{"id":"c0fa9950-a1af-46d0-9fbe-6622382783f0","arxiv_id":"2501.09933","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A selective-inference method, SI-SeqFS-DA, computes valid p-values for sequential feature selection after optimal-transport domain adaptation, with false positive rate controlled at the nominal level.","lead":"Sequential feature selection after optimal-transport domain adaptation can return spurious features, and ordinary p-values ignore the search over features. This paper constructs selective-inference p-values that control the false positive rate at the target level and reports higher power than data splitting and Bonferroni correction.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Algorithm 2 may not compute the exact truncation region Z when Zu or Zv is a union of intervals, so the p-value in Eq. (16) is not exactly uniform and Lemma 1's exact FPR control is unproven.","rationale":"The reader's weakest assumption was known covariance; that is a practical limitation, but it does not challenge the proof under the paper's stated assumptions. The deeper risk is internal: exactness of the truncation region is a premise of Lemma 1, and Algorithm 2 is not shown to deliver it in general. Lemmas 3 and 4 produce quadratic inequalities whose solution sets need not be intervals, yet Algorithm 2 and the accompanying footnote assume or force interval structure. If Z is mis-identified, the p-value construction is not exact. This is a correctness risk, not a mere consensus disagreement, and it can be settled by the proposed brute-force comparison. Because the issue is potentially fixable by computing unions of intervals exactly, the conditional verdict remains appropriate; the paper should be revised to either prove single-interval structure in the settings considered or replace Algorithm 2 with an exact union computation. I therefore keep the reader's CONDITIONAL verdict.","tokens_in":22483,"tokens_out":12226,"duration_ms":124411,"concrete_test":"Construct a small instance (p=3, ns=20, nt=10, known Σ, e.g., Σ=I) and choose X, a, b so that for some i the reduced-cost quadratic p_i+q_i z+f_i z^2 with f_i>0 is positive on two disjoint intervals, making Zu a union. For z on a fine grid over [-20σ,20σ], compute the true set Z_true={z: M(a+bz)=Mobs} by brute-force SeqFS-DA on the line. Run Algorithm 2 from the released code and compare its Z and the Eq. (16) p-value to the brute-force versions. If Z differs by more than 1e-6 in measure or the p-value differs by more than 1e-3, the exactness premise of Lemma 1 is violated; repeat over 100 random seeds to confirm the discrepancy is systematic.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 1's proof (Appendix 7.1) requires the truncation region Z in Eq. (15) to be identified exactly. Algorithm 2 is the only mechanism for computing Z, and §3.3.3 explicitly assumes that each of Zu and Zv consists of a single interval, adding that if they are unions, one should select the interval containing the current data point. Lemmas 3 and 4 characterize Zu and Zv by systems of quadratic inequalities in z (p+qz+fz^2 ≥ 0 and w+rz+oz^2 ≤ 0). The superlevel set of a convex quadratic with f>0 is a union of two rays, so these sets are not guaranteed to be intervals; the algorithm's single-interval handling can therefore miss parts of the true conditional region. If Z is replaced by a strict subset or a different set, the conditional law of η_j^T (Y_s, Y_t) given {M=Mobs, Q=Qobs} is a truncated normal over the wrong truncation set, so the p-value in Eq. (16) is not exactly Uniform(0,1) and P(p≤α)=α fails. The finite search range [zmin,zmax]=[-20σ,20σ] introduces an additional approximation. This is an internal gap in the proof of the central claim, independent of the known-covariance assumption.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a selective-inference method, SI-SeqFS-DA, for testing features selected by sequential feature selection (SeqFS) after optimal-transport-based domain adaptation (OT-DA). The method conditions on the selection event (the selected feature set and its order) and on the OT transportation event, parameterizes the data along a one-dimensional line defined by the test statistic and its nuisance component, and characterizes the resulting truncation region Z via systems of quadratic inequalities. The selective p-value is then computed from the truncated normal distribution of the test statistic over Z. The authors claim exact false-positive-rate control at any level alpha (Lemma 1) and provide extensions to backward SeqFS and to model-selection criteria (AIC, BIC, adjusted R2). Experiments on synthetic and real-world data compare the method with data splitting, Bonferroni correction, a na\"ive p-value, and an over-conditioning baseline, and report FPR control and improved power for the proposed method.","tokens_in":22770,"tokens_out":7959,"duration_ms":74883,"significance":"If the central validity claim holds, this would be a useful and nontrivial contribution: it is the first SI framework specifically for SeqFS under OT-based DA, an increasingly common pipeline in transfer learning. The paper provides proof sketches for the lemmas, a reproducible implementation (GitHub link), and extensive experiments including high-dimensional settings. The novelty relative to the authors' prior work (Loi et al. 2024, Duy et al. 2024) is a genuine extension from convex selection problems (Lasso) or anomaly detection to the non-convex, greedy SeqFS procedure, with the new selection-event characterization in Lemmas 3-5 being the main technical content. However, the exact-FPR claim depends on an exact identification of the truncation region Z, and the implementation contains an acknowledged single-interval approximation that is not justified. The known-covariance assumption and the circular definition of the model-selection events further temper the strength of the claims. Overall, the paper is worth pursuing, but the load-bearing gaps need to be resolved or explicitly downgraded to approximate control.","major_comments":[{"comment":"The implementation of the truncation region is not exact. Lemma 3 characterizes Zu as {p+qz+fz^2 >= 0}, which for f>0 is the union of two rays (an outer interval), and Lemma 4 characterizes Zv as {w+rz+oz^2 <= 0}, which for o<0 is also a union of two rays. Footnote 2 says that when Zu or Zv is a union of intervals, the algorithm selects the interval containing the observed data point. Consequently, Algorithm 2 can return a set Z' that is a strict subset of the true region Z in Eq. (15). The p-value in Eq. (16) is then a truncated normal tail probability over the wrong truncation set, so it is not exactly Uniform(0,1) and Lemma 1's equality P(p<=α)=α fails. The finite search range [zmin,zmax]=[-20σ,20σ] adds a further approximation even if the outside mass is small. The authors need to either prove that the omitted intervals have negligible effect under the conditional distribution, implement exact union-of-interval arithmetic, or explicitly state that the method targets approximate rather than exact FPR control.","section":"§3.3.3, Eq. (15)-(16), Lemma 3 and Lemma 4, Algorithm 2"},{"comment":"The exact FPR control is stated for known covariance matrices Σs and Σt. In Section 2 the text says these are 'known or estimable from independent data,' but the derivation of the line direction b (Eq. 12) and the variance ηj^T Σ ηj both treat Σ as fixed and known. If Σ is estimated from finite independent data, then b and the variance are themselves random, the truncated normal calibration is only approximate, and the claimed equality in Lemma 1 does not hold exactly. The paper provides no sensitivity analysis, no correction term, and no experimental study varying the accuracy of the covariance estimate. This is a load-bearing limitation because the exact control claim depends on it; at minimum the manuscript should clearly state that exactness requires a known Σ, and should report simulations under covariance misspecification to indicate the practical impact.","section":"§2 and Lemma 1 (Appendix 7.1)"},{"comment":"The characterizations of the model-selection events appear to be circular. The text says 'Let us denote by M_hatK_AIC is the optimal model selected by AIC,' but M_hatK_AIC is not defined independently of z; if it is the minimizer of AIC at Y_tilde(z), then the displayed inequality AIC(M_hatK_AIC, Y_tilde(z)) <= AIC(M_K, Y_tilde(z)) for any K is tautologically true for all z, making Z_AIC_u the entire real line and imposing no restriction. If instead M_hatK_AIC is intended to be the fixed optimal model at the observed data, that must be stated, and the equivalence between the event {argmin_K AIC = M_obs} and the quadratic-inequality set needs to be proved. As written, the validity of the AIC, BIC, and adjusted-R2 extensions is not established.","section":"§4.2, definitions of Z_AIC_u, Z_BIC_u, Z_adj_u"}],"minor_comments":[{"comment":"The references Duy et al. (2022a) and Duy et al. (2022b) are identical entries; one of them appears to be a different paper and must be corrected and cited distinctly.","section":"References"},{"comment":"The phrase 'the the cost matrix' contains a duplicated article; it should read 'the cost matrix.'","section":"§2.1"},{"comment":"The x-axis of the high-dimensional FPR plot is labeled '# total features - p' but the stated experimental settings are p ∈ {1000, 1500, 2000, 2500}, which does not match the axis ticks (500, 1000, 1500, 2000); please clarify the axis values.","section":"§5.1, Fig. 12a"},{"comment":"The search range [zmin, zmax] is set to [-20σ, 20σ] but σ is not defined there; it should be stated as the standard deviation of the test statistic, σ = sqrt(ηj^T Σ ηj).","section":"§3.3.3"}],"recommendation":"major_revision","confidential_remarks":"The paper is closely related to the authors' own prior work (Loi et al. 2024; Duy et al. 2024), and the incremental novelty is the adaptation of the OT-event characterization to SeqFS. That is a legitimate contribution, but the relationship should be made very explicit. The main technical gap (truncation region approximation) is fixable in principle, but without a fix the central exact-control claim is not supported. I would also ask the editor to verify the duplicated reference entry and the definition of the model-selection events before accepting any version of the manuscript."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper delivers a valid selective-inference method for sequential feature selection after OT-based domain adaptation. The core is standard Lee et al. machinery applied carefully to a new pipeline, and it holds up under the stated Gaussian, known-covariance assumptions. I read the stress-test about Algorithm 2 not computing the full truncation region; it does not break validity. Over-conditioning on an interval containing the observed z is a standard valid SI move, so the p-value remains uniform conditional on that interval, and marginalizing over the selection event preserves FPR control. The real soft spots are covariance estimation and the finite search range, plus some experimental reporting.\n\nWhat is new and what is done well: the paper characterizes the SeqFS selection event after OT-DA as systems of quadratic inequalities in the scalar parameter z, splits the problem via divide-and-conquer, and extends to backward SeqFS and model-selection criteria such as AIC, BIC, and adjusted R2. This is the first FPR-controlling method for SeqFS-DA, as far as I know. The proof of Lemma 1 follows the usual truncation argument, and the code is available. Experiments span synthetic and real data, including high-dimensional settings, and the results show the proposed method controls FPR while beating baselines on TPR. That is solid, reproducible work.\n\nSoft spots, in proportion. First, the union-of-intervals issue is not a validity gap, but it does undercut the power claim. If Zu or Zv is a union, selecting the interval containing the observed z is valid but conservative; the reported TPR numbers may not reflect the best achievable power. The authors should either prove that their divide-and-conquer recovers the full region in the cases they run, or state clearly that these are over-conditioning results. Second, the known-covariance assumption is load-bearing: the line direction b and the truncation region both depend on Sigma = blockdiag(Sigma_s, Sigma_t). If Sigma is estimated, exact FPR control fails, and there is no sensitivity analysis. This is common in SI but deserves explicit discussion. Third, the finite search range [-20 sigma, 20 sigma] makes the p-value approximate; the paper claims exact control, so this should be acknowledged. Fourth, the figures lack error bars, and the high-dimensional FPR plot shows only the proposed method, not baselines. With 120 repetitions, error bars are easy to add.\n\nThe citation pattern is fine. The authors lean on their own prior OT-event characterizations, but those are the right references, and the new SeqFS event derivation is self-contained.\n\nBottom line: this is a useful, incremental contribution for researchers who need inference after a DA pipeline. It is not a field reorganizer, but it is coherent and serious. I would send it to a referee who knows selective inference and ask them to scrutinize the region-finding algorithm's power behavior and the covariance assumption.\n\nRecommendation: accept for peer review, expect revision.","headline":"Valid selective p-values for SeqFS after OT-based DA; the known-covariance assumption and finite search range are the real caveats, not the interval/union concern.","tokens_in":23302,"tokens_out":6609,"would_cite":true,"duration_ms":75412,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62F03","62J05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper shows that features selected by sequential feature selection after optimal-transport domain adaptation can be tested with p-values that control the false positive rate exactly.","keywords":["sequential feature selection","domain adaptation","optimal transport","selective inference","statistical hypothesis testing","p-value","false positive rate","truncated normal distribution"],"falsifier":"Simulate the paper's setup under Gaussian noise with a deliberately misspecified covariance matrix, such as a plug-in estimate with 20% error in operator norm, run the proposed p-value at α = 0.05 over 10,000 replicates, and check whether the empirical false-positive rate stays at 0.05; any systematic deviation would show the known-covariance premise is load-bearing.","tokens_in":22249,"feed_emoji":"🎯","tokens_out":11289,"duration_ms":107108,"temperature":0.7,"pith_summary":"When target data are scarce, a common workaround is domain adaptation: align a related source dataset to the target using optimal transport, a distribution-alignment technique, and then run sequential feature selection on the combined data. The paper shows that the seemingly natural p-values from this pipeline are invalid, because they ignore both the adaptation and the fact that features were chosen by looking at the data. Its proposed method, SI-SeqFS-DA, computes a selective p-value for each chosen feature by conditioning on the entire selection event. The central theorem states that, under Gaussian responses with known covariance, this p-value has exactly a uniform distribution under the null hypothesis, so the false positive rate equals any chosen significance level. That gives practitioners a way to trust feature-selection decisions in limited-data, cross-domain regression.","feed_headline":"Selective p-values control false positives in domain adaptation","feed_subtitle":"A new construction keeps the false-positive rate at its nominal level even though features are chosen adaptively.","key_machinery":"The central object is the truncation region Z = {z ∈ R : M_{a+bz} = M_obs} on the one-dimensional line a+bz, where a is the observed nuisance component and b = Ση_j / (η_j^T Σ η_j) is the covariance-weighted direction of the test statistic. Conditioning on Z turns the test statistic into a truncated normal, so computing Z is the whole problem. The paper's divide-and-conquer decomposition splits Z into sub-problems indexed by a fixed optimal transport transportation T_u and a fixed ordered selection path O_v; both sub-regions are shown to be solutions of systems of quadratic inequalities in z, so their intersections are intervals and Algorithm 2 stitches the intervals together to form Z. This converts an intractable high-dimensional conditioning event into a calculation over finitely many one-dimensional intervals.","core_discovery":"The paper's central claim is that valid inference for sequential feature selection after optimal-transport-based domain adaptation is possible by conditioning on the event that produced the selected feature set. Concretely, it takes the least-squares coefficient of a selected feature as the test statistic, writes it as a linear contrast, and conditions on the nuisance component so the data are restricted to a one-dimensional line Y(z)=a+bz. On that line, the condition that SeqFS after domain adaptation selects the observed set M_obs becomes a union of intervals, called the truncation region Z. The paper proves each piece of Z is described by quadratic inequalities: one set for the optimal transport transportation matrix and one for the ordered sequence of feature sets chosen by SeqFS. A divide-and-conquer algorithm collects the intervals whose final set matches M_obs, and the selective p-value is the tail probability of a truncated normal distribution over Z. Lemma 1 states this p-value is valid, P_{H0,j}(p ≤ α) = α for all α, which is the exact false-positive-rate control the paper claims in its experiments and extensions.","pith_inferences":["The known-covariance assumption is the main practical gap: in real applications the covariance is usually estimated, and a misspecification study would show how much false-positive inflation occurs when plug-in estimates are used instead of the true matrices.","The divide-and-conquer interval construction is not tied to optimal transport in principle; any domain-adaptation procedure whose selection event reduces to polynomial inequalities in z could inherit the same p-value machinery, and a sampling-based approximation could extend it to deep-learning-based adaptation, as the paper itself notes as a possible route.","The same template should generalize to other greedy feature-selection wrappers after domain adaptation, such as forward stagewise or orthogonal matching pursuit, because their selection events also reduce to quadratic inequalities in z."],"forward_implications":["At any significance level α, the method keeps the false positive rate at exactly α for forward SeqFS after optimal-transport-based domain adaptation, so irrelevant features are rejected at the nominal rate rather than an inflated one.","Using the full truncation region Z instead of a single sub-problem interval raises the true positive rate (lowers the false negative rate) while preserving false-positive control, which the experiments show against data splitting and Bonferroni correction.","The same construction applies to backward elimination and to settings where the number of selected features is chosen by AIC, BIC, or adjusted R-squared, with the same exact-control guarantee.","Because each sub-problem is solved by closed-form quadratic inequalities, the runtime grows linearly in the source sample size, and the experiments demonstrate the method on datasets with thousands of features."],"supporting_citations":[{"why":"Establishes the selective-inference recipe: condition on the selection event, derive the truncated-normal law of the test statistic, and use its tail probability as a valid p-value.","marker":"Lee et al., 2016"},{"why":"Defines the optimal-transport-based domain adaptation whose transported source data enter SeqFS in this paper's setup.","marker":"Flamary et al. 2016"},{"why":"Contributes the divide-and-conquer scheme and the quadratic-inequality characterization of the OT transportation event that Lemma 3 adapts.","marker":"Duy et al. 2024"},{"why":"Gives the closest prior selective-inference method for feature selection after OT-based DA for Lasso; its KKT-based argument does not apply to SeqFS, which motivates the new construction.","marker":"Loi et al. 2024"},{"why":"Provides exact post-selection inference for sequential regression procedures, the foundation for the SeqFS truncation-region derivation.","marker":"Tibshirani et al. 2016"},{"why":"Shows that using the full selection region instead of over-conditioning raises power in stepwise feature selection, the rationale for the union over sub-problems.","marker":"Sugiyama et al. 2021a"}],"fun_headline_variants":["Selective p-values tame false positives in domain adaptation","Valid inference for feature selection after domain adaptation","Truncated normal p-values ensure FPR control after DA","Exact FPR control for post-adaptation feature selection"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes the noise in each domain is Gaussian with a known spread; if the spread is estimated from data instead, the exact false-positive guarantee is not established.","fun_headline_variants_meta":{"raw":{"variants":["Selective p-values tame false positives in domain adaptation","Valid inference for feature selection after domain adaptation","Truncated normal p-values ensure FPR control after DA","Exact FPR control for post-adaptation feature selection"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00017,"raw_usage":{"total_tokens":1263,"prompt_tokens":937,"completion_tokens":326,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":553,"completion_tokens_details":{"reasoning_tokens":261}},"tokens_in":553,"tokens_out":326,"duration_ms":4128,"temperature":1.0,"reasoning_tokens":261,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T19:32:42.914304+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the paper's setup under Gaussian noise with a deliberately misspecified covariance matrix, such as a plug-in estimate with 20% error in operator norm, run the proposed p-value at α = 0.05 over 10,000 replicates, and check whether the empirical false-positive rate stays at 0.05; any systematic deviation would show the known-covariance premise is load-bearing.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the optimal-transport-based domain adaptation whose transported source data enter SeqFS in this paper's setup."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Contributes the divide-and-conquer scheme and the quadratic-inequality characterization of the OT transportation event that Lemma 3 adapts."},{"cited_title":"Statistical Inference for Feature Selection after Optimal Transport-based Domain Adaptation","cited_arxiv_id":"2410.15022","evidence_quote":"Gives the closest prior selective-inference method for feature selection after OT-based DA for Lasso; its KKT-based argument does not apply to SeqFS, which motivates the new construction."}],"review_version":1}