{"id":"66be7cc1-eb57-4520-a902-b1eb728fddff","arxiv_id":"2412.12844","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Square federated learning with fuzzy cognitive maps lets participants with different samples and different features train one shared model without sharing private data.","lead":"This paper combines two kinds of federated learning, horizontal and vertical, so that different hospitals or banks can train one shared fuzzy cognitive map model without pooling their private data. It is worth reading because it shows a possible path for collaborative AI when partners have different patients, different features, and different data distributions.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Averaging locally-learned FCM adjacency matrices is meaningful only if shared features denote identical concepts across participants; the experiments guarantee this by construction, so the central claim is not yet supported for real non-IID feature sets.","rationale":"I read the paper in good faith: the contribution is a framework (not a SOTA accuracy claim) for federated FCM training under both horizontal and vertical data heterogeneity, with experiments on four aggregation strategies. The experimental numbers genuinely show that when participants split one dataset, train local PSO-FCMs, and aggregate adjacency matrices with the tested weights, the mean test accuracy and usually F1 improve. The concern I identify is the one the reader also flagged: the aggregation step assumes shared feature nodes have the same semantic meaning and comparable local weight estimates. That assumption is guaranteed in the experiments because they derive all participants from a single dataset, so feature columns are identical by construction in every participant that keeps them. In the paper's own target setting — partial overlapping attribute skew, where only some features are shared and the rest differ — the protocol must establish or enforce semantic alignment before averaging. The paper neither argues for that alignment nor tests a case where it is violated. A second, more tractable internal concern is that Section 4.2's prose contradicts Eqs. 7-9: the text says the scheme gives inversely proportional weight to performance, while the equations weight by the normalized performance, favoring higher-quality local models. This is a genuine algorithm-description inconsistency that should be corrected, and it means the experiments' interpretation of the accuracy/AUC/precision-based strategies is not exactly what the method text describes. It does not, by itself, invalidate the framework: the equations are the concrete implementable definition, and the prose can be fixed. Additional weaknesses (using the same test set to compute aggregation weights and report final metrics; no seeds, code, or significance tests) are real but secondary and addressable. The authors acknowledge the dependency on FCMs and the need for further investigation in the conclusions, which is an honest limitation statement. Because the central novelty claim — concurrent vertical and horizontal FL with FCMs — remains plausible and the identified gap is a missing experiment rather than a demonstrated failure, I would keep the CONDITIONAL verdict and add the semantic-mismatch test as a required condition. This is not a prosecution: if the test passes, the concern is largely resolved; if it fails, the central claim must be narrowed to aligned-feature aggregation.","tokens_in":18381,"tokens_out":2436,"duration_ms":22212,"concrete_test":"Construct a non-IID feature-set test in which participants touch the same feature index but the feature refers to semantically different local concepts (e.g., participant A's 'temperature' is its own numeric reading while participant B's 'temperature' is a different sensor's reading; or the same conceptual variable is present under two different names across clients). Run the same four aggregation strategies from Section 5 on participants that share only the label and some semantically mismatched features, then examine the federated model's accuracy and F1 relative to the participants' local pre-FL performance.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that square federated learning with FCMs yields effective training when both sample and feature spaces vary across participants. The mechanism's load-bearing step is Eq. 2: the server produces an augmented adjacency matrix by averaging the entries of independently PSO-learned local FCMs, and the resulting federated model then improves local accuracy. This operation silently presupposes that any feature shared by two participants labels one and the same concept with one and the same causal meaning, and that the locally learned weight is a comparable estimate of that common causal relation. In the experiments, that alignment is built in: every split is derived from a single PMLB dataset, shared columns retain identical semantics, and the three removed features are simply absent. Hence the aggregation is applied only under conditions where it is least likely to fail. The paper identifies partial overlapping attribute skew as the target scenario (Section 4.1), but provides no experiment and no argument for the case in which the same string happens to label semantically different variables, or in which the same concept is labeled differently, or in which local PSO solutions are not aligned by the objective function. The prose in Section 4.2 compounds the problem: it announces the aggregation in the accuracy-based strategy as 'inversely proportional to performance,' yet Eq. 7 divides by the sum of accuracies, which gives greater weight to the more accurate model — the stated motivation, giving more help to weaker models, is not what the implementation does. That is a significant correctness-relevant inconsistency in the method description, and the reader identified it. I do not treat disagreement with broad FL consensus as a flaw; the issue is internal: the paper's own stated target (non-aligned, partial-overlap features) is not what is tested.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes \"square federated learning,\" a framework in which multiple participants with partially overlapping feature spaces and different sample spaces train local fuzzy cognitive maps (FCMs) via particle swarm optimization and send their adjacency matrices to a server. The server aggregates these matrices using one of four weighting schemes—constant, accuracy-based, AUC-based, and precision-based—via the augmented adjacency-matrix construction of Eq. (2). The authors report experiments on 21 PMLB classification datasets (though the text says 22) with five participants per dataset, and they claim that mean post-federation accuracy improves over pre-federation accuracy in all four strategies while F1 improvements depend on the strategy. The paper frames this as the first approach to concurrently address horizontal and vertical federated learning with FCMs.","tokens_in":18537,"tokens_out":4327,"duration_ms":44679,"significance":"The conceptual idea is attractive and potentially useful: FCMs have a natural aggregation operation through augmented adjacency matrices, so they are a plausible model class for federated settings with heterogeneous feature spaces. A working square-FL method for FCMs would be a genuine contribution to the federated-learning literature, and the paper explicitly and honestly states that its goal is not to beat centralized baselines but to enable training under distributed, non-IID feature sets. However, the paper's central empirical claim is currently supported mainly by point estimates without variance, significance tests, or a disclosed evaluation protocol that prevents the aggregation weights from being derived from the same test labels used for evaluation. The manuscript also contains a direct internal contradiction between the stated intent of the weighting strategies and the equations that define them. If these issues are fixed, the contribution could be valuable; in its present form, the quantitative evidence is not yet convincing.","major_comments":[{"comment":"The aggregation weights in Eqs. (7)-(9) are computed from each participant's test set, and the same test set is then used to report the post-FL accuracy and F1 values in Tables 3-6. This makes the reported improvements an in-sample evaluation: the federated model is partially selected to perform well on the exact labels used for the headline numbers. The authors should compute the weighting metrics on a separate validation set and reserve a disjoint test set for final evaluation, or at minimum demonstrate that the improvement persists under such a protocol.","section":"§4.2 and §5, Eqs. (7)-(9), Tables 3-6"},{"comment":"The prose after Eqs. (7), (8), and (9) states that the aggregation gives more weight to less accurate or less precise models, and that individual models contribute \"inversely proportional to their performance metric.\" The equations, however, divide by the sum of the metric values, so a participant with higher accuracy, AUC, or precision receives a larger weight. This is the opposite of the stated intent and directly affects how the experimental comparisons among strategies are interpreted. The text or the equations must be corrected.","section":"§4.2, Eqs. (7)-(9)"},{"comment":"Every reported result is a single mean over five participants, with no standard deviation, no number of PSO runs, no seeds, no train/test split details, and no code. Since PSO is stochastic and the random removal of features changes the participants' feature spaces, the claim that accuracy improves \"in all cases\" is not supported without evidence that the differences are stable across runs. The authors should report distributions over repeated runs, provide significance tests or confidence intervals, and release code or at least exact seeds and split procedures for reproducibility.","section":"§5, Tables 3-6"},{"comment":"The augmented-matrix aggregation in Eq. (2) is only semantically meaningful when shared feature nodes denote identical concepts across participants and when the locally learned weights estimate the same causal relations. In the experiments this alignment is guaranteed by construction: all participants are derived from a single PMLB dataset and shared columns are identical by definition. The paper identifies partial overlapping attribute skew as the target scenario in §4.1 but provides no experiment or argument for cases where feature names align while meanings differ, where the same concept is labeled differently, or where independently optimized PSO solutions are not aligned by the objective. The authors should either add experiments with deliberately misaligned or renamed shared features or clearly scope the claim to feature-compatible settings.","section":"§4.1 and Eq. (2)"}],"minor_comments":[{"comment":"The text says the experiments use \"twenty-two open classification datasets,\" but Tables 2 through 6 each contain only 21 dataset rows. Please correct the count or add the missing dataset.","section":"§5.2, Tables 2-6"},{"comment":"Eq. (2) is difficult to parse: the symbol M is used without being defined, and the dimensions of the submatrices are not stated precisely. Please define M and clarify the indexing so that the averaging over existing entries is unambiguous.","section":"§3.1.1, Eq. (2)"},{"comment":"The centralized neural-network baseline is described only as having \"6 hidden layers and dropout\" with no optimizer, learning rate, epochs, or architecture details. This is not load-bearing for the main claim, but the baseline would be more interpretable if those details were provided.","section":"§5.1, Table 2"},{"comment":"The notation for the direct-sum operator and the augmented aggregation operator is inconsistent with the later use of Eq. (2); please unify the notation across the section.","section":"§3.1.1"},{"comment":"The paper repeatedly emphasizes the semantic significance and interpretability of FCM weights. Given that emphasis, a short discussion of whether averaging independently learned weights preserves semantic meaning under non-IID data would strengthen the presentation.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The idea is worth pursuing, and I do not see a fundamental flaw in the framework itself. The load-bearing problems are evidential and presentational: the test-set weighting scheme biases the headline numbers, the text contradicts the equations, and the stochastic experimental setup is reported without variance or reproducibility artifacts. These are fixable within the scope of a revision. I would also ask the editor to ensure that the related-work claim of being \"the first\" to combine horizontal and vertical FL with FCMs is checked carefully, since the paper's comparison to the broader square-FL literature is brief."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this paper's 'square federated learning' idea is real. Using the augmented adjacency matrix from earlier FCM work to aggregate models when participants differ in both sample and feature space isn't in the cited prior art, and the experiments show a consistent mean accuracy gain across all 22 PMLB datasets and all four weighting strategies. That is a genuine, if modest, result.\n\nThe method is straightforward and clearly explained: local PSO-trained FCMs send their adjacency matrices to a server, which averages them (weighted or not) and sends back an aggregated matrix; repeat. The authors are honest that the goal isn't beating centralized baselines, and the baseline comparison is useful.\n\nNow the soft spots. Section 4.2 has a direct contradiction: the prose says accuracy/AUC/precision weights favor less accurate models, but the equations divide by the sum of the metric, which gives more weight to the better models. That's a load-bearing description error. The experiments also sidestep the hard case. All splits come from a single PMLB dataset with features randomly removed, so shared features are semantically identical and identically distributed by construction. The paper claims to target partial overlapping attribute skew, but the test never exercises the situation where feature names align but meanings or distributions don't. The same test set is used both to compute the aggregation weights and to report post-FL accuracy, so the gains aren't an independent estimate.\n\nThe privacy claim in the abstract is unsupported: sending adjacency matrices can leak information, and the paper doesn't use or analyze any privacy mechanism. There's no code, seeds, or split details, which hurts reproducibility.\n\nNone of this is fatal. The core idea is worth pursuing, and the empirical pattern is consistent. But the manuscript needs a major revision: fix the weighting description, run a genuinely heterogeneous feature experiment, report variance across seeds, and either drop or substantiate the privacy claim.\n\nI'd send this to a referee who knows FCMs and FL. The square FL concept could be a useful contribution if it survives that scrutiny. I wouldn't cite the current version for the specific numbers, but the framework idea might be citable after revision.","headline":"Square FL with FCMs is a genuinely novel idea, but the weighting descriptions contradict the equations and the evaluation avoids the hard feature-alignment case.","tokens_in":19249,"tokens_out":3442,"would_cite":false,"duration_ms":32743,"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":"A new federated learning framework built on fuzzy cognitive maps lets participants train together when neither their samples nor their features are the same, and the paper's experiments show that the participants' mean accuracy improves…","keywords":["federated learning","fuzzy cognitive maps","square federated learning","vertical federated learning","horizontal federated learning","non-IID data","privacy-preserving machine learning","aggregation strategies"],"falsifier":"Run the square federated protocol with two participants whose feature sets are completely disjoint, then compare each participant's test accuracy before and after federation; because the augmented matrix has zero blocks connecting the feature sets, no cross-participant information exists, so any accuracy gain would be an averaging artifact rather than evidence that the method exploits differing feature spaces.","tokens_in":18065,"feed_emoji":"🧠","tokens_out":10448,"duration_ms":88297,"temperature":0.7,"pith_summary":"The paper proposes a federated learning framework that handles both horizontal and vertical data partitions at once, using fuzzy cognitive maps (FCMs) as the shared model. It introduces \"square federated learning,\" in which each participant trains a local FCM on its own samples and features, sends only the learned adjacency matrix (not raw data) to a server, and the server merges the matrices into one augmented FCM that covers the union of all participants' feature nodes. The authors test four aggregation strategies—constant, accuracy-based, AUC-based, and precision-based weights—on twenty-two binary classification datasets, with five participants that each lose three randomly chosen features. They report that the mean accuracy of the participants' local models improves after federation under all four strategies, while the F1-score improvement is more sensitive to the aggregation choice. A sympathetic reader cares because this is a step toward privacy-preserving collaborative training in settings where participants' data schemas only partially overlap, as is common in healthcare and finance.","feed_headline":"Square federated learning lifts accuracy when features vary","feed_subtitle":"Combines horizontal and vertical setups, improving mean accuracy across all four aggregation strategies.","key_machinery":"The central object is the augmented adjacency matrix of a fuzzy cognitive map (FCM). Each participant trains a local FCM with particle swarm optimisation, producing an $n\\times n$ weight matrix $W_i$; the server aggregates these with Eq. 2, computing for each cell the mean over all participants of that participant's weight if the edge exists in its model and zero otherwise. The result is a matrix with rows and columns for every node appearing in any participant's map, so models with different feature sets can be married into one global FCM; this is what lets horizontal and vertical federation proceed simultaneously. The four federation strategies differ only in the weights of the averaging: constant $1/n$, or normalised accuracy, normalised AUC, or normalised precision of each participant's local model on its test set.","core_discovery":"The central claim is that fuzzy cognitive maps make it possible to federate models across participants whose data differs in both the sample dimension (horizontal federation) and the feature dimension (vertical federation), a combination the paper calls square federated learning and presents as the first such framework for FCMs. Because an FCM is an adjacency matrix whose entries are learned causal weights between concept nodes, the server can merge participants' local matrices into a single augmented adjacency matrix, averaging the weights on nodes that appear in several participants and placing participant-specific nodes in disjoint blocks. The experiments use five participants drawn from the same datasets, with each participant missing three features, and all four aggregation strategies raise the mean accuracy of the local models after federation; the constant-weight strategy also improves mean F1 score on 20 of 22 datasets, accuracy-based weights on 19, precision-based weights on 18, and AUC-based weights on fewer than 70% of datasets. The authors do not claim to beat centralized training, and note that the value of the proposal lies in enabling training in distributed settings where centralised training is not possible.","pith_inferences":["Because Eq. 2 divides by the total number of participants even when only some participants contain a given edge, participants with heavily pruned feature sets are effectively diluted in the shared nodes; averaging only over the participants that possess the edge might change results on skewed overlaps.","A stress test the paper does not run is to vary the number of features removed per participant; the experiments fix three removals, so behaviour at extreme vertical splits with only one shared feature remains unknown.","The aggregated FCM could double as a diagnostic artifact: edges with large post-federation weights may reveal which concepts consistently matter across participants, a privacy-preserving byproduct the paper does not discuss.","If the square idea transfers to other model families whose parameters can be aligned and averaged (such as sparse linear models with a shared index space), the approach might generalize beyond FCMs; the paper explicitly leaves this generalization open."],"forward_implications":["Participants with partially overlapping data schemas can train a shared FCM without any raw data leaving their premises, since only learned adjacency matrices are exchanged.","The federation step does not require a central server: the paper notes the aggregation can be performed by one participant or by all participants in a peer-to-peer arrangement.","A single augmented FCM replaces the usual separation of horizontal and vertical federated pipelines, so consortia whose members' data differs along both axes can be handled by one protocol.","If F1 score is the target metric, the constant-weight aggregation is the safest of the four tested strategies, while AUC-based weighting improves mean F1 on fewer than 70% of the experimental datasets."],"supporting_citations":[{"why":"defines the federated averaging process that the framework extends to simultaneous horizontal and vertical splits.","marker":"[1]"},{"why":"supplies the augmented FCM aggregation method (Eq. 2) that lets local maps with different node sets be merged.","marker":"[26]"},{"why":"provides the PSO-based FCM learning algorithm used to train each participant's local model.","marker":"[21]"},{"why":"prior federated FCM work with the aggregation strategies that this paper adapts and compares.","marker":"[2]"},{"why":"prior federated FCM work, specifically on vertical federated learning without an initial model, which this work generalizes.","marker":"[67]"},{"why":"the source for the non-IID data setting and the taxonomy of attribute skew the square setting addresses.","marker":"[90]"},{"why":"the open benchmark datasets used in all square federated learning experiments.","marker":"[92, 93]"}],"fun_headline_variants":["Square federated learning: fuzzy maps for horizontal and vertical data","Fuzzy cognitive maps combine horizontal and vertical federated learning","Square FL: one fuzzy-map framework for horizontal and vertical data","Fuzzy maps enable square federated learning for mixed feature sets","Square federated learning with fuzzy cognitive maps lifts accuracy"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The method assumes that a feature with the same name means the same concept in every participant and that averaging the independently learned weights of such features yields a global model at least as good as the local ones; if participants differ in how they encode or interpret a shared column, the aggregation loses its meaning.","fun_headline_variants_meta":{"raw":{"variants":["Square federated learning: fuzzy maps for horizontal and vertical data","Fuzzy cognitive maps combine horizontal and vertical federated learning","Square FL: one fuzzy-map framework for horizontal and vertical data","Fuzzy maps enable square federated learning for mixed feature sets","Square federated learning with fuzzy cognitive maps lifts accuracy"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000849,"raw_usage":{"total_tokens":3667,"prompt_tokens":892,"completion_tokens":2775,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":508,"completion_tokens_details":{"reasoning_tokens":2692}},"tokens_in":508,"tokens_out":2775,"duration_ms":19798,"temperature":1.0,"reasoning_tokens":2692,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T13:40:30.721040+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the square federated protocol with two participants whose feature sets are completely disjoint, then compare each participant's test accuracy before and after federation; because the augmented matrix has zero blocks connecting the feature sets, no cross-participant information exists, so any accuracy gain would be an averaging artifact rather than evidence that the method exploits differing feature spaces.","supporting_citations":[{"cited_title":"Salmeron, Augmented fuzzy cognitive maps for modelling lms critical success factors, Knowledge-Based Systems 4 (22) (2009) 275–278","cited_arxiv_id":null,"evidence_quote":"supplies the augmented FCM aggregation method (Eq. 2) that lets local maps with different node sets be merged."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"prior federated FCM work, specifically on vertical federated learning without an initial model, which this work generalizes."}],"review_version":1}