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Statistical Inference for Sequential Feature Selection after Domain Adaptation

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2501.09933 v1 pith:3DNC7WRW submitted 2025-01-17 stat.ML cs.LG

Duong Tan Loc , Nguyen Thang Loi , Vo Nguyen Le Duy This is my paper · ORCID
classification stat.MLcs.LG MSC 62F0362J05
keywords sequentialfeatureselectiondomainadaptationoptimaltransportselectiveinferencestatisticalhypothesistestingp-valuefalsepositiveratetruncatednormaldistribution
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Formalized claims in Lean

  1. Claim #1: 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 res

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [§3.3.3, Eq. (15)-(16), Lemma 3 and Lemma 4, Algorithm 2] 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.
  2. [§2 and Lemma 1 (Appendix 7.1)] 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.
  3. [§4.2, definitions of Z_AIC_u, Z_BIC_u, Z_adj_u] 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.
minor comments (4)
  1. [References] 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.
  2. [§2.1] The phrase 'the the cost matrix' contains a duplicated article; it should read 'the cost matrix.'
  3. [§5.1, Fig. 12a] 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.
  4. [§3.3.3] 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).

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the central Lemma 1 is the standard selective-inference probability-integral-transform argument, and the new truncation-region lemmas are derived in the appendix rather than assumed. A minor self-citation in Lemma 3 is not load-bearing because the proof is reproduced, though an exactness gap in Algorithm 2 is a correctness concern, not a circularity.

full rationale

The central validity claim, Lemma 1, is not circular. The selective p-value in Eq. (11) is defined as the tail probability of the test statistic under the law conditional on the selection event and the nuisance component; Appendix 7.1 shows that this law is truncated normal on the region Z and that uniformity follows from the probability integral transform. This is exactly the standard SI argument of Lee et al. (2016), and it does not assume the conclusion it proves. The genuinely novel part, the characterization of the truncation region Z for SeqFS-DA, is established in Lemmas 3-5 with derivations in Appendices 7.3-7.5. Lemma 3 begins with 'Building on the results from Duy et al. (2024)...', which is a self-citation, but the appendix then gives the full parametric-LP derivation of the quadratic inequalities, so the argument does not reduce to an unverified citation; the prior result is also peer-reviewed and parameter-free with stated assumptions. The only substantive concern is an exactness gap, not a circularity: Section 3.3.3 states 'For simplicity, we focus on the case where each of Zu and Zv consists of a single interval', and footnote 2 adds 'If Zu or Zv is a union of intervals, we can select the interval containing the data point that we are currently considering'. Lemma 1 requires the exact region Z of Eq. (15), and if Algorithm 2 returns only a sub-interval, the implemented p-value may not be exactly the p-value analyzed in Lemma 1. This is a proof-of-exactness risk for the claimed FPR control, but it is not a case of a prediction reducing to a fitted input or to a self-citation chain. Experimental comparisons use external baselines (Naive, Bonferroni, data splitting) and no fitted parameter is renamed as a prediction. Overall circularity burden is low.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The method introduces no new physical or model entities; its assumptions are the Gaussian linear model, known covariances, a unique OT plan, and tie-free SeqFS. The main implementation-dependent parameter is the finite search range used to approximate the truncation region.

free parameters (1)
  • Search range [zmin, zmax] = [-20σ, 20σ]
    Algorithm 2 restricts the line search to this hand-chosen interval; the paper states probability outside is negligible, but the resulting p-value is approximate rather than exactly valid on the full real line.
assumptions (5)
  • domain assumption Ys and Yt are Gaussian with known covariance matrices Σs and Σt
    Stated in Section 2; the truncated-normal null distribution of the test statistic and the line direction b in Eq. (12) both require Gaussianity and known Σ.
  • domain assumption Feature matrices Xs and Xt are non-random and fixed
    Stated in Section 2; all conditioning is on the observed features, and the selection event depends on them.
  • ad hoc to paper The OT optimal transportation matrix is unique and determined by a single optimal basis Bu
    Lemma 3 partitions z-space by optimal basis; degeneracy or multiple optimal transports would make the event {T_{a+bz}=Tu} ill-defined. The paper does not discuss this.
  • ad hoc to paper The SeqFS argmin has no ties
    The selection event in Section 2.2 defines jk = argmin without a tie-breaking rule; ties would make M a set-valued function and break the conditional inference framework.
  • ad hoc to paper Zu and Zv are assumed to be single intervals for the implementation
    Section 3.3.3 and footnote 2 state the implementation focuses on the single-interval case; if they are unions, the algorithm selects only the interval containing the current point, so the computed region may be a subset of Z.

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Pith. "Pith review of Statistical Inference for Sequential Feature Selection after Domain Adaptation." pith.science (2026). https://pith.science/paper/3DNC7WRW

@misc{pith2026250109933,
  author       = {Pith},
  title        = {Pith review of: Statistical Inference for Sequential Feature Selection after Domain Adaptation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3DNC7WRW}},
  note         = {Machine review of arXiv:2501.09933}
}
abstract

In high-dimensional regression, feature selection methods, such as sequential feature selection (SeqFS), are commonly used to identify relevant features. When data is limited, domain adaptation (DA) becomes crucial for transferring knowledge from a related source domain to a target domain, improving generalization performance. Although SeqFS after DA is an important task in machine learning, none of the existing methods can guarantee the reliability of its results. In this paper, we propose a novel method for testing the features selected by SeqFS-DA. The main advantage of the proposed method is its capability to control the false positive rate (FPR) below a significance level $\alpha$ (e.g., 0.05). Additionally, a strategic approach is introduced to enhance the statistical power of the test. Furthermore, we provide extensions of the proposed method to SeqFS with model selection criteria including AIC, BIC, and adjusted R-squared. Extensive experiments are conducted on both synthetic and real-world datasets to validate the theoretical results and demonstrate the proposed method's superior performance.

Figures

Figures reproduced from arXiv: 2501.09933 by the authors.

Figure 1
Figure 1. Illustration of the proposed SI-SeqFS-DA method. When SeqFS-DA is per [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Illustration of the SI-SeqFS-DA method. First, we transform the data using [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. FPR and TPR in the case of Forward SeqFS [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figures from the paper (19 more)
Figure 4
Figure 4. Figure 4: FPR and TPR in the case of Backward SeqFS [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]
Figure 5
Figure 5. Figure 5: FPR and TPR in the case of Forward SeqFS with AIC [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]
Figure 6
Figure 6. Figure 6: FPR and TPR in the case of Backward SeqFS with AIC [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: FPR and TPR in the case of Forward SeqFS with BIC [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: FPR and TPR in the case of Backward SeqFS with BIC [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 9
Figure 9. Figure 9: FPR and TPR in the case of Forward SeqFS with adjusted [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]
Figure 10
Figure 10. Figure 10: FPR and TPR in the case of Backward SeqFS with adjusted [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]
Figure 11
Figure 11. Figure 11: Computational cost of the proposed SI-SeqFS-DA [PITH_FULL_IMAGE:figures/full_fig_p020_11.png]
Figure 12
Figure 12. Figure 12: SI-SeqFS-DA in high-dimensional setting domain comprised data from riboflavin. We set p = 4088 and randomly selected instances from the source and target domains, with ns ∈ {40, 60, 80, 100}. 5.2 Results on Real-World Datasets We conducted comparisons on three real-wo…
Figure 13
Figure 13. Figure 13: Diabetes dataset. The source domain consists of “people over 50 years old”, [PITH_FULL_IMAGE:figures/full_fig_p021_13.png]
Figure 14
Figure 14. Figure 14: Heart Failure dataset. The settings for the source and target domains are [PITH_FULL_IMAGE:figures/full_fig_p021_14.png]
Figure 15
Figure 15. Figure 15: Seoul Bike dataset. The source domain is “people who rent bikes on regular [PITH_FULL_IMAGE:figures/full_fig_p022_15.png]
Figure 16
Figure 16. Figure 16: Backward SeqFS after DA on real-world datasets. [PITH_FULL_IMAGE:figures/full_fig_p030_16.png]
Figure 17
Figure 17. Figure 17: Forward SeqFS after DA with AIC on real-world datasets. [PITH_FULL_IMAGE:figures/full_fig_p031_17.png]
Figure 18
Figure 18. Figure 18: Forward SeqFS after DA with BIC on real-world datasets. [PITH_FULL_IMAGE:figures/full_fig_p032_18.png]
Figure 19
Figure 19. Figure 19: Forward SeqFS after DA with adjusted R2 on real-world datasets [PITH_FULL_IMAGE:figures/full_fig_p033_19.png]
Figure 20
Figure 20. Figure 20: Backward SeqFS after DA with AIC on real-world datasets. [PITH_FULL_IMAGE:figures/full_fig_p034_20.png]
Figure 21
Figure 21. Figure 21: Backward SeqFS after DA with BIC on real-world datasets. [PITH_FULL_IMAGE:figures/full_fig_p035_21.png]
Figure 22
Figure 22. Figure 22: Backward SeqFS after DA with adjusted R2 on real-world datasets [PITH_FULL_IMAGE:figures/full_fig_p036_22.png]

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.