{"id":"28d6ed82-4741-4e2c-b361-1d52e9b3f580","arxiv_id":"2601.06315","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Variational Bayesian inclusion-flag estimates, thresholded into a directed graph, select a smaller Koopman dictionary while leaving output influence paths intact.","lead":"The paper gives a way to automatically prune the dictionary of a Koopman model: a Bayesian model estimates the probability that each dictionary function is needed, and a graph search keeps only the functions that can affect the outputs. The resulting smaller models are better conditioned and usually as accurate, which matters for system identification and model predictive control.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Algorithm 2's 'outputs unaffected' guarantee depends on Γ_ε being a faithful support graph; Γ_ε is never validated against true supports and ε is tuned per system, so a false exclusion silently breaks the guarantee.","rationale":"I read the paper as claiming a model-reduction method: use VB spike-and-slab to estimate which Koopman matrix entries exist, threshold to a graph, and keep only output-reachable ancestors. The graph algorithm itself is correct in the ideal case: if Γ_ε has an edge exactly when the corresponding regression coefficient is nonzero, then the retained nodes form a closed subsystem and outputs are unaffected. The soft spot is the premise. The posterior inclusion probabilities from mean-field VB are approximate, no ground-truth support is available in the experiments, and ε is a free parameter chosen per dataset. The paper's own results contain a red flag: on the Wiener-Hammerstein benchmark, the proposed method's test NMSE worsens after reduction (Table 1), so the empirical claim 'preserve or improve' is not uniformly met. The conditional verdict is appropriate; additional validation on synthetic ground-truth systems would settle whether the support graph is reliable. I do not see an internal inconsistency in the graph construction itself, and the conditioning improvements (Figures 8-9) are credible. The reader's weakest_assumption matches my assessment; I would not change the verdict.","tokens_in":14118,"tokens_out":10580,"duration_ms":114098,"concrete_test":"Generate a synthetic system whose dictionary is exactly Koopman-invariant and whose K_true is known and sparse (e.g., a random linear map on a polynomial dictionary). Simulate training data, run the proposed VB inference, threshold Γ at the paper's ε, and compare Γ_ε with the true support of K_true (precision/recall). Then simulate the full and reduced models from identical initial conditions and check whether the output trajectories diverge. If any truly nonzero edge is missed or any discarded node has a path to an output, the 'unaffected outputs' claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central reduction claim is conditional on the thresholded inclusion matrix Γ_ε correctly representing every regression edge that can influence an output (Algorithm 2, Section 4.2). The paper never checks this support-graph assumption: there is no comparison of Γ_ε to ground-truth nonzero coefficients in any experiment, and the threshold is set separately for each system (ε=0.01 Lorenz, 0.25 USV, 0.1 Wiener-Hammerstein). If a coefficient that is actually needed for output prediction has posterior inclusion probability below ε, then the corresponding edge is absent from Γ_ε, the discarded node is not an ancestor, and the output evolves differently after reduction. The guarantee is therefore not a property of the graph algorithm alone; it is a property of the unverified inference output. Moreover, the paper re-estimates the reduced model from data rather than projecting the full K̂_F, so even with a faithful Γ_ε the reduced-model outputs can differ from the full-model outputs; Table 1 shows the proposed method's test NMSE rising from 4.6×10^-3 to 7.3×10^-3 on Wiener-Hammerstein, which is direct evidence that 'output variables remain unaffected' is not what the experiments demonstrate. The paper's own conclusion acknowledges tuning and mean-field limitations but does not address this support-graph validity.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a hierarchical Bayesian spike-and-slab model for identifying finite-dimensional Koopman operator approximations. Each column of the EDMD regression is treated independently; the model includes inclusion flags gamma, coefficients beta, noise precision rho, and their hyperparameters, estimated by mean-field variational Bayes. The posterior inclusion matrix Gamma is thresholded at eps to produce a binary adjacency matrix over dictionary observables. Algorithm 2 condenses the resulting directed graph into SCCs and retains the SCCs that contain the output observables plus all their ancestors; the reduced dictionary is then used to re-estimate the Koopman model. Experiments on Lorenz, an underactuated USV, and the Wiener-Hammerstein benchmark compare the full and reduced dictionaries for four identification methods. The reduced dictionary improves conditioning of pseudoinverse and thresholded least-squares baselines and often preserves one-step NMSE; the proposed VB method itself matches SBL on the Lorenz and USV examples but underperforms SBL on the real Wiener-Hammerstein dataset.","tokens_in":14500,"tokens_out":13816,"duration_ms":131690,"significance":"The central idea using posterior inclusion probabilities to build a reachability graph over observables and reduce the dictionary while preserving output-reachable structure is interesting and potentially useful for Koopman model identification and control. The VB update derivations are transparent and the algorithms are specified in enough detail to be reimplemented. However, the main contribution as stated ('output variables remain unaffected by the reduction') is not established: it holds only if the thresholded inclusion matrix is a faithful support graph, which is never validated, and the paper's re-estimation protocol can change outputs (Table 1 shows a degradation for the proposed method on Wiener-Hammerstein). With additional validation and more honest framing, this could be a practical tool for dictionary reduction in EDMD.","major_comments":[{"comment":"The guarantee that the reduced dictionary leaves output variables unaffected depends on Gamma_eps being a faithful support graph: every coefficient that can influence an output must have posterior inclusion probability >= eps, and no discarded coefficient may affect the output through an unmodeled path. The paper never validates Gamma_eps against ground-truth supports (e.g., synthetic regressions with known sparse coefficients) or against the full K_F. The threshold eps is set ad hoc (0.01 Lorenz, 0.25 USV, 0.1 Wiener-Hammerstein), and Figures 2 and 7 show the reduced size depends on eps. A false exclusion would silently remove an ancestor and change the outputs. Please add a validation of the inferred support (recovery of known nonzeros) and a sensitivity analysis over eps, or explicitly frame the guarantee as conditional.","section":"§4.2, Eq. (34); contribution bullet in §1"},{"comment":"After computing the reduced dictionary, the paper re-estimates the Koopman model from the reduced dictionary rather than deriving the reduced model by restricting the full K_F to the kept observables. Even with a perfect support graph, re-estimation can change the outputs. Table 1 directly demonstrates this: for the proposed method, the test one-step NMSE on the Wiener-Hammerstein system increases from 4.6e-3 (full) to 7.3e-3 (reduced). This contradicts the contribution bullet stating the outputs 'remain unaffected by the reduction.' To support the graph guarantee, report the outputs of the reduced model obtained by projecting the full K_F onto the kept dictionary (submatrix), and compare those with the full model; present the re-estimated results as practical performance, not as the guarantee.","section":"§5.3, Table 1; §4.2, Algorithm 2"},{"comment":"The update for the noise precision rho uses b_bar = 1/2 ||t - Phi(gamma_hat * beta_hat)||^2 + b. Under the mean-field posterior, the exact expectation of ||t - Phi(gamma * beta)||^2 contains the variance of each product gamma_i beta_i; because q factorizes, this adds sum_i gamma_hat_i[(1 - gamma_hat_i) beta_hat_i^2 + sigma_i^2] ||phi_i||^2, which is not accounted for. The update is therefore an approximation, not the exact VB update promised in the contributions. Please either derive the exact expectation or explicitly denote the point-estimate approximation and discuss its effect on the inferred inclusion probabilities.","section":"§4.1, Eq. (17d)"},{"comment":"The performance evaluation uses only the one-step prediction NMSE, while the paper itself cites [8] to note that one-step performance can be misleading for Koopman models. Since the stated motivation is model predictive control and long-horizon prediction, please report multi-step or long-horizon NMSE for the full and reduced models, especially for the Wiener-Hammerstein experiment where the one-step metric already degrades for the proposed method. Without this, the claim that the reduction 'preserves or improves prediction performance' is not supported for the regimes of interest.","section":"§5, Eq. (35)"},{"comment":"The inclusion-probability threshold eps is hand-tuned per system (0.01, 0.25, 0.1) and there is no criterion for its selection. Figures 2 and 7 show that the reduced dictionary size and hence the identified model depend strongly on eps and on the noise level. The paper calls the thresholding 'scale-free,' but the actual choice remains a user-defined free parameter. Please provide a principled selection rule (cross-validation, stability selection, or similar) and use it consistently, or explicitly list eps as a tuning parameter of the method.","section":"§5.1–§5.3"}],"minor_comments":[{"comment":"No code or data repository is provided, which limits reproducibility despite the detailed algorithmic description.","section":"General"},{"comment":"There is a notation inconsistency: in Section 3, gamma_j is a vector of length L+l, but in Eq. (33) and Algorithm 1, gamma_hat_i is used for the scalar posterior inclusion probability. Please disambiguate.","section":"§3, §4.1"},{"comment":"Minor typos and grammar: 'Koopman operator recently gained' in the abstract should be 'has recently gained'; 'overparameterized' and 'overfitting' are used inconsistently. Please proofread.","section":"§1, Abstract"},{"comment":"The heatmaps use a logarithmic color scale spanning many orders of magnitude; consider adding annotations or a reproducible colorbar to make the comparison of sparsity patterns easier.","section":"Figures 8–9"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern about the support-graph validity is on target. The central claim of the paper is conditional on an unvalidated thresholded inclusion matrix, and the re-estimation protocol in Section 5 further breaks the 'unaffected outputs' guarantee. The paper should be given the opportunity for major revision; the graph-reduction idea is sound in principle and the VB derivations are mostly standard, but the main contribution needs re-framing and validation. The paper's own conclusion recommending SBL as the final model also weakens the standalone contribution of the proposed identification model."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague—\n\nThis is a genuine method paper, not a repackaging. The new bit is combining spike-and-slab inclusion flags from variational Bayes with a graph condensation/ancestor search to prune a Koopman dictionary while keeping the output-reachable states. That combination isn't in the cited literature, and it's a sensible preprocessing step: it shrinks dictionaries, improves conditioning, and can be bolted onto standard EDMD/SINDy/SBL identifiers. The VB updates are standard conjugacy and look right; the damping and clipping details are pragmatic. The paper earns credit for testing on Lorenz, a controlled USV sim, and real Wiener-Hammerstein data, and for making the honest observation that one-step NMSE can be misleading.\n\nThe main soft spot is exactly what the stress test flags. Algorithm 2's guarantee that outputs are unaffected is conditional on Gamma_epsilon being a faithful support graph, and that is never checked against any ground truth. Epsilon is hand-picked per system (0.01, 0.25, 0.1), and the reduced model is re-estimated from data, so the final outputs can drift — the Wiener-Hammerstein test NMSE goes from 4.6e-3 to 7.3e-3 for method IV. The paper's own text says outputs remain unaffected but the experiments don't show that at the level of individual trajectories or supports; they show preserved NMSE on the reported systems. That is still a useful result, but the guarantee should be softened or validated, e.g. by synthetic dictionaries with known sparsity patterns.\n\nOther soft spots are minor in comparison: the proposed identifier itself is beaten by SBL on real data, which the conclusion admits, and no code or data are released, which makes the per-system thresholds hard to reproduce. The citation pattern is fine; reference [26] is the natural SBL baseline and using it as such isn't a problem.\n\nOverall: the central idea is sound, the presentation is readable, and the limitations are mostly acknowledged. I'd send this out for review with a request to validate the support-graph assumption and release code. It deserves referee time.","headline":"A genuine method paper: VB spike-and-slab plus SCC graph pruning for Koopman dictionaries; worth refereeing, but the output-unaffected guarantee needs validation against true supports and the per-system epsilon tuning documented better.","tokens_in":14913,"tokens_out":2111,"would_cite":true,"duration_ms":24182,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93B30","93C55","62F15","05C20"],"pacs":[],"model":"deepseek-v4-flash","headline":"A probabilistic Koopman model with inclusion flags can be shrunk to the observables that actually feed the outputs, without hurting prediction accuracy.","keywords":["Koopman operator","variational Bayesian inference","spike-and-slab prior","dictionary reduction","graph condensation","strongly connected components","system identification","EDMD"],"falsifier":"Build a synthetic linear system with a known sparse dictionary where a small genuine coefficient connects a state to the output. Run the proposed VB inference with a modest threshold, then check whether that coefficient's posterior inclusion probability falls below epsilon. If it does and the reduced dictionary excludes that state, one-step prediction will degrade on test data, refuting the 'outputs unaffected' claim.","tokens_in":14043,"feed_emoji":"🕸️","tokens_out":3114,"duration_ms":32712,"temperature":0.7,"pith_summary":"The paper tries to establish that a Koopman model built from an overcomplete dictionary can be reduced to a much smaller dictionary without changing its output behavior. It does this by treating every regression coefficient as having an inclusion probability, estimated through variational Bayesian inference, and then using those probabilities as a directed graph. Nodes that cannot reach any output are removed. In three systems — a chaotic Lorenz attractor, an underactuated surface vehicle, and a real Wiener–Hammerstein circuit — the reduced models keep one-step prediction error at the same level while fixing ill-conditioned matrices. If true, this gives practitioners a principled way to prune dictionaries for EDMD-type models.","feed_headline":"Graph search shrinks Koopman models without touching outputs","feed_subtitle":"Keep prediction error and fix ill-conditioning by pruning only dead observables.","key_machinery":"A spike-and-slab prior over Koopman regression coefficients, inferred by variational Bayesian updates, produces an inclusion matrix Γ whose entries are posterior probabilities of each coefficient being active. Thresholding Γ at ε turns it into the adjacency matrix of a directed graph over observables; strongly connected component condensation and ancestor search then identify the minimal dictionary needed to predict the outputs. This graph interpretation is what carries the reduction argument.","core_discovery":"The paper's central claim is that the posterior inclusion matrix Γ, after thresholding at ε, encodes a directed graph whose nodes are dictionary observables, and that keeping only the strongly connected components that contain or feed the output observables yields a reduced dictionary that leaves the output dynamics unaffected. The authors derive variational Bayesian updates for a spike-and-slab model in which each Koopman regression coefficient is multiplied by a Bernoulli inclusion flag, so that weight magnitude and inclusion in the model are separated. The reduced dictionary is then obtained by graph condensation and ancestor search on the thresholded inclusion graph. The paper argues tha","pith_inferences":["If the inclusion graph faithfully reflects true causal dependencies, the same graph-based pruning could be applied to other dictionary-learning or encoder-decoder Koopman pipelines, not just the authors' VB model.","The method's reliance on a fixed threshold ε is a natural place to add uncertainty quantification; a data-driven ε or a threshold based on posterior credible intervals could make the reduction more robust.","On systems with hidden or weakly excited modes, a coefficient with genuine but small influence could fall below ε and be silently dropped, so the 'outputs unaffected' guarantee is only as strong as the support-graph fidelity.","A natural next test is a high-dimensional system with known sparse structure, where the reduced dictionary can be compared against the ground-truth generative support to measure how often true edges are discarded."],"forward_implications":["Overcomplete Koopman dictionaries can be pruned automatically without manually re-selecting observables, reducing model complexity for downstream control design.","Ill-conditioned Koopman matrices, which often plague pseudoinverse and thresholded least-squares fits, become numerically stable after reduction, as shown by the Wiener–Hammerstein heatmaps.","The reduced dictionary preserves one-step prediction NMSE for the systems tested, so the smaller model is not trading away predictive content.","The reduction works as an aiding module: other identification methods benefit from the reduced dictionary even though the inclusion matrix was produced by the proposed Bayesian model.","Threshold choice has little effect on the USV results, suggesting the inclusion probabilities converge to near-zero or near-one values rather than remaining indeterminate."],"fun_headline_variants":["Graph search trims Koopman dictionary, keeps accuracy","Variational Bayes and graph search reduce Koopman states","Koopman reduction via graph search and inclusion flags","Prune unused observables to fix Koopman conditioning","Inclusion flags enable dictionary reduction for Koopman"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The guarantee that outputs remain unaffected depends on the thresholded inclusion matrix being a complete map of every regression coefficient that can influence an output; if a coefficient that genuinely matters has posterior inclusion below epsilon, it gets silently pruned and the output guarantee fails.","fun_headline_variants_meta":{"raw":{"variants":["Graph search trims Koopman dictionary, keeps accuracy","Variational Bayes and graph search reduce Koopman states","Koopman reduction via graph search and inclusion flags","Prune unused observables to fix Koopman conditioning","Inclusion flags enable dictionary reduction for Koopman"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000454,"raw_usage":{"total_tokens":2087,"prompt_tokens":680,"completion_tokens":1407,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":424,"completion_tokens_details":{"reasoning_tokens":1338}},"tokens_in":424,"tokens_out":1407,"duration_ms":10290,"temperature":1.0,"reasoning_tokens":1338,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T11:26:16.269586+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Build a synthetic linear system with a known sparse dictionary where a small genuine coefficient connects a state to the output. Run the proposed VB inference with a modest threshold, then check whether that coefficient's posterior inclusion probability falls below epsilon. If it does and the reduced dictionary excludes that state, one-step prediction will degrade on test data, refuting the 'outputs unaffected' claim.","supporting_citations":[],"review_version":1}