{"id":"7bbbe5cb-bf1a-4603-8822-5e19dbc9bb21","arxiv_id":"2602.07269","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A budget-constrained greedy plus iterative algorithm selects cheap/expensive sensors to maximize Bayesian D-optimality and beats random designs in benchmark state estimation.","lead":"This paper shows how to choose a mix of cheap and expensive sensors under a fixed budget so that noisy measurements reveal the most about an unknown physical field. The algorithms are tested on sea-surface temperature and fluid-flow benchmarks, where they beat random sensor placement.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Ignored truncated-SVD error in the dictionary model may dominate the reported reconstruction gains; the D-optimality comparison runs entirely in the reduced space.","rationale":"The reader's weakest_assumption identifies exactly the same structural risk: the reduced dictionary u ≈ Φm with truncated-SVD error ignored enters the observation model (2) and the entire D-optimality objective, so the sensor-placement comparison is made in a model that may not reflect true state-estimation performance. I agree this is the most load-bearing concern, and it is not resolved by the paper's experiments because those experiments measure Φ_D in the reduced space and report reconstruction errors of the same order as the truncation floor. The proposed 99.9%-energy rerun is a direct sensitivity check: if the conclusions are robust to tightening the basis, the concern is mitigated for these benchmarks; if not, the central claim needs to be reworded or accompanied by a truncation-error correction. I did not elevate secondary issues (missing code, no error bars, no comparison to CELF/Clark et al.) because they affect confidence but not the logical core of the algorithms. The verdict should remain CONDITIONAL, as the reader set; no adjustment is needed beyond making the truncation stress test an explicit condition.","tokens_in":21924,"tokens_out":16715,"duration_ms":171414,"concrete_test":"Rerun the SST and cylinder experiments (Table 2 and Experiment 3, Figure 5) with the POD basis constructed at 99.9% cumulative singular-value energy instead of 99%, keeping all other settings fixed. If the D-optimality advantage of greedy/iterative over random designs, and the reported relative reconstruction errors, change materially (e.g., iterative no longer dominates greedy, or the RE gap versus random shrinks to within the basis-truncation floor), then the ignored truncated-SVD error in Section 2.1 is load-bearing for the central claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central empirical claim is evaluated inside the reduced dictionary model u ≈ Φm (Section 2.1), where the authors explicitly write 'we have ignored the error coming from the truncated SVD.' This is not a benign modeling detail. The reconstruction errors in Table 2 (RE ≈ 0.10 for SST and ≈ 0.086 for cylinder) are of the same order as the error floor expected from retaining only 99% of the singular-value energy: a rank-ℓ approximation with 99% energy leaves roughly 1% of the squared energy, i.e., about 10% relative ℓ2 error, uncaptured. Thus the reported reconstructions may be dominated by basis truncation rather than by sensor placement. Moreover, Experiment 3 (Section 5.2.3, Figure 5) compares greedy and iterative methods against random designs only in terms of Φ_D computed in the reduced model; it does not measure reconstruction error against the true state. If a large part of the true state lies outside span(Φ), sensor configurations that maximize D-optimality for the reduced model need not improve the true Bayesian state estimate, and random designs may achieve comparable reconstruction quality. The acknowledgment in Section 2.1 is therefore the weakest load-bearing point of the paper: it threatens the objective itself, not just the size of the empirical gain.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript formulates a budget-constrained multifidelity D-optimal sensor placement problem for Bayesian state estimation using a reduced-order dictionary model. It proposes a cost-normalized greedy algorithm accelerated by Sherman–Morrison rank-one updates (Algorithm 2), an iterative allocation-refinement algorithm (Algorithm 4), proves monotonicity/submodularity of the objective, gives a counterexample showing that the greedy method has no constant-factor approximation guarantee, and reports experiments on sea surface temperature and flow-past-a-cylinder benchmarks.","tokens_in":22235,"tokens_out":13181,"duration_ms":119322,"significance":"If the claims hold, the paper makes a useful contribution to an underexplored area: efficient multifidelity sensor selection under a budget. The Sherman–Morrison updating scheme in Section 3.4 is correctly derived and reduces the per-iteration cost from O(kMℓ^2(k+ℓ)) to O(kMℓ), a genuine practical improvement. The extension of monotonicity/submodularity to the multifidelity setting (Proposition 3.2) is correct, and the explicit counterexample in Lemma 3.4 is a valuable negative result. However, the experimental validation is conducted entirely inside the truncated-reduced model, the theoretical corroboration in Appendix B contains an invalid inference, and the iterative algorithm has feasibility gaps in its pruning or termination logic. These issues are fixable but currently affect load-bearing claims.","major_comments":[{"comment":"The observation model (2) explicitly ignores truncation error from the SVD basis. All D-optimality comparisons and all reported reconstruction errors are therefore computed inside the reduced model. The relative errors in Table 2 (0.1038 for SST, 0.0864 for cylinder) are of the same magnitude as the ~10% relative-error floor expected from retaining 99% of the singular-value energy. As a result, the experiments do not establish that the selected sensors improve state estimation for the true system; they only demonstrate improvement of the reduced-model objective. Section 5.2.3 compares greedy and iterative against random designs only in Φ_D, not in true-state reconstruction error. Please quantify the truncation error for the two datasets, report true-state reconstruction comparisons against random designs (and, if feasible, against existing multifidelity methods such as [6,19]), and tempe","section":"§2.1, Table 2, §5.2.3"},{"comment":"The derivation of the critical cost/noise ratio is invalid. The text claims that the displayed inequalities imply a_ch^T B^{-1} a_ch = a_exp^T B^{-1} a_exp. From log(1+σ_ch^{-2} a_ch^T B^{-1} a_ch) ≤ log(1+σ_exp^{-2} a_exp^T B^{-1} a_exp) one can only conclude σ_ch^{-2} a_ch^T B^{-1} a_ch ≤ σ_exp^{-2} a_exp^T B^{-1} a_exp, not equality of the quadratic forms. A concrete counterexample is B=I, σ_ch=1, σ_exp=0.1, a_ch^T a_ch=100, a_exp^T a_exp=1, which satisfies both displayed inequalities while the quadratic forms are 100 and 1. Thus the subsequent Taylor approximation and the claimed theoretical corroboration of the empirical regimes are not supported. Either correct the derivation or remove the claim in Section 1.1 bullet 5 that the experiments are 'theoretically corroborated.'","section":"Appendix B"},{"comment":"The main loops in Algorithms 1 and 2 terminate only when b_curr < c_ch; they do not check whether any candidate location remains. In the proof of Lemma 3.4, M=1, and after the greedy step selects the only location, b_curr = 1 − ε/2 ≥ c_ch = ε/2 for ε ∈ (0,1), so the next iteration would attempt argmax over J × ∅. This makes the counterexample formally dependent on undefined behavior. The pseudocode should include the condition I ≠ ∅ (or an equivalent break) in the while loop.","section":"Algorithms 1 and 2, Lemma 3.4"},{"comment":"The pruning step sets k_ch = floor((b − c_exp k_exp)/c_ch) but does not enforce k_ch ≤ M − k_exp. When k_ch + k_exp exceeds the number of candidate locations, the allocation is infeasible. Moreover, the dominance argument based on monotonicity assumes that an additional sensor can always be placed; if k_ch + k_exp = M, that assumption fails, so smaller-k_ch allocations cannot be discarded by this argument. Please cap k_ch by M − k_exp and revisit the pruning claims; otherwise the iterative algorithm may return infeasible sensor configurations.","section":"§4.1, Algorithm 4, Phase I"}],"minor_comments":[{"comment":"Typographical/numbering inconsistencies: 'Theorem 3.4' should be 'Lemma 3.4'; 'Theorem 2.3' and 'Theorem 2.4' should be 'Lemma 2.3' and 'Corollary 2.4'; Section 4.1 and Algorithm 4 refer to 'Theorems 3.2 and 3.3' though these are propositions.","section":"§3.5, §3.2"},{"comment":"Case (3d): 1/3.85 ≈ 0.2597, which is slightly above the critical value 0.25, not 'slightly below' as stated in the table caption. The text should also clarify the meaning of the symbol ⪅ in Figure 3(d).","section":"Table 3, §5.2.1"},{"comment":"The comparison against random designs reports only point values of Φ_D in Table 4. Please report the spread of the random-design histograms (e.g., interquartile range or standard deviation) so that 'considerably better' is substantiated quantitatively.","section":"§5.2.3, Figure 5"},{"comment":"The prior-scaling parameter λ is set to 0.01 without any sensitivity study. Since the D-optimality objective and the selected sensors can depend on the prior scale, a brief sensitivity check or a statement of how λ was chosen would strengthen the empirical claims.","section":"§2.1, Eq. (3)"},{"comment":"The termination condition in lines 14 and 19 compares the new objective with the previous one and breaks on decrease. Since equal objective values also indicate no progress, the condition could be '≤' rather than '<' for consistency with the stated 'objective function decreases' criterion.","section":"Algorithm 4"}],"recommendation":"major_revision","confidential_remarks":"The core algorithmic contributions and the matrix-algebra derivations are sound and worth publishing after revision. The main risks are the unquantified reduced-model truncation error in the experiments, the invalid inference in Appendix B, and the feasibility/termination gaps in the pseudocode. None of these appear to be fatal, but they are load-bearing enough that I cannot recommend acceptance in the current form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Thanks for the report. Here's my read.\n\nThe paper's real contribution is algorithmic: a cost-normalized greedy for multifidelity D-optimal sensor placement with a Sherman-Morrison update that makes each marginal-gain evaluation O(ℓ) instead of O(ℓ^3), and an iterative fidelity-alternating scheme that is, to my knowledge, new. The submodularity/monotonicity proof for the two-fidelity objective is correct, the expensive-sensor-dominance lemma is right, and the counterexample showing no constant-factor guarantee for the greedy is valid in spirit, though the pseudocode has a minor edge-case bug when the candidate location set empties (Algorithm 1/2 would attempt argmax over ∅). The complexity analysis is honest.\n\nThe soft spot is the reduced-order dictionary model. The paper acknowledges it in one sentence: \"we have ignored the error coming from the truncated SVD.\" That error is not negligible. Retaining 99% singular-value energy leaves about 10% relative ℓ2 error. The reported reconstruction errors in Table 2 are 0.104 and 0.086. Those numbers are at the truncation floor. So the reconstructions do not demonstrate that sensor selection matters; a random set of sensors would likely give roughly the same error. Experiment 3 compares greedy and iterative against random designs using Φ_D computed entirely in the reduced space. It never measures reconstruction error against the true state. So the abstract's claim that the methods \"empirically demonstrate improved performance over random designs\" is only true for the surrogate objective, not for the Bayesian state estimation problem the paper is ostensibly about.\n\nThere are other, smaller issues: no code is shipped, no error bars or seeds are reported, and the baseline is only random — not the existing multi-fidelity methods (Clark et al., CELF, or convex relaxation). The Phase-I pruning is clever but the cost table in the paper shows only the number of allocations, not the actual runtime advantage.\n\nNone of this breaks the algorithmic core. The iterative method is plausible and the theory is sound. But the empirical section needs real work: include a truncation-error term or test with a basis that captures more energy, compare reconstruction error to random baselines, and benchmark against the existing literature. As it stands, the practical claims outrun the evidence.\n\nThis is a paper a serious editor should send to review. The algorithms are novel enough to merit referee time, and the theoretical portion is solid. But I'd expect major revisions, mainly on the experiments. I'd cite it only after those are addressed.","headline":"Genuinely novel iterative multifidelity greedy and a useful Sherman-Morrison implementation, with clean theory; but the experiments are undercut by the truncated-SVD error floor and reduced-space-only evaluation.","tokens_in":22717,"tokens_out":3973,"would_cite":false,"duration_ms":37151,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62K05","93E10","65F30","49N45"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that a new iterative refinement algorithm for budget-constrained multifidelity sensor placement matches or beats a Sherman–Morrison-accelerated greedy algorithm on D-optimality benchmarks, with both clearly outperforming ra","keywords":["Bayesian state estimation","sensor placement","D-optimality","multifidelity","greedy algorithms","optimal experimental design","column subset selection","Sherman–Morrison update"],"falsifier":"Compute the exact D-optimal solution by exhaustive search on a small problem (M ≤ 20 candidate locations, two fidelities) and show either that the iterative algorithm's final design has lower Φ_D than the greedy design, or run the iterative algorithm on a problem where the true state is a localized traveling pulse absent from the training snapshots and show the chosen sensors yield higher average reconstruction error than randomly placed sensors.","tokens_in":21832,"feed_emoji":"📡","tokens_out":3849,"duration_ms":35901,"temperature":0.7,"pith_summary":"This paper tackles the problem of where to place sensors of different quality—cheap, noisy ones and expensive, accurate ones—when there is a fixed budget, with the goal of minimizing posterior uncertainty in a Bayesian state estimate. The authors formulate the problem as a budget-constrained multifidelity D-optimal experimental design and link it to column subset selection. They contribute a greedy algorithm accelerated by Sherman–Morrison rank-one updates and a new iterative algorithm that alternately refines the positions of each sensor fidelity for every feasible budget allocation. On sea surface temperature and flow-past-a-cylinder benchmarks, the iterative algorithm matches or beats greedy in D-optimality, and both clearly beat random sensor placement while keeping reconstruction errors low. The practical point is that a budget can be spent on a mix of fidelities in a principled way that scales to large candidate sets.","feed_headline":"Iterative sensor placement beats greedy under a budget","feed_subtitle":"Two fast algorithms pick the best mix of cheap and precise sensors; the iterative one wins in D-optimality tests.","key_machinery":"The D-optimality objective written as a log-determinant of a rank-one sum, B(S)=I+Σ_j (A_j S_j)(A_j S_j)^T; the marginal gain of adding a sensor collapses to log(1 + [A_j]_:^i B^{-1} [A_j]_:i), so Sherman–Morrison updates maintain B^{-1}A_j dictionaries at O(Mℓ) per step. This converts the combinatorial search into an efficient greedy/iterative sweep. The allocation pruning rests on monotonicity and the 'expensive-sensors-are-more-informative' proposition.","core_discovery":"Using the Bayesian D-optimality criterion, the paper expresses the objective as Φ_D(S)=log det(I + Σ_j (A_j S_j)(A_j S_j)^T), where A_j = σ_j^{-1} Σ_pr^{1/2} Ψ^T are scaled restrictions of the reduced-order basis to candidate locations. Because adding a sensor is a rank-one update, the matrix determinant lemma and the Sherman–Morrison formula let the marginal information gain be computed in O(ℓ) flops, giving a greedy algorithm with total O(kMℓ) complexity. The paper proves monotonicity and submodularity of the multifidelity objective, shows expensive sensors are always at least as informative as cheap ones at the same location, and uses these properties to prune the set of budget-feasible (","pith_inferences":["The paper's critical-ratio condition suggests a testable design rule: when c_ch/c_exp < σ²_exp/σ²_ch, cheap sensors dominate; a practitioner could validate this against exhaustive search on small instances.","If the POD truncation error is significant (e.g., localized features absent from training data), the D-optimal designs for the reduced model may not improve true-state reconstruction; the paper flags but does not test this.","The alternating greedy refinement is coordinate ascent on the combinatorial selection space and may converge to different local optima depending on initialization; starting from the greedy solution instead of all-expensive might change outcomes.","The approach generalizes to more than two fidelities by extending the allocation pruning; the complexity bound would scale with the number of fidelities."],"forward_implications":["If the empirical ordering holds, the iterative algorithm is a drop-in replacement for greedy multifidelity selection, giving better D-optimality at comparable cost.","The O(kMℓ) complexity makes budgeted sensor placement feasible for large spatial grids (tens of thousands of candidate locations), as demonstrated on a 44,219-point SST grid.","The cost-to-noise ratio c_ch/c_exp compared to σ²_exp/σ²_ch gives a practical rule for which fidelity a greedy search will favor.","Because no constant-factor guarantee exists, users should run both algorithms and choose the best, or use the iterative output as a warm start.","The same rank-one update machinery carries over to other budgeted submodular objectives with a log-det structure, such as information-theoretic sensor placement."],"fun_headline_variants":["Iterative sensor placement outdoes greedy under budget","Multifidelity sensor placement: iterative beats greedy","Budget-capped sensor pick improved by iterative loop","New iterative method outperforms greedy for sensor budgets","Faster sensor placement with rank-one updates"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The true state is assumed to be well approximated by the leading POD modes of the training data, u ≈ Φm, and the truncation error from the SVD is ignored when computing the D-optimality objective.","fun_headline_variants_meta":{"raw":{"variants":["Iterative sensor placement outdoes greedy under budget","Multifidelity sensor placement: iterative beats greedy","Budget-capped sensor pick improved by iterative loop","New iterative method outperforms greedy for sensor budgets","Faster sensor placement with rank-one updates"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000147,"raw_usage":{"total_tokens":1003,"prompt_tokens":702,"completion_tokens":301,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":446,"completion_tokens_details":{"reasoning_tokens":231}},"tokens_in":446,"tokens_out":301,"duration_ms":3472,"temperature":1.0,"reasoning_tokens":231,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T03:38:12.869451+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the exact D-optimal solution by exhaustive search on a small problem (M ≤ 20 candidate locations, two fidelities) and show either that the iterative algorithm's final design has lower Φ_D than the greedy design, or run the iterative algorithm on a problem where the true state is a localized traveling pulse absent from the training snapshots and show the chosen sensors yield higher average reconstruction error than randomly placed sensors.","supporting_citations":[],"review_version":1}