Model-Agnostic FDR Control via Group Gaussian Mirror and Permutation SHAP

Jiaan Han; Junxiao Chen; Yanzhe Fu

REVIEW 4 major objections 3 minor 60 references

Model-Agnostic FDR Control via Group Gaussian Mirror and Permutation SHAP

T0 review · 4 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Block-level Gaussian mirrors and Permutation SHAP derivatives extend FDR control to grouped and sequential features.

desk verdict Linear grouped Gaussian mirror is a real, defensible extension; the neural PSGM FDR control is not established — Theorem 3 assumes the fitted model already ignores the null group. read the letter →

arxiv 2608.00989 v1 pith:WTMUFPP5 submitted 2026-08-02 stat.ML cs.LG

Jiaan Han , Junxiao Chen , Yanzhe Fu This is my paper

classification stat.MLcs.LG MSC 62J0562H1568T07

keywords FDRcontrolgroupedfeatureselectionGaussianMirrorPermutationSHAPsequentialneuralnetworksstatisticshigh-dimensionalinferencetimeseriesmodels

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

The paper claims that false-discovery-rate control can be moved from one-feature-one-weight testing to grouped-feature testing, where one original feature is represented by a block of lags, recurrent states, or attention sub-features. For grouped linear models it constructs block-level mirror variables and proves the resulting block mirror statistics are symmetric under the grouped null, yielding FDR control in both low and high dimensions. For sequential neural networks it uses derivatives of Permutation SHAP values as block-level importance scores, selects the matrix-valued perturbation through a kernel-based conditional-dependence objective, and proves asymptotic symmetry of the smoothed derivative estimator under a fixed fitted nonlinear model. If correct, this gives practitioners a model-agnostic way to select grouped features with controlled FDR in lagged and attention-based models, without specifying the covariate distribution.

What carries the argument

The engine is the block-level mirror construction: for each grouped feature $j$, form mirror copies $X_j^\pm = X_j \pm \widetilde{Z}_j G_j$ and combine their fitted importance vectors through the block mirror statistic $M_j$. In the linear case the perturbation is fixed by residualized block orthogonality, with $G_j = (R_j^\top R_j)^{1/2}$ for the residualized group block $R_j$, making the two mirror coefficient vectors independent under the null. In the neural case the perturbation matrix is chosen by minimizing the kernel conditional-dependence score $I_j^K(G_j)^2$, and the importance vector is $\psi_j(x) = \nabla_{x_j} \Phi_j(x)$, the derivative of the Permutation SHAP value with respect

What would settle it

Generate a grouped dataset with a known null block, train an LSTM, GRU, or Transformer on the mirror-augmented inputs, and compute the PSGM mirror statistic for that null block over many noise realizations and sample sizes. If the empirical distribution of $M_j$ is not symmetric about zero, for example if $\sup_{t>0} |P(M_j > t) - P(M_j < -t)|$ does not go to zero as $n$ grows, the central symmetry claim fails.

Watch

Extended reading notes

Core claim

The central claim is that a grouped null hypothesis $H_{0,j}: \beta_j = 0$ can be tested with a mirror statistic built from two perturbed copies of the whole feature block, $X_j^\pm = X_j \pm \widetilde{Z}_j G_j$, where $\widetilde{Z}_j$ is a normalized Gaussian noise block and $G_j$ an $m\times m$ perturbation matrix. Under a grouped linear model, the two fitted coefficient vectors $\widehat{\beta}_j^+$ and $\widehat{\beta}_j^-$ are independent centered Gaussian vectors with equal covariance under the null, so the block statistic $M_j = \operatorname{sign}(\langle \widehat{\beta}_j^+, \widehat{\beta}_j^-\rangle)(\|\widehat{\beta}_j^+\|_2 + \|\widehat{\beta}_j^-\|_2)$ is symmetric about zero

Load-bearing premise

The neural guarantee assumes the fitted model is null at the model level, meaning $f(x)=g(x_{-j})$ for a null group $j$; for a network trained on the mirror-augmented inputs, the paper does not prove that the fitted network avoids dependence on null blocks, so null symmetry of PSGM statistics is assumed rather than established.

Editorial extensions

If this is right

If the linear results are correct, grouped lag models can be first screened by Group Lasso on one data chunk and then mirrored on a second, yielding asymptotic FDR control for the selected feature blocks.
For LSTM, GRU, and Transformer models, the framework provides feature selection whose FDR is controlled at the target level when the fitted model does not depend on null blocks.
Because the method is model-agnostic and does not require specifying the covariate distribution, it applies to any architecture for which Permutation SHAP derivatives can be computed.
The grouped null answers the scientific question for lagged time series and attention models: whether a feature contributes anywhere across its block, not whether a single sub-weight is zero.
Empirically, the method maintains FDR control and often improves power relative to baselines, particularly when signals are spread across correlated feature blocks.

Reading between the lines

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

Editorial inference: the neural FDR guarantee rests on a model-level null assumption that is not proved for networks trained on the mirror-augmented data; a direct empirical check would be to test sign symmetry of the PSGM statistic on simulated null blocks while varying network depth, regularization, and attention structure.
Editorial inference: the same block-mirror machinery could be combined with other swap-equivariant attribution maps, suggesting the essential role of Permutation SHAP here is to provide a symmetric block-level importance signal rather than a uniquely necessary one.
Editorial inference: the framework works best with sparse support and moderate block size; for dense supports or very long lag windows, the separation between null and non-null mirror statistics may weaken, and future procedures could adapt by pairing the group mirror with group knockoffs or more conservative thresholding.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Read it. The linear half is genuinely useful; the neural half is sold with a theorem that assumes the conclusion.

The new material is the grouped linear extension: block-level mirror variables with matrix-valued perturbations and a block statistic built from the two fitted coefficient vectors. The low-dimensional derivation in Theorem 1 is clean — the block-orthogonality algebra goes through, the symmetry argument is honest, and the GLS whitening in Proposition 2 is a nice addition. The high-dimensional screen-then-mirror version (Theorem 2) is plausible under sure-screening and data-splitting assumptions. The linear experiments look decent: FDR is near the target with full power in most settings, and the method clearly beats Group Lasso as an FDR procedure. The Permutation SHAP derivative is a real modeling idea, and the reduction to the linear coefficient in Proposition 4 is neat. The citation of prior work is fair; this is a natural escalation from Gaussian Mirror and Neural Gaussian Mirror.

The soft spot is load-bearing and it is the neural half. Theorem 3 starts from 'there exists g such that f(x)=g(x_-j)' — that is, it assumes the fitted model ignores the null group. But in the actual algorithm, the network is trained on the augmented mirror inputs, and the whole question is whether training yields a model with that property. The kernel objective is built to make the two mirror branches conditionally independent of each other, not to stop the network from fitting spurious dependence on the null block. So the theorem is circular on the central claim. Theorem 5 still assumes odd/even structure and swap-equivariance instead of deriving them from the training dynamics. The simulations back up the concern: PSGM FDR exceeds the nominal 0.1 in most neural rows of Tables 2–4, reaching 0.283 for one Transformer setting. The paper's own Section 7 admits exactly this, noting null symmetry 'may be weakened if the network learns spurious correlations.' The honest summary is that the model-agnostic neural FDR guarantee is not established.

Minor: the high-dimensional time-series chunking leans on mixing-type asymptotic independence that isn't examined in the experiments, and I don't see code or data shipped.

Who should read this: anyone working on grouped FDR in linear lagged models — the GGM construction is worth their time. The neural part is a useful cautionary example of how to set up these claims. I'd send it to peer review; the linear contribution is solid, and the neural gap is precisely what referees should wrestle with. My own recommendation: use the linear method, don't lean on the neural guarantees.

Referee Report

4 major / 3 minor

Summary. The paper proposes group-level FDR-controlled feature selection for grouped and sequential models. For grouped linear models, it constructs block mirror variables X_j^± = X_j ± eZ_j G_j with matrix-valued perturbations, proves null symmetry of the resulting block mirror statistic (Theorem 1), and gives a two-stage screen-then-mirror high-dimensional extension (Theorem 2). For neural models, it optimizes a kernel conditional-dependence objective to choose G_j, trains a network on augmented mirror inputs, and uses smoothed Permutation SHAP derivatives as block-level importance scores; Theorems 3 and 5 are claimed to give asymptotic symmetry and hence FDR control. Experiments cover grouped linear models, LSTM/GRU/Transformer architectures, and a CausalRivers real-data application.

Significance. If the neural claim were established, the framework would be a useful extension of mirror-statistic FDR control to grouped and sequential deep models, and it would be genuinely model-agnostic. Credit is due for the clean linear construction (Theorem 1), which requires no covariate-distribution specification and yields a parameter-free matrix-valued perturbation, and for Proposition 4, which shows Permutation SHAP derivatives reduce to the linear coefficient vector. The experiments are broad. However, the paper's headline neural FDR guarantee is not proved: the proofs assume the very property that the training procedure must deliver, and the reported simulations show FDR often exceeding the nominal level. The linear part alone is a moderate, well-executed contribution, but the manuscript as a whole does not support the neural claims as stated.

major comments (4)

[Theorem 3 / Appendix C.7 / Section 7] The proof of Theorem 3 begins: 'We interpret the null at the model level: there exists a measurable function g such that f(x)=g(x_-j).' This is an assumption about the fitted model, not a consequence of the data-generating null β_j=0. In Algorithm 3, f is trained on augmented inputs [U_j, V_j, X_-j], and nothing in the training procedure prevents the network from using U_j or V_j through spurious correlations. Thus Theorem 3 does not establish null symmetry for the actual PSGM pipeline. Section 7 concedes this: 'Null symmetry of PSGM mirror statistics may be weakened if the network learns spurious correlations or assigns systematic directional importance to null grouped features.' The empirical FDRs in Tables 2–4, e.g., 0.283 for Transformer strong high-dimensional, are consistent with this gap. This is the central load-bearing issue for the neural FDR claim.
[Theorem 5 / Appendix C.10 / Section 4.3] Theorem 5 assumes eZ_{j,λ} is odd in Z_j, bG_j is even in Z_j, and the downstream importance map is swap-equivariant. Oddness of eZ_{j,λ} follows from linearity in Z_j, but the evenness of bG_j is not derived: bG_j is a minimizer of I^K_j(G_j)^2, and no invariance argument is given for the objective or for the selection rule. More importantly, swap-equivariance of the full pipeline—including training a neural network with stochastic optimization and random initialization—is asserted rather than proven. These are conditions, not consequences of Algorithm 4. Hence the high-dimensional neural symmetry-safe claim is unsupported.
[Theorem 3 / Section 4.1] Even if the model-level null were granted, Theorem 3 proves only marginal asymptotic symmetry of each individual sub-feature derivative estimator. The mirror statistic M_j = sign(<ψ_j^+, ψ_j^->)(||ψ_j^+||+||ψ_j^-||) requires joint distributional symmetry or exchangeability of the plus and minus derivative vectors. Appendix C.7 does not establish joint asymptotic normality or exchangeability of (ψ_j^+, ψ_j^-). Thus null symmetry of M_j does not follow from the stated theorem even under its own assumptions.
[Section 3.2 / end of Theorem 1] The paper claims FDR control for the low-dimensional Group Gaussian Mirror, but no theorem proves that the mirror threshold τ_q controls FDR. Theorem 1 establishes null symmetry of M_j, and the text says 'we use the same mirror thresholding and selection rule as in 2.1 to perform FDR Control,' but no grouped analogue of the Gaussian Mirror FDR lemma is stated or proved. Since the grouped statistics are correlated across j, a formal argument is needed. The low-dimensional linear FDR claim is therefore not fully supported as written.

minor comments (3)

[Theorem 3 / Appendix C.7] The notation in Theorem 3 does not distinguish the plus and minus branch estimators ψ^+ and ψ^-, even though the mirror statistic uses both. This should be made explicit.
[Tables 2–4] No standard errors or confidence intervals are reported. With 50 repetitions, the binomial standard error at FDR 0.1 is about 0.042, so some deviations are interpretable, but reporting uncertainty would help the reader.
[Theorem 3] The regularity conditions are stated only as 'under some regularity conditions.' For a theorem carrying the paper's neural FDR claim, these conditions should be listed precisely.

Circularity Check

2 steps flagged · score 6.0 of 10

The neural FDR guarantee is partially circular: Theorem 3 and Theorem 5 assume the very null-symmetry properties the algorithm must establish, while the linear GGM derivation is genuine.

self definitional [Appendix C.7, proof of Theorem 3]
"Fix a null grouped feature j. We interpret the null at the model level: there exists a measurable function g such that f(x)=g(x_-j),∀x. Hence the fitted model does not depend on any sub-feature in the grouped block j. ... Therefore, we have ∆j(x;π,X′)=0 ... Φj(x)=0 ... bψj,k(x0) is symmetric around zero."

The data-level null is H0,j : β_j = 0 in the generative model, but the PSGM procedure trains a network on augmented mirror inputs U_j, V_j. Null symmetry of the mirror statistic requires that the fitted network be insensitive to the null group. Instead of proving that training on augmented inputs yields such a network, the proof simply declares the null to be the model property f(x)=g(x_-j). Under that assumption the Permutation SHAP marginal contribution is identically zero by definition, so the asymptotic symmetry is a tautology. The load-bearing step — that a null feature is invisible to the fitted model — is assumed rather than derived.
self definitional [Section 4.3 and Appendix C.10, Theorem 5]
"Assume eZj,λ is odd and bGj is even in Zj, if the downstream importance map is swap-equivariant, then the mirror outputs O±j are exchangeable and symmetry-safe, namely, (O+j,O−j)=d(O−j,O+j)."

Exchangeability of the mirror outputs is exactly the property needed for FDR control. The theorem places that property into its hypotheses: oddness of eZj,λ, evenness of bGj, and swap-equivariance of the importance map. In the actual algorithm, bGj is the minimizer of the kernel conditional-dependence objective, and the paper does not prove this minimizer is even in Zj. The statement 'the ridge-type projection preserves all the geometry required' only addresses eZj,λ; it does not establish the remaining hypotheses. The theorem therefore restates the required symmetry conditions as the conclusion rather than certifying that the construction satisfies them.

full rationale

The linear half of the paper is a genuine derivation. Theorem 1 constructs eZj and Gj so that the residualized mirror blocks are block-orthogonal, and then proves E[bβ+j]=E[bβ−j]=0, equal covariances, and zero cross-covariance under the grouped null. This is a parameter-free, self-contained argument against the Gaussian linear model; Theorem 2 extends the symmetry to the two-stage screened version under stated conditions. No circularity appears there. The neural half is where the derivation chain breaks. The central advertised claim is model-agnostic FDR control for LSTM, GRU, and Transformer models via PSGM. Theorem 3's proof begins by redefining the null as 'there exists f(x)=g(x_-j)', which is precisely the model-level independence that the mirror construction must enforce through training but never proves. Under this assumption, the Permutation SHAP marginal contribution is zero for every permutation and background draw, so the smoothed derivative is trivially symmetric. This is a reduction of the desired null symmetry to an equivalent assumption, not a derivation from the data-generating null. Theorem 5 is similarly conditional: exchangeability of the mirror outputs is assumed via odd/even symmetries and swap-equivariance, and the paper only argues informally that the ridge projection is odd, leaving bGj evenness and the trained importance map's swap-equivariance unverified. Section 7 itself acknowledges the gap: 'Null symmetry of PSGM mirror statistics may be weakened if the network learns spurious correlations or assigns systematic directional importance to null grouped features.' This is the admission that the neural proof's key antecedent is not guaranteed by the procedure. The empirical tables corroborate the fragility: in Tables 2–4, several PSGM FDR values exceed the nominal 0.1, including 0.283 in one Transformer row. These observations do not by themselves prove circularity, but they confirm that the unproven assumption is load-bearing. There is no damaging self-citation problem here: references [53] and [54] are prior Gaussian Mirror/NGM work by different authors, and the paper extends rather than reduces to them. The circularity is confined to the neural theorems, but those theorems carry the paper's headline neural FDR claim. Hence a score of 6, reflecting partial circularity: the linear construction is sound, while the neural 'prediction' of FDR control reduces in part to an assumed symmetry property.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

No new physical entities are introduced; the matrix-valued perturbation and Permutation SHAP derivative are algorithmic constructs, not latent entities. The linear theorem is mostly parameter-free, but the neural pipeline introduces several hand-chosen or optimized construction parameters. The load-bearing assumptions for the neural part are in the axioms: Theorem 3 assumes model-level null independence, and Theorem 5 assumes the symmetry properties of the construction and the importance map.

free parameters (5)

Perturbation matrix G_j = argmin_{G_j} I^K_j(G_j)^2 per group
Optimized against the kernel conditional-dependence objective rather than prescribed by the null-symmetry theorem; its symmetry properties are assumed in Theorem 5.
RBF kernel bandwidth = median pairwise distance after column standardization
Bandwidths for K_U, K_V, K_W are chosen by a data-dependent heuristic in all neural experiments.
Monte Carlo permutation draws M_n = 10
Fixed by sensitivity analysis in Appendix D; the asymptotic theorem requires M_n to diverge.
Local smoothing bandwidth h_n = ceil(0.3 n) nearest neighbor
Tricube local-quadratic smoothing bandwidth; the asymptotic argument requires h_n -> 0, but this fixed fraction does not.
Ridge penalties lambda_W, lambda_Z = 0.1 n
Hand-set values for the high-dimensional ridge projection in Algorithm 4.

assumptions (7)

domain assumption Errors in the grouped linear model are Gaussian and i.i.d., epsilon ~ N(0, sigma^2 I).
Theorem 1 relies on joint Gaussianity of the fitted mirror coefficient vectors; the GLS extension handles correlated errors only through whitening.
domain assumption R_j^T R_j and Q_j^T Q_j are nonsingular.
Needed for G_j and tilde Z_j to be well-defined in Theorem 1.
domain assumption The first-stage screener satisfies group-level sure screening, P(S1 subset of hat A) -> 1.
Theorem 2's asymptotic null symmetry depends on this property.
domain assumption The two data chunks are independent, or asymptotically independent under stationarity and mixing.
Used in Theorem 2 so that the first-stage selection event does not contaminate the second-stage mirror statistics.
ad hoc to paper The fitted nonlinear model satisfies f(x)=g(x_-j) for a null group j.
Appendix C.7, proof of Theorem 3, assumes the null group is absent from the fixed fitted model. This is exactly the property that would make Permutation SHAP derivatives zero, and it is not established for networks trained on augmented mirror inputs.
ad hoc to paper The ridge projection is odd in Z_j, hat G_j is even in Z_j, and the downstream importance map is swap-equivariant.
Theorem 5 states FDR safety conditional on these symmetry properties; no proof is given that trained neural networks or Permutation SHAP derivative maps satisfy them.
standard math The log-density lies in a tensor-product RKHS and the likelihood-ratio score formalism applies.
Inherited from Neural Gaussian Mirror and used for the grouped kernel objective in Theorem 4.

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Cite this review

Pith. "Pith review of Model-Agnostic FDR Control via Group Gaussian Mirror and Permutation SHAP." pith.science (2026). https://pith.science/paper/WTMUFPP5

@misc{pith2026260800989,
  author       = {Pith},
  title        = {Pith review of: Model-Agnostic FDR Control via Group Gaussian Mirror and Permutation SHAP},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WTMUFPP5}},
  note         = {Machine review of arXiv:2608.00989}
}

read the original abstract

Most FDR-controlled feature selection methods are designed for coordinate-wise hypotheses, where each feature has a single weight or importance score. This abstraction fails in sequential and grouped models, where one original feature is represented by a block of sub-features, such as lags, recurrent states, or attention-based interactions. We propose a grouped-feature FDR control framework for such settings. For grouped linear models, we construct null-symmetric block-level mirror statistics with matrix-valued perturbations. For neural sequential models, we combine Permutation SHAP derivatives as model-agnostic block-level importance scores with kernel-based dependence measure. The framework is model-agnostic across network architectures, does not require specifying the covariate distribution, and reduces to Gaussian Mirror or Neural Gaussian Mirror when the block size is one. We prove FDR control for low- and high-dimensional grouped linear models and asymptotic symmetry of smoothed Permutation SHAP derivatives under fixed fitted nonlinear models. Experiments on simulated and real-world datasets show reliable FDR control and improved power under correlated grouped-feature signals.

Figures

Figures reproduced from arXiv: 2608.00989 by the authors.

**Figure 1.** Flowchart of GGM and PSGM C Proof of Theorems and Propositions C.1 Proof of Proposition 1 Proposition 1. In a grouped design X = [X1, . . . , Xp] with Xj ∈ R n×m and βj ∈ R m, applying the Gaussian Mirror separately to each sub-feature of Xj targets the coordinate-wise nulls H0,(j,ℓ) : βj,ℓ = 0 for ℓ = 0, . . . , m − 1, rather than the grouped null H0,j : βj = 0. Proof. In the grouped design, the j-th original featu… view at source ↗

**Figure 2.** Convergence curve of the kernel-based objective optimization [PITH_FULL_IMAGE:figures/full_fig_p028_2.png] view at source ↗

**Figure 3.** Graph-proxy relevance construction example for Target ID 176 [PITH_FULL_IMAGE:figures/full_fig_p030_3.png] view at source ↗

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