{"id":"2b6695c1-4c2a-4378-bc4d-e2722467f8fc","arxiv_id":"2608.02769","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":7,"one_line_summary":"DAIF adaptively selects fusion granularity via CKA clustering and clusterwise empirical Bayes AMP denoising, with consistency, state-evolution, and Bayes-optimality theorems.","lead":"DAIF learns which modalities to fuse by clustering them with a kernel dependence measure, then denoises each cluster with an approximate message passing algorithm before supervised prediction. The paper provides asymptotic theory and shows competitive results on a single-cell protein prediction task and a breast cancer survival task.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Clustering consistency theorem assumes a hard CKA separation that is not derived from the factor model; small cross-cluster dependence breaks the guarantees, and real-data K choices do not validate the data-adaptive claim.","rationale":"The reader's weakest_assumption and my analysis point to the same place: Theorem 6.1 and its corollary are only as strong as the clique-product prior and the Delta_sep separation in Assumption H.1. I find no internal inconsistency in Theorem H.1's proof under that assumption, and Lemma H.2 gives a genuine convergence result for CKA-based affinities. The weakness is that Delta_sep is not derived from the factor model and there is no robustness or finite-sample guarantee for small or overlapping cross-cluster dependence; the real-data experiments do not provide independent support because the gap statistic did not select the best-performing K in either dataset. This does not falsify the method, but it means the headline 'learns fusion granularity from data' is conditional on a structure that is plausible yet unverified. The paper's theoretical appendices are otherwise substantive, and the simulations are reproducible in conception, so a CONDITIONAL recommendation remains appropriate.","tokens_in":47053,"tokens_out":9437,"duration_ms":94699,"concrete_test":"Modify the Section 4 simulation to include a small shared factor: for each row i, draw g_i ~ N(0,1), set U_{3,i} = sqrt(1-eps)*U_{3,i}^{orig} + sqrt(eps)*g_i and similarly perturb U_{1,i}, U_{2,i} with the same g_i, so the 'independent' modality 3 has population CKA of order eps with cluster {1,2}. For eps in {0, 0.01, 0.05, 0.1} and n in {3000, 5000}, repeat 50 runs and record the fraction of runs where hierarchical clustering recovers the true {1,2}|{3} partition, as well as the reconstruction errors E_1, E_2, E_3 from Table 1. If clustering fails or E_k degrades sharply for eps as small as 0.05, then the exact-zero between-cluster assumption is load-bearing and the theorem's premise is not robust to slight partial dependence.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theoretical chain (Theorems 6.1–6.3) depends on exact recovery of the modality partition. Under the product prior (2.5), between-cluster population CKA is exactly zero for a characteristic kernel, so the non-vacuous part of Assumption H.1 is the within-cluster lower bound Delta_sep>0 together with the exact clique-product structure. Lemma H.2 proves only that empirical CKA converges to population CKA; it does not derive Delta_sep from the factor model (2.1)–(2.5), nor does the paper quantify how small Delta_sep can be before finite-sample clustering fails. More importantly, real modalities with overlapping or weak partial dependence violate (2.5): between-cluster CKA becomes positive, Theorem H.1's induction argument breaks, and the estimated clusters pC_k need not equal C_k. Then pµ from (3.5) is maximized over the wrong coordinates, and Theorems 6.2 and 6.3 lose their oracle basis. The empirical section shows the selection mechanism itself is fragile: on TEA-seq the gap statistic chose K=1 although K=3 had slightly lower RMSE (2.6854 vs 2.6867), and on TCGA-BRCA it chose K=2 although K=3 had the best test C-index (0.7375 vs 0.7359). Thus the data-adaptive granularity claim is neither proven for realistic dependence nor validated by the real-data K choices.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes DAIF, a pipeline that learns the granularity of multimodal fusion from data. It models the latent subject signatures as arising from a prior that factorizes over an unknown partition of the modalities (2.5), estimates the partition by hierarchical clustering on CKA-based dissimilarities, estimates clusterwise priors by empirical Bayes, and then applies AMP-based denoising within each estimated cluster. The main theoretical results are Theorem 6.1 (weak consistency of the estimated prior, including the estimated clusters), Theorem 6.2 (state evolution of the AMP iterates coincides with that of oracle iterates), Theorem 6.3 (Bayes-optimal recovery of the signal matrices), and Theorem 6.4 (Gaussian approximation for training and test embeddings used in downstream prediction). Simulations and two real-data applications (TEA-seq and TCGA-BRCA) are used to support the framework.","tokens_in":47449,"tokens_out":6779,"duration_ms":65014,"significance":"If the theoretical chain holds, the paper gives a principled interpolation between early fusion (OrchAMP) and late fusion (EB-PCA), with nonparametric clustering replacing an a priori architecture choice. The appendices contain real derivations rather than sketches: Theorem H.1 is proved by an induction on the agglomerative merges, and Theorem 6.2 transfers state evolution from Nandy and Ma (2024) once the estimated priors are consistent. However, the headline \"data-adaptive granularity\" claim is only established under a hard clique-independence condition, and the real-data K-selection results do not currently validate the adaptive choice. The contribution is therefore best viewed as a sound intermediate-fusion framework under well-separated cluster structure, with the practical benefit of adaptation still needing stronger evidence.","major_comments":[{"comment":"Assumption H.1 is the load-bearing premise for Theorem 6.1, but it is not derived from the factor model (2.1)-(2.5). Under the product prior (2.5) and a characteristic kernel, between-cluster population CKA is zero by independence, so the substantive content is the within-cluster gap Delta_sep > 0. Lemma H.2 only proves convergence of empirical affinities to population affinities; it does not express Delta_sep in terms of the model parameters, nor does the paper provide a robustness statement for small positive between-cluster affinities. If the dependence structure is only approximately clique-disjoint, the population between-cluster affinity is positive, Theorem H.1's induction (which relies on an empirical affinity gap of Delta_sep/3) can fail, and then the estimator in (3.5) is maximized over the wrong coordinates. This would invalidate the oracle-equivalence basis of Theorems 6.2 and 6.3. Please provide a quantitative misspecification analysis or state the separation condition as a direct assumption with a discussion of its scope.","section":"Section 6.1 / Appendix H.2 (Assumption H.1)"},{"comment":"The real-data K-selection results do not support the claim that DAIF learns the optimal fusion granularity. In the TEA-seq analysis, the gap statistic selected K=1 even though K=3 had a lower test RMSE (2.6854 vs 2.6867). In the TCGA-BRCA analysis, the gap statistic selected K=2 even though K=3 had the best test C-index (0.7375 vs 0.7359). No uncertainty quantification is reported for these RMSE or C-index differences, so the observed gaps may be within noise, but as presented the data-adaptive choice is never the best-performing granularity in either real-data experiment. The paper either needs error bars or bootstrap intervals that demonstrate the selection procedure is not worse than the best fixed granularity, or the claims about data-adaptive granularity should be softened.","section":"Sections 5.1 and 5.2 (Tables 6 and 7)"},{"comment":"The proof of Theorem 6.3 is a single sentence: it invokes Theorem 6.2 and the techniques of Theorem 5.3 of Nandy and Ma (2024), together with fixed-point equations from (5.13) of that paper. Since Bayes-optimal recovery of the signal matrices is a headline theoretical contribution, the proof should verify that the clusterwise empirical-Bayes priors estimated from the data satisfy the required fixed-point equations, and that the limiting recursions in Theorem 6.2 converge to those fixed points. As written, this step is asserted rather than demonstrated, which makes the Bayes-optimality claim not fully supported within the manuscript.","section":"Section 6.2 (Theorem 6.3)"}],"minor_comments":[{"comment":"The sentence \"Since the gap statistic only compares K=1,2, we separately evaluate K=3\" is confusing, because the standard gap statistic can be evaluated over an arbitrary range of K; the authors should clarify whether they restricted the search and why.","section":"Section 5.1"},{"comment":"The prior class P(R^{r_k}) in (3.5) is not defined in the main text, and the likelihood L_k is only introduced in Appendix H.4. The conditions needed for MLE consistency, such as compactness or moment bounds, should be stated explicitly as part of Theorem 6.1.","section":"Equation (3.5) and Appendix H.4"},{"comment":"The notation \\hat D_h for the estimated singular-value matrix in (3.3) is easy to confuse with the true diagonal signal matrix D_h and with the principal-component matrix D^pc_h used in the same display; a clearer notation such as \\tilde D_h would improve readability.","section":"Equation (3.3)"},{"comment":"The RMSE and C-index values are reported without replication or bootstrap uncertainty; given that the K-selection differences are very small, each table should indicate whether the reported differences are stable across runs or splits.","section":"Tables 6 and 7"}],"recommendation":"major_revision","confidential_remarks":"This is a competent paper with real derivations in the appendices, and the reliance on the authors' prior AMP work is legitimate. The main reason for major revision rather than rejection is that the central data-adaptive granularity claim is not yet supported by the real-data K-selection results, while the theory requires a hard separation assumption that is not derived or robustified. I would encourage the authors to add a misspecification analysis and uncertainty quantification for the real-data comparisons."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a real contribution: the first AMP pipeline I know of that learns fusion granularity from data via CKA-based clustering and then proves oracle equivalence given exact cluster recovery. The theory is honest about its own footing. Theorem 6.1 is explicitly conditional on a hard separation gap (Assumption H.1) with between-cluster affinity exactly zero, and they don't pretend otherwise. Theorem 6.2 leans on Nandy-Ma and Zhong et al., but that's legitimate—published proofs used as lemmas—and the appendices contain real derivations, not hand-waving. What's genuinely new is the combination: EB-PCA and OrchAMP as special cases, a data-driven clustering step in the middle, and consistency results that transfer to AMP optimality. The simulations are carefully designed with heterogeneous priors and show clear gains over AJIVE/MCCA/GCCA. For someone working on high-dimensional multimodal integration, this is a useful piece of machinery. The soft spots are real but not fatal. The load-bearing premise is that true dependence is a disjoint union of cliques with a hard CKA gap. That is assumed, not derived from the factor model, and the paper never quantifies how small the gap can be before finite-sample clustering fails. If real modalities have overlapping or partial dependence, the clustering consistency theorem loses its foundation, and everything downstream—prior estimation, state evolution, Bayes optimality—is built on oracle clusters. The real-data section doesn't rescue this: on TEA-seq the gap statistic chose K=1 while K=3 had slightly lower RMSE (2.6854 vs 2.6867), and on TCGA-BRCA it chose K=2 while K=3 had the best test C-index (0.7375 vs 0.7359). So the central data-adaptive granularity claim is weakly validated. There is also no code or data repository, and the experiments use single train/test splits; the feature selection choices (e.g., picking 5 CNV genes by marginal variance) feel ad hoc. These are addressable. Bottom line: this deserves a serious referee. It is a well-executed paper with an honest theoretical core and a clear niche. I would bring it to a reading group and would cite the clustering-consistency theorem if I worked on AMP. With code release, a more careful real-data analysis, and some frank discussion of when the clique assumption fails, it could be a solid publication.","headline":"A substantive AMP-based fusion method with honest theory, but the data-adaptive fusion claim rests on a hard clique-separation assumption the real-data experiments only weakly support.","tokens_in":684,"tokens_out":787,"would_cite":true,"duration_ms":27247,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62H30","62H25","62F12"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that learning fusion granularity from data costs nothing asymptotically: clustering modalities with CKA and running clusterwise empirical-Bayes AMP recovers the same oracle-level latent representations and predictions.","keywords":["data adaptive intermediate fusion","multimodal supervised learning","approximate message passing","empirical Bayes","centered kernel alignment","hierarchical clustering","state evolution","factor model"],"falsifier":"Generate data with three modalities where two are strongly dependent and the third has small but nonzero population dependence with one of them, so the separation gap is absent, increase the sample size, and check whether hierarchical clustering still recovers the true group structure and whether the DAIF state evolution still matches an oracle that knows the correct non-clique dependence; if the oracle and DAIF divergences persist at large n, the separation condition is not merely technical.","tokens_in":46854,"feed_emoji":"🔗","tokens_out":8897,"duration_ms":76933,"temperature":0.7,"pith_summary":"The paper tries to show that the right granularity for multimodal fusion—which modalities to integrate and which to keep separate—can be chosen from the data itself without sacrificing statistical optimality. It proposes DAIF, a pipeline that measures pairwise dependence between modalities with a kernel-based score, clusters the modalities by that score, estimates a prior over latent signals inside each cluster, and then denoises the features with approximate message passing. The payoff, if the theorems hold, is that early fusion and late fusion become special cases of one adaptive procedure rather than competing architectural choices. The method is aimed at settings like single-cell multi-omics, where RNA, chromatin, and protein data are unequally dependent and a fixed fusion choice helps some cell types while hurting others.","feed_headline":"Learned fusion granularity costs nothing asymptotically","feed_subtitle":"Clustering modalities by dependence plus clusterwise empirical-Bayes AMP matches an oracle that knows the fusion structure.","key_machinery":"The central object is the clusterwise product prior $\\mu = \\bigotimes_{k=1}^K \\mu_k$ over the latent signatures $(U_1,\\ldots,U_m,\\tilde U_1,\\ldots,\\tilde U_{\\tilde m})$, under the factor model $X_h = n^{-1/2} U_h D_h V_h^\\top + Z_h$ for high-dimensional modalities and $\\tilde X_\\ell = \\tilde U_\\ell L_\\ell^\\top + \\tilde Z_\\ell$ for low-dimensional ones. The mechanism that carries the argument has four moving parts: centered kernel alignment (CKA), a normalized kernel dependence measure, converted into a dissimilarity matrix for average-linkage hierarchical clustering; a clustered empirical-Bayes maximum likelihood estimator that estimates each $\\mu_k$ from the leading singular vectors inside the estimated cluster; AMP recursions that alternate Onsager-corrected power iteration with posterior-mean denoising under the estimated clusterwise prior; and state-evolution recursions that track the covariance of the iterates. The key identity driving the theory is that after the Onsager correction the iterates behave like the true signals plus Gaussian noise with known covariance, so posterior-mean shrinkage is the Bayes-optimal denoiser, and the estimated-prior version converges to the oracle version by the uniform Lipschitz regularity of the denoisers.","core_discovery":"The central claim is that the granularity of multimodal fusion can be learned from data and that learning it costs nothing in the large-sample limit. Under a Bayesian factor model in which the cross-modal dependence of the latent subject signatures is exactly a partition into independent cliques, DAIF (1) estimates pairwise modality dependence with centered kernel alignment, (2) recovers the cliques by average-linkage hierarchical clustering, (3) estimates each cluster's prior by empirical Bayes, and (4) runs an approximate message passing (AMP) denoiser with clusterwise priors. The main theorems state that the estimated prior converges weakly to the true prior (Theorem 6.1), that the AMP iterates have the same asymptotic state evolution as oracle iterates built from the true priors (Theorem 6.2), and that the limiting reconstruction error equals the Bayes-optimal error, i.e., the error of the posterior expectation of the signal given all modalities (Theorem 6.3). A fourth theorem (Theorem 6.4) gives the error-in-variables representation of the rescaled training embeddings and the OLS test projections, which is what makes supervised prediction valid. Early fusion and late fusion appear as the boundary cases of one cluster and one cluster per modality.","pith_inferences":["An implication the paper leaves implicit is that its clique-partition assumption is also a testable design target: for data with overlapping or partial dependence, the separation gap fails, and the consistency theorems do not cover that regime; a natural extension would replace hard clusters with soft or hierarchical affinity thresholds.","The per-modality signal-strength threshold suggests an immediate extension: if individual modalities are below the threshold but jointly identifiable, a spectral initialization that pools modalities before PCA could lower the effective signal threshold, a direction the paper lists as future work and one that would broaden DAIF to weak-modality settings.","The error-in-variables representation (3.14) is generic enough that the Cox-survival loss used on one real dataset should extend to other generalized linear or censored outcomes with the same Gaussian-perturbation correction.","Because the framework is modular in the dependence measure, the same pipeline should work with distance correlation or nearest-neighbor measures; the theory only needs uniform convergence of the affinity estimates, so the practical effect of measure choice is testable without changing the AMP core."],"forward_implications":["Users no longer need to commit a priori to early or late fusion: DAIF's boundary cases are a jointly denoising variant with one cluster and per-modality empirical-Bayes PCA with one cluster per modality, with every intermediate number of clusters selected from the data.","When the true dependence is a separated clique partition, the estimated clusters and priors are asymptotically harmless: the AMP iterates' state evolution matches that of an oracle that knows the true clusters and priors.","The recovered embeddings are Bayes-optimal in the double limit as iterations and sample size grow, so downstream prediction is built on the best possible linear-Gaussian features for this factor model.","The OLS test projection and the rescaled training embeddings share the same asymptotic Gaussian perturbation distribution, which justifies training a predictor on one and applying it to the other even when the test sample is small.","In simulations, intermediate fusion beats both full fusion and no fusion on latent reconstruction and prediction error, and the gain is largest when only a subset of modalities are dependent."],"supporting_citations":[{"why":"Defines centered kernel alignment (CKA), the dependence measure DAIF converts into the dissimilarity matrix for clustering.","marker":"Kornblith et al. (2019)"},{"why":"Provides the empirical-Bayes PCA baseline (late fusion) and the consistency lemmas for empirical-Bayes prior estimation that Theorem 6.1 adapts.","marker":"Zhong et al. (2022)"},{"why":"Supplies the orchestrated AMP recursion, state evolution, and oracle-optimality machinery that DAIF extends from one cluster to estimated clusters.","marker":"Nandy and Ma (2024)"},{"why":"Establishes the AMP state-evolution framework for low-rank matrix estimation on which the asymptotic characterization of the iterates rests.","marker":"Montanari and Venkataramanan (2021)"},{"why":"Gives the spiked-model singular-vector asymptotics behind the principal-component initialization and the Gaussian-perturbation approximation.","marker":"Benaych-Georges and Nadakuditi (2012)"},{"why":"Provides the TEA-seq trimodal single-cell dataset used to test CD45RA protein prediction.","marker":"Swanson et al. (2021)"},{"why":"Provides the TCGA-BRCA multi-omic dataset via the Xena platform for the survival-prediction experiment.","marker":"Goldman et al. (2020)"},{"why":"Supplies the cooperative-learning baseline against which DAIF's prediction performance is compared.","marker":"Ding et al. (2022)"}],"fun_headline_variants":["Fusion granularity learned from data, no asymptotic cost","Data-driven fusion with AMP: cluster, estimate, denoise","Learn fusion structure, match oracle in the limit","Multimodal fusion adapts to dependence, not fixed","DAIF: Data-adaptive fusion via clustering and AMP"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole guarantee rests on the true dependence graph among modalities being exactly a collection of disjoint cliques: within a cluster the population affinity is at least Delta_sep, between clusters it is exactly zero, and the empirical affinity estimates converge uniformly to those population values.","fun_headline_variants_meta":{"raw":{"variants":["Fusion granularity learned from data, no asymptotic cost","Data-driven fusion with AMP: cluster, estimate, denoise","Learn fusion structure, match oracle in the limit","Multimodal fusion adapts to dependence, not fixed","DAIF: Data-adaptive fusion via clustering and AMP"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000635,"raw_usage":{"total_tokens":3004,"prompt_tokens":1096,"completion_tokens":1908,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":712,"completion_tokens_details":{"reasoning_tokens":1828}},"tokens_in":712,"tokens_out":1908,"duration_ms":13052,"temperature":1.0,"reasoning_tokens":1828,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:00:04.678338+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Generate data with three modalities where two are strongly dependent and the third has small but nonzero population dependence with one of them, so the separation gap is absent, increase the sample size, and check whether hierarchical clustering still recovers the true group structure and whether the DAIF state evolution still matches an oracle that knows the correct non-clique dependence; if the oracle and DAIF divergences persist at large n, the separation condition is not merely technical.","supporting_citations":[],"review_version":1}