'superior performance compared with several other causal discovery algorithms', the absence of these group-specific competitors weakens the empirical contribution. Please add at least one such baseline to the synthetic study, or justify why it is not applicable.","section":"§4, Experiments"}],"minor_comments":[{"comment":"The abstract and introduction state that 'in general, the causal directions can be identified in the group setting'; this should be qualified to models satisfying Condition 1.","section":"Abstract and Section 1"},{"comment":"In the definition of m*, the displayed formula uses m* on the right-hand side ('Σ_{l=1}^{m*}') where the running variable m should appear; the same typo appears in Theorem 3.1 and Algorithm 3. Please correct to Σ_{l=1}^m.","section":"§3.3, Theorem 3.1 and Algorithm 3"},{"comment":"The sentence 'choosing u(k) and v_h(k) as above guarantees that Σ|u|(k) ≤ 1 and Σ|u|(k) ≤ 1' repeats the same condition; the second inequality should concern the vector v^{(k)}, e.g., Σ_h (v_h^{(k)})^2 ≤ 1.","section":"Appendix B, proof of Proposition 1"},{"comment":"All performance numbers are averages over 20 runs, but Figures 3 and 4 do not show variability; since the supplementary material contains boxplots, please report confidence intervals or spread in the main text for at least the headline AAID comparison.","section":"§4, Results"}],"recommendation":"major_revision","confidential_remarks":"The paper has a useful algorithmic contribution and a plausible empirical demonstration, but the identifiability theorem's premise, Condition 1, remains uncharacterized and unverified for the paper's own experiments. The most important task for the authors is to resolve the status of Condition 1: prove genericity, provide examples, or explicitly weaken the theoretical claims. The Corollary 1 proof and the MURGS update's zero-covariance assumption are also load-bearing and should be addressed. If these can be fixed, the paper would be a strong contribution; I would not recommend rejection at this stage."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper gives the vector-valued generalization of the ANM identifiability theorem and a new group-sparse additive model (MURGS) for pruning. The theorem is real but its premise, Condition 1, is left unexamined. The algorithm is the stronger part.\n\nThe grouped identifiability theorem (Thm 2.1) is a genuine extension of Hoyer et al. and the Corollary to the multivariate case follows the standard Peters et al. route. The proof of Thm 2.1 is written out in the appendix and the logic is sound given Condition 1. MURGS is new: the sup-norm over response components plus group-lasso style penalty is a clean way to prune at the group level, and the closed-form backfitting update is derived properly under the zero within-group covariance assumption. The code is public and the simulations are reasonably thorough, with a good metric battery (SHD, SID, AAID, OAID) and a real manufacturing application. The main results are reproducible.\n\nSoft spots, in order of importance. First, Condition 1 is a negative condition on a tensor differential equation, and the paper gives no example, no genericity argument, and no characterization. The remark after Eq. (3) explicitly defers the role of group sizes to future work. That means the identifiability theorem does not yet tell us which models it covers, and the simulations are not checked against it. This is a real gap, but not fatal: the theorem is correctly stated as conditional, and the scalar analogue's condition is known to rule out a measure-zero set. The burden is on the authors to show the group case inherits that genericity.\n\nSecond, Corollary 1's proof leans on an unstated Proposition 29 from Peters et al. 2014; I had to go to the source to verify the step. It's probably correct, but the appendix should cite it explicitly and outline the argument.\n\nThird, the experiments omit error bars and do not include GroupDirectLiNGAM in the synthetic comparison (it appears only in the real data). Also, the HSIC comparability issue they flag in the Remark is real: comparing HSIC values across regressions with different output dimensions is not obviously meaningful, and the paper does not address it beyond the remark.\n\nThe bottom line: this is a solid, useful paper for anyone doing causal discovery on grouped data. The algorithmic contribution stands on its own, and the theory is a genuine but conditional extension. It deserves a serious referee. I would send it out and ask the authors to characterize Condition 1 and clean up the Corollary 1 proof.","headline":"Grouped ANM identifiability is conditional on an uncharacterized Condition 1, but the MURGS pruning method is a real contribution and worth referee time.","tokens_in":29067,"tokens_out":2315,"would_cite":true,"duration_ms":26992,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62H22","62G08","62J07"],"pacs":[],"model":"deepseek-v4-flash","headline":"The causal graph of a grouped additive noise model is identifiable when a differential-equation condition holds, and a new two-step algorithm recovers it.","keywords":["grouped causal discovery","additive noise models","random vectors","identifiability","sparse additive models","HSIC independence test","RESIT","MURGS"],"falsifier":"A concrete test would be to construct a smooth vector-valued GANM whose triple is known to solve Eq. (3), generate data from it, and show that the reverse orientation also admits an additive noise fit, causing GroupRESIT-MURGS to return the wrong direction; alternatively, computing the Lebesgue measure of the set of triples satisfying Eq. (3) and finding it to be generic would show the identifiability theorem covers almost no models.","tokens_in":28092,"feed_emoji":"🔗","tokens_out":7302,"duration_ms":71561,"temperature":0.7,"pith_summary":"This paper tackles causal discovery when the causal entities are groups of variables, such as brain regions or production-line process cells, rather than individual scalars. It claims that for grouped additive noise models, where each group's noise vector is independent of other groups but may be dependent within the group, the causal graph is identifiable from the observational distribution provided a differential-equation condition fails to hold. On the algorithmic side, it introduces GroupRESIT-MURGS, a two-phase procedure that first recovers a causal order by regressing each group on the rest and testing independence with the Hilbert-Schmidt criterion, then prunes edges using a new multi-response group sparse additive model. The paper reports that the method outperforms grouped versions of PC, GraN-DAG, and LiNGAM in simulations, and on real assembly-line data it recovers a graph with only one edge contradicting the known process order.","feed_headline":"Grouped variables still yield identifiable causal graphs","feed_subtitle":"Additive noise models for random vectors reveal cause and effect in brain-region and production-line data.","key_machinery":"The load-bearing object is Condition 1, a tensor-valued differential equation written in terms of the log-densities and the Jacobian and Hessian tensors of the group regression function; identifiability holds exactly when the triple $(f_g, P_{X_j}, P_{X_g})$ does not satisfy it, because satisfying it is what permits an additive noise representation in the reverse direction. The estimation machinery is the two-phase GroupRESIT pipeline: multi-output neural network regressions combined with HSIC independence tests to identify sink nodes, followed by MURGS, which penalizes a sum of sup-norms over response coordinates and a group-lasso-type norm over parent groups, and whose optimization uses a derived closed-form soft-thresholding backfitting update.","core_discovery":"The paper's central claim is that grouped additive noise models (GANMs) are identifiable: when each group $X_g$ is generated from its parent groups by a smooth nonlinear function plus a noise vector, with noise vectors independent across groups, the direction of causation is read off from the joint distribution. The proof proceeds by showing that any backward model of the same form forces the triple of the regression function, the marginal density of the cause, and the noise density to satisfy a specific tensor differential equation, Condition 1. Ruling out that equation makes the causal direction unique for each bivariate pair, and conditioning on non-descendants lifts the result to full DAGs. The paper further contributes MURGS, a sparse additive model whose penalty zeroes out whole parent groups, fit by a closed-form soft-thresholding block-coordinate update.","pith_inferences":["The paper does not assess how often Condition 1 holds; a reader should suspect that identifiability may fail for generic unbalanced group sizes, since the tensor contractions in Eq. (3) change with the dimensions of the groups.","MURGS's penalty structure could be transferred to time-series or conditional independence settings where groups are lagged versions of the same variables, though the paper does not explore this.","The practical caveat in the paper's own remark, that HSIC values are comparable only when dimensions and scales match, suggests that the Phase I ordering may be unstable when groups have very different sizes; testing this would require a dedicated simulation study."],"forward_implications":["Causal order among groups can be learned without reducing groups to scalar summaries or treating members as independent scalars.","MURGS can serve as a plug-in pruning module for other order-based causal discovery methods on grouped data.","The identifiability theory extends the scalar ANM results of Hoyer et al. and Peters et al. to vector-valued settings, so future work on nonlinear causal discovery with grouped variables can build on this foundation.","On real production-line data, the method yields a graph that is almost entirely consistent with the known sequential order of process cells, indicating practical value for manufacturing quality control."],"supporting_citations":[{"why":"Derives the scalar additive noise identifiability differential equation that Condition 1 generalizes to vectors.","marker":"Hoyer et al. [2008]"},{"why":"Provides the RESIT two-phase causal order and pruning framework that GroupRESIT extends to groups.","marker":"Peters et al. [2014]"},{"why":"Defines the Hilbert-Schmidt independence criterion used to test residual independence in Phase I.","marker":"Gretton et al. [2005]"},{"why":"Introduces the group lasso penalty that MURGS extends to the functional multi-response setting.","marker":"Yuan and Lin [2006]"},{"why":"Supplies the sparse additive model and GCV criterion adapted for MURGS tuning.","marker":"Liu et al. [2007]"},{"why":"Provides the GraN-DAG baseline whose grouped version, GGraN-DAG, is outperformed in simulations.","marker":"Lachapelle et al. [2020]"},{"why":"Contributes the grouped linear causal ordering approach that serves as an alternative Phase I and is pruned with MURGS.","marker":"Entner and Hoyer [2012]"},{"why":"Defines the AAID and OAID graph-distance metrics used to evaluate causal-order recovery.","marker":"Henckel et al. [2024]"}],"fun_headline_variants":["Causal identifiability extended to grouped random vectors","Grouped data still reveal true cause-effect direction","Additive noise models crack causal order for grouped variables","Vector groups don't hide causal links: new method proves it","From brain regions to assembly lines: grouping data still yields causal order"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The identifiability theorem applies only when Condition 1 holds, yet the paper does not characterize which functions and densities satisfy that differential equation, gives no examples of it, and does not argue that it is generic; the supplementary text explicitly leaves the role of group sizes for future work.","fun_headline_variants_meta":{"raw":{"variants":["Causal identifiability extended to grouped random vectors","Grouped data still reveal true cause-effect direction","Additive noise models crack causal order for grouped variables","Vector groups don't hide causal links: new method proves it","From brain regions to assembly lines: grouping data still yields causal order"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000213,"raw_usage":{"total_tokens":1360,"prompt_tokens":825,"completion_tokens":535,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":441,"completion_tokens_details":{"reasoning_tokens":455}},"tokens_in":441,"tokens_out":535,"duration_ms":6262,"temperature":1.0,"reasoning_tokens":455,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:24:03.269483+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete test would be to construct a smooth vector-valued GANM whose triple is known to solve Eq. (3), generate data from it, and show that the reverse orientation also admits an additive noise fit, causing GroupRESIT-MURGS to return the wrong direction; alternatively, computing the Lebesgue measure of the set of triples satisfying Eq. (3) and finding it to be generic would show the identifiability theorem covers almost no models.","supporting_citations":[],"review_version":1}