$\text{G}^2\text{M}$: A Generalized Gaussian Mirror Method to Boost Feature Selection Power
Authors: Hongyu Shen
Zhizhen Zhao
Venue: NeurIPS 2025 poster ()
Keywords: Feature Selection
FDR Control
Hypothesis Testing; Gaussian Mirror; Boosting Power
Primary area: probabilistic_methods

Abstract: Recent advances in false discovery rate (FDR)-controlled feature selection methods have improved reliability by effectively limiting false positives, making them well-suited for complex applications. A popular FDR-controlled framework called data splitting uses the "mirror statistics" to select features. However, we find that the unit variance assumption on mirror statistics could potentially limit the feature selection power. To address this, we generalize the mirror statistics in the Gaussian mirror framework and introduce a new approach called "generalized Gaussian mirror" ($\text{G}^2\text{M}$), which adaptively learns the variance and forms new test statistics.  We demonstrate both theoretically and empirically that the proposed test statistics achieve higher power than those of Gaussian mirror and data splitting. Comparisons with other FDR-controlled frameworks on synthetic, semi-synthetic, and real datasets highlight the superior performance of the $\text{G}^2\text{M}$ method in achieving higher power while maintaining FDR control. These findings suggest the potential for the $\text{G}^2\text{M}$ method for practical applications in real-world problems. Code is available in https://github.com/skyve2012/G2M.

G A Generalized Gaussian Mirror Method to boost feature selection power
Hongyu Shen', Zhizhen Zhao'
Department of Electrical and Computer Engineering University Illinois Urbana-Champaign (hongyu2, zhizhenz] 0illinois...i.. edu
Abstract
Recent advances in false discovery rate (FDt))controlled feature selection methods have improved reliability by effectively limiting false positives, making them well- suited for complex applications. A popular FDR-controlled framework called data splitting uses the "mirror statistics" to select features. However, we find that the unit variance assumption on mirror statistics could potentially limit the feature selection power. To address this, we generalize the mirror statistics in the Gaussian mirror framework and introduce a new approach called "generalized Gaussian mirror" (G²M), which adaptively learns the variance and forms test statisties. We demonstrate both theoretically and empirically that the proposed test statistics achieve higher power than those of Gaussian mirror and data splitting. Comparisons with other FDR-controlled frameworks on synthetic, semi-synthetic, and real datasets highlight the superior performance the G2M method in achieving higher power while maintaining FDR control. These findings suggest the potential for the G M method for practical applications in real-world problems. Code is available at: pattps://gitain com com/skyyv2012/G22
Introduction
The complexity and high dimensionality of data in the era of big data make it challenging develop a universal feature selection algorithm that performs well in diverse datasets [18. An algorithm that works effectively for one type of data may fail for another. This variability complicates the verification of feature selection results, thereby hindering the practical use of selected features. Such challenges especially prevalent in fields like healthcare, where there is guarantee that identified genes or metabolites [ correlate well with response variables (e.g., diseases), making downstream applications such as drug discovery even more difficult.
To address these challenges, recent research has focused on false discovery rate (FDR)-controll) feature selection methods 4 R. These methods limit the number of false positives during feature selection, providing a guarantee of the maximum number of incorrect selections. This property is particularly advantageous in titr--iigh-dimmesioonal settings, such genetic RNA-sequencing data.
Over the past decade, various methods have been proposed. Model-X knockoff's 4 8 framework designed to select relevant features while controlling the FDR. It generates "knockoff" variables references to the original design matrix and forms knockoff statistics. The symmetry- about-zero property of these statistics for null features enables FDR control under specified nominal levels. Although theoretical guarantees exist for Gaussian design matrices g 37]. challenges remain for non- Gan-Gaussian settings due to the lack of theoretical assurances. Methods like Deep Knockoff 30,, KnockoffGAN [1], and DeepDRK 24l have been proposed to enhance selection power while
is novel
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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controlling FDR in non-Gaussian data. However, as highlighted in Shen et al. [4], balancing reconstructability (which enhances power) and the swap property (which ensures FDR control) remains empirically difficult, and theoretical guarantees are still lacking.
Another line of work is the conditional randomization test (CRT), first introduced in Candès al. [8] CRT relies on conditional sampling for each feature given the others to compute p-value statisties, which are then used with the Benjamini Benjamini-Hochberg (BH) procedure to control FDR. Unlike knockoff statistics, CRT avoids the reconstructabilitty issue by sequentially considering individual features. However, its computational bottleneck arises from the need to repeatedly generate conditional samples, especially for non- Non--aussian data where Markov Chain Monte Carlo (MCMC) techniques are required. This substantially increases the computational cost. Recent methods such as HRT [40], and Distillation-CHR [19] have sought to improve sampling accuracy and efficiency. Despite these advances, the inherent challenges of onn-aassiann data remain unresolved.
Distinet from these approaches, data splitting 9 and Gaussian mirror statistics" with properties similar to knockoff statisties but without relying on the swap property. This avoids the trade-off identified in Shen et al. [34]. The key difference between the two methods lies in how the mirror statisties are constructed. Data splitting divides the design matrix and response into two parts, generating paired estimates for each feature, which are then used to compute the mirror statisties. In contrast, the Gaussian mirror introduces two Gaussian noise perturbations to create additional columns in the design matrix, replacing the original feature column These perturbed columns are used to compute the mirror statistics. However, on the one hand, the data-splitting approach is constrained by its unit-variance assumption for the mirror statistics. On the other hand, the Gaussian mirror paper does not provide a discussion on the statistical power of the proposed method.
44 introduce a "mirror
To address these limitations, we generalize the variance assumptions in the mirror statistics introduced in the data-splitting framework and propose a generalized Gaussian mirror test statistics that considers the variance information of the fitting coefficients an intermediate component that forms the mirror statistics. We demonstrate that this statistics is the entry-wise uniformly most powerful test statistics, leading to improved feature selection power compared to both the Gaussian mirror and data- splitting test statistics. We call the proposed method "generalized Gaussian mirror" (G²M) With experiments considering synthetic, semi-synthetic, and real datasets, we demonstrate the practical value of the proposed approaches over set of benchmarking methods.
2 Related Work
2.1 Data Splitting
Data splitting is feature selection method designed to control the FDR (9. This method partitions the design matrix X and the response vector y into two disjoint datasets, denoted as xx+,y** and (X-, . ). Separate models then fitted each dataset using the same approach (e.g., both use ordinary least squares), yielding the coefficients 37 and 3 from two respective models for each feature zy. For each pair (8, 8.7 ), the method constructs test statistics w/ which is a function of these coefficients and reflects the importance of the corresponding feature Zz-
Notably, the statistics wy is designed features (j E Sp) while maintaining positive values for non-null features (j E Si). After calculating wy for all features, the selection procedure identifies the set of significant features S1 = [j wy 2 JyP. where the threshold Tq is determined based on the desired FDR control.
"mirror statistics," ensuring symmetry about 0 for null
1+ Kj : wy S -tH max 1, Kj w, 2 ()) < }
min 0
This setting controls feature selection FDR level at g according to Theorem 2.1
'It is noteworthy that the data-splitting (Section 1 and Gaussian mirror (Section 2.2 methods are closely related to the proposed G method. We describe their methods individually and discuss the linkage to our
proposed approach in Section 2
'To the best of our knowledge. there are limited works concerning improvements of data splitting or Gaussian mirror. Ge et al. [12] considers extending an uneven data splitting with cox model. Dai et al. [10], Xing et al. 44 consider nonlinearity in test statistics. These are orthogonal to focus, therefore omitted.

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Theorem 2.1. For a given set of mirror statistics {w, Pi= the following properties hold:
1. if is a null feature index, then for any real number t, P(w; 2t) = P(w, S -t).
2. if is a non-null feature index, then P(w; 2 0) > P(w; S 0).
Then, for given nominal FDR level q, the feature selection set
== { E [I......p: : w 2 T
controls the FDR, where Ty is chosen via Eq.
Proof. Given the above two properties for the mirror statistics, we have P(w, 2 1) P(w, S -() due to the symmetry-aboutt abrut-zero property of the mirror statistics wy under the null distribution. As result, the selection rule $ {j W Ty) (equivalently, the denominator of Eq. (I) selects features based on the level-g quantile on the right side of the null distribution, ensuring the FDR level controlled most q.
2.2 Gaussian Mirror
The Gaussian mirror [45] is a different approach from the data splitting method given how 3 and BJ are generated. Specifically, given an index j, the Gaussian mirror proposes two perturbed variables in place of the original ry. resulting
new design matrix X (af 25 xx-)), where 2j = zj + Cj2g, 2j = Ij cj²3, 2j ~ (O, I,), and cj is a scalar in R. 13 and $J are the coefficient estimations for 2,t and 5 respectively.
The Gaussian mirror adopts similar notion of mirror statistics as the data splitting method to select features with controlled FDR. With c
||PL,,;;1 || Xing et al. (45 shows that /8/ and $ are
independent, allowing the choice of a specific form of mirror statistics wy 18 + + - 1 BJ1
to perform EDR-controleed feature selection at level q, according to Eq. (T. Here, P1-3 denotes the projection in the null space of X
2.3 Connection between Data Splitting and Gaussian Mirror in the G²M setting
In this paper, the proposed G test statistics is based on crrtance-generalize uniformly most powerful (UMP) statistics under the mirror statistics setting in the data splitting paper [9. Specifically, adopts the 3. and 83 from the Gaussian mirror paper [45 to form a generalized mirror statistics (see Lemma. 3.5) over the one from the data splitting paper. The proposed statisties considers the variance information of f35 and 97 resulting in higher feature selection power compared to that of the data splitting and Gaussian mirror. More importantly, it is theoretically verifiable. In addition, We provide evidence on the no-constant variance of 35 and $8 in Appendix A the rationale why such variance generalization is necessary.
3 Method
This section details the proposed approach in this paper. The arrangement is follows:
Section 3.2 considers a more general setup of Gaussian mirror variable n:ft and 27. And provide the related form of 3 and SJj as weighted sum of the true B, coefficient and the noise variables. We also provide two useful properties about 35t and BJ whose proofs are omitted in the original Gaussian mirror paper 45
Section 3.3 presents the first major result (i.e., Lemma 3.5) on the UMP test statistics given the feature index j. The setup we consider is more realistic compared the one considered in Dai et al. [9, resulting in higher test power.
Section 3.4 introduces the second major result that proves the proposed G test statistics (see Proposition 3.Tp is more powerful test statisties compared to the mirror statistics in Xing et al. [45] Dai et al. [9].
3

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In Section 3.5 we combine the results discussed in the previous sections and propose two algorithms one exact algorithm that characterizes an ideal scenario, and one estimation algorithm that can be used in practice. Experiments in Section 4 are conducted w.r.t. the estimation version.
3.1 Notation
We first introduce the notation for the paper. Let X E RRxP be the design matrix, 2, (Zij...., Fny)" be the j-th column of X, and X-3 be the submatrix of X with the j-th column removed. the sample size and p is the feature dimension. Following the setting in Xing al. 45), we assume that X is normalized such that Σ Iij = 0 and | n, E [l..... p). Let (y1..... Vn.) be the vector of n independent responses. We assume that the response variable S1 and the task of feature selection is to
only depends on subset of features Xs, {X, identify the set Sj. And the response follows a linear model: y = X + where 3 = (3,..... 9p) indicates the coefficients for the features x; Let So = (1..... p} Si be the index set of the null features. Let Po |Sol and = |S be the number of the null and the relevant features, respectively.
......
3.2 A General form of Gaussian Mirror Coefficient Estimation
We propose to represent 8 and Bj as function of the true By and the noise vector given generalized Gaussian mirror setup that considers additional variables (i.e., dj and g;). This improves flexibility on the representation of 17 and 37- Details are included in Proposition 3.1 which helps understand the behavior of 855 and 87 in terms of gj. 2j. Cj. dj. 3g and
Proposition 3.1. Consider generalized Gaussian mirror setup where x z 2j and Jj = =j dj- qn and 2j and qj are n dimensional i.i.d. standard Gaussian. Cj and dj are in IR Then the least square regression coefficients for a.z and x; have the following forms:
BJ = a. + T
+ --
where stands for n-dimensional i.i.d. standard Gaussian vector. And a, 7 S. and r7 are functions of Cj. dj. 2ÿ, and qj (see Appendix B. for forms).
In addition Proposition 3.1 introduce three corollaries that reveal certain properties about 85 and 87 and terms associated with them (see Eq. (2). These properties are crucial developing the estimation algorithm in Section 3.5
Corollary 3.2. Given žj リE + 1, where and are defined in Proposition 3.1
Corollary 3.3. Given Cj = dj. and 2j Oz. = 0.5, where a and Ç are defined in Proposition 3.1
Specifically, Corollary 3.2 describes property on the behavior of a and that sum to one long as the Gaussian mirror perturbation vectors zj = Qj. Notice that this does not require 6, d, indicating a more generalized setup beyond the Gaussian mirror. On the other hand Corollary 3.3 indicates that under the vanilla Gaussian mirror setup (see Section 2.2 and Xing et (45),, the mean of B8 and 3, is 0.5,3. Note that the result is only mentioned in Xing et al. [45) without any proof. Here we provide formal proof of this fact, complementing the original Gaussian mirror work. The proof of Corollary 3.2 and 3.3 can be found in Appendix B
Corollary 3.4. Var(3;) = 1r 2 and Var(37) = ||n|
The proof straightforward by realizing that is the only term that introduces randomness to 33 and 95 Hence Var(3 Var(> T e) = | Similar reasoming can be applied to Var(3,)
3.3 Entry-wise Uniformly Most Powerful Mirror Statisties
According to Section 2.3 and Appendix is clear that the variances 33 and 99, need not be This violates the condition for the UMP mirror test statistics proposed in Dai et al. (9]. To consider

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more realistic setup. we present the first result (i.e., Lemma 3.5 that reveals a mirror test statistics that a function of the variance 8 and 5j. We show that the proposed test statistics is UMP for every
pair of 15/t and 8-- Note that we do not assume universal distribution for 3 and 3 across all j's. This flexibility directly leads to Theorem 3.6 which proves that the test statistics in Lemma 3.5 achieves higher power during feature selection compared to the existing mirror statistics [45 9
Lemma 3.5. Suppose that the set of coefficients (5) ,57) is independent for all j E fl,.... p). Furthermore, suppose:
(a) For j E St, the two coefficients 37 and Bj follow Nww, (w,,a)) Nww, o independently. And W 8 Rademacher((.5) (6 0), where E R is the absolute magnitude of w
(b) For j E So, the two coefficients B3 and Bj follow N(0, o and NO0, of)) independentl;;
(c) Po
+ r as
Then the optimal choice of in the mirror statistics (i.e., sign( (7) that yields the highest power follows the form:
f(a,b) b) = [ (-) P- exp( S+)],
(3)
where S_
5(6-2a)
5(6-26) 205
S+ = 5(5+2a)
+
6(64226)
and
20
20
20
( 1- (
(
P4 = ①
P- = 1 $
PE|H.
P- + P+
where & is the cumulative density function for the standard normal distribution.
Lemma 3.5 reveals closed-form test statistics that is UMP. This is used in developing the proposed feature selection algorithms in Section 3.5
3.4 More Powerful Test Setting
Combining the results from Proposition 3.1 and Lemma 3.5 we present the following Theorem 3.6 that indicates a more powerful test statistics of the proposed over the existing ones in Xing et al.
45), Dai et al. 9
Theorem 3.6. The proposed test statistics f(a, b) in Eq. (3) achieves the highest power compared
to the Gaussian mirror test statistics: la + la bl (45), and the data splitting test statistics: sign(ab) + [6] [9], under the same nominal FDR level q.
In Section we will empirically observe evidence that supports Theorem 3.6 Theorem 3.6 can be found in Appendix B
And the proof of
3.5 Algorithm
With all the ingredients introduced in the previous sections, we present two algorithms: 1. the exact algorithm (Algorithm I when one has access to the scale of the true coefficients for the nonnull By 's: or 8, if is index- index-dependent; 2. the estimation algorithm (Algorithm) 2 that essentially proposes an estimation of 8 or dz. resulting generalized likelihood ratio test statistics given the estimation. Note that the proposed algorithm considers the vanilla Gaussian mirror setup to take advantage the nice properties indicated in Corollary 3.2 and 3.3 This means Cj dj and 2j = qj for all j's. And
9 [P P following the setup Xing et al.
45
=
Algorithm introoucces the exact algorithm when true å presents. This is rarely the case as often we do not have prior knowledge about true By 's. To overcome this issue, we propose an algorithm to estimate 6, which is Algorithm 2
Essentially, the estimation algorithm employs k-means-based algorithm find & potential modes for 5 quantity that is assumed to be given in Algorithm T
yet unavailable practice. We
We adopt similar setting in Proposition in Dai al. to define w.
5

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Algorithm Exact algorithm
1: Input: Design matrix X, response nominal FDR level q and true scale å or dj for the Bj.
j E St.
2: Output: S [j wy 2 TgJ
3: for j = 1 to do
4: Sample 2, (., I.)
5: Calculate cj = |
P and form 25 = z; + Cz.j. 3 Ijj - Cjej.
6: Obtain the ordinary least squares estimator of 3 and 37:
(8) 05 )-z) arg min "5.
ll - X x X- 2 3, 1
7: Calculate Var(05)= 11 and Var($,) 1/
8: Compute Wy - 56g $j (8 G,), where fla, b) is specified in Eq. 43)
9:
end for
Calculate the threshold given the FDR level q:
min t: 1 [j Wy < -tH Tq
max 1, Kī : W 2 +H =g)
choose k-means given it has been widely considered as an efficient unsupervised algorithm due to no
accessible to true values of 8y. First, combining the results of Corollary and and 3.3 the algorithm
obtains an unbiased estimate of the absolute value of 3, (j E (I..... p in Step 7 of Algorithm 2
This is because and are both equal to 0.5 based values represent possible values of 6. Then we utilize the k-means algorithm with & chosen via the silhouette score common technique for choosing the number of clusters. Note that the k: identified modes include the case for the null 8y 's where & = O, due to not knowing which variable is null and which is nonnull. Luckily, which corresponds to the mode for the null 3 ss resulting in k 1 candidates. For every calculated pair of 3J and 87 ōy is chosen to be the closest mode among the k candidates, compared to
Corollary 3.2 and 3.3 The p calculated
can simply exclude the smallest mode (e.g., the closest one zero),
the value 9, + With Algorithm 2 we can perform feature selection without accessing the true information. In Section 4 all results about the proposed GPM method are based the estimation algorithm (Algorithm
Algorithm 2 Estimation algorithm
1: Input: Design matrix X, response y, nominal FDR level and the number of modes for 8:
k Σ p.
2: Output: Algorithm
X, 1 4
3: j to p do
4: Sample 2, (0,
5: Calculate = TPI-,
[|
and form 2² = + Cj 2tj = =x - Cz3j.
6: Obtain the ordinary least squares estimator of 3 and 3/
= arg min
l x xー 2 B3 22,
07 17
7: Calculate the value Uy 1
8: end for
9: Run k-means algorithm to find modes given fo, 1i represented as {mi)k.>
10: for j = to p do
11: 8, = minm; [0g - mil I (l....,k).
12: end for

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4 Experiment
In this section, we evaluate the proposed G approach using three types of datasets, following the experimental setup described in Shen et al. [4 given its comprehensibility Specifically, we consider: 1. Fully symthetie datasets, where the design matrix X and the response y are generated from known distributions; 2. Semi-synthetic datasets, where the design matrix is extracted from real-world data, and the response is generated using a predefined model; 3. A real-world case study, where both X and y are fully unknown. The experimental details and corresponding results are summarized in Section [4.1, Section 4.2 and Section 4.3 respectively. In each section, we provide details on the benchmarking methods, the configuration of the dataset, and the results. All experiments were carried out on an IBM AC922 server with 2x 20 core IBM POWER9 CPU @ 2.4GHz.
Table 1: FDR and power with Gaussian design matrix X for p 0.6 (left) and p = 0.7 (right). Bold entries indicate the highest power given controlled FDR level 0.1. Blue for the second best, and Red for FDR> 0.1.
0.6
0.7
Method
OL,
Ridge
LASSO
OLS
Ridge
LASSO
FDR
Power
FDR
Prwer
FDR
Power
Power
Power
Power
CRT [
0.00
0.13
0.07
0.69
0.07
0.77
0.00
0.02
0.13
0.51
0.11
0.56
Distilled-CRT
0.53
0.98
0.27
0.98
0.13
0.97
0.58
0.95
0.25
0.93
0.11
0.87
Gaussian Minror
55
0.09
0.54
0.05
0.67
0.04
0.74
0.05
0.18
0.04
0.51
0.23
0.48
Gaussian Mirori
45
0.10
0.61
0.05
0.71
0.04
0.76
0.13
0.49
0.06
0.64
0.04
0.61
Data Splitting
0.00
0.00
0.06
0.66
0.04
0.50
0.00
0.00
0.06
0.56
Data Splitting 9
0.72
0.07
0.08
0.71
0.17
0.70
0.73
0.07
0.07
0.66
0.25
0.70
HRT [4D)
0.01
0.23
0.00
0.21
0.00
0.22
Powerful Knockoff 37
0.00
0.00
0.05
0.48
0.06
0.69
0.01
0.00
0.04
0.29
0.05
0.55
Powerful Knockoff
0.75
0.08
0.08
0.06
G (ours)
0.08
0.65
0.05
0.82
0.05
0.82
0.08
0.36
0.03
0.67
GMM (ours)
0.08
0.72
0.04
0.80
0.06
0.84
0.10
0.55
0,03
0.67
0.04
0.70
4.1 Synthetic Data
To thoroughly evaluate performance, we adopt the experimental framework outlined in Shen et al. [34] to generate diverse synthetic datasets defined by (X, 3/) In this setup, X E R2 represents dependent variables drawn from predefined distributions, and y IR serves the response variable. We model y using a linear response framework: y ~ N(XT 3, 1). The true underlying / is a p-dimensional vector with entries independently sampled from the distribution /5./ Rademacher((.5). This
choice differs from the commonly used scaling factor Rademacher((..) more challenging evaluation configuration.
38 22
enabling a
In addition the Copula and Gaussian mixture models considered in Shen al. 34 that quantifies the nonlinear design matrices, we include the Gaussian setting, exploring various correlation levels among the features. Since methods such knockoffs (8) and the conditional randomization test (CRT) [8] have closed-form solutions in Gaussian settings, incorporating this setup offers valuable insights into the consistency between empirical results and theoretical guarantees. Due to space limit, we defer the introduction of these settings in Appendix D
Benchmarking Methods & Settings + Given the linear and nonlinear nature of the design matrix, we utilize two sets of benchmarking methods. The first set comprises methods that are theoretically proven control the false discovery rate (FDR), including knockoff [8), conditional randomization
test (CRT) 88, data splitting [9] Gaussian mirror [45), distilled-CRT [19] HRT (40), and powerful
knockoff 37 In addition, we also consider another rule choose Tr (Eq. (4) according to Ren and
Barber 29 to improve the power for certain applicable methods and mark them with "t":
= min {1:
1+ 10 wj S -()
<q or Σ 1 : my 2 < 1
(4)
1>0 max (1, w +
jE(p]
where "1" is the indicator function. Essentially, the two equations differ only when mine :
max(1, (f-6 g} > min Σteb) [ my 2 () < 2 This means when g is small or
1+1
*We include discussion about computation complexity for non non-deep- deep-leamiing based based methods. We defer the discussion in Appendix Cliivnn this is not the major focus this work
7

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the fraction of non-nulls is small, then the first term usually finds a larger T.. in favor of the FDR control. In this case, the power is pretty low, as when Ty large, there are only limited non-nulls selected. Instead of choosing Tq with the first term (e.g., Eq. (]) the second term chooses smaller Tq. which results in higher power.
The second set focuses on deep learning-based methods designed to handle nonlinear design ma- trices, such as Deep Knockoff (30] DDLK 38 DeepDRK 34 14
KnockoffGAN [14] sRMMD 22 and
Table 2: FDR and power with Copula and Gaussian Mixture design matrix X. Bold entries indicate
the highest power with controlled FDR at level 0.1. Blue for the second best, and Red for FDR>> 0.1.
Method
Citation
Gaussian
Mixture
Clayton & Exp.
Clayton & Gamma
Joe & Exp.
Joc & Gamma
FDR
Power
FDR
Power
FDR
Power
FDR
Power
FDR
Power
CRT
0.00
0.21
0.08
0.69
0.05
0.91
0.08
0.45
0.07
0.87
Distilled-CRT
19
0.17
0.22
0.06
0.43
0.06
0.36
0.05
0.26
0.04
0.26
Gaussian Mirror
45
0.05
0.83
0,07
0.54
0.07
0.89
0,06
0.34
0.08
0.86
Gaussian Mirrori
45
0.04
0.83
0.07
0.70
0.08
0.92
0.09
0.52
0.09
0.84
Data Splitingg
0.07
0.72
0.07
0.52
0.07
0.79
0.06
0.28
0.08
0.76
Data Splittingt
2
0.09
0.76
0.09
0.62
0.08
0.81
0.10
0.49
0.09
0.78
HRT
40
0.00
0.01
0.15
0.50
0.01
0.09
0.02
0.50
Powerful Knockoff
0.08
0.62
0.03
0.17
0.05
0.38
0.04
0.11
0.07
0.39
Powerful Knockofff
0.09
0,40
0.06
0.51
0.14
0.39
0.07
0.53
Deep Knockoff
500
0.74
1.00
0.29
0.88
0.25
0.95
0.40
0.86
0.26
0.94
DDLK
0.79
0.13
0.30
0.27
0.66
0.04
0.00
KnockoffGAN
0.44
0.99
0.07
0.35
0.09
0,70
0.05
0.17
0.09
0.60
sRMMD
0.72
1.00
0.29
0.88
0.24
0.94
0.31
0.78
0.26
0.93
DeepDRK
34
0.10
0.83
0.07
0,35
0,08
0.78
0.10
0.42
0.09
0.70
G'M (ours)
0.07
0.86
0.09
0.58
0.10
0.94
0.10
0.32
0.10
0.89
GPM (ours)
0.06
0.92
0.06
0.75
0.09
0.95
0.10
0.61
0.10
0.91
For the Gaussian setup, we focus solely on the first set of methods, as they align with the theoretical guarantees for Gaussian designs. We use three different fitting methods to generate the estimation coefficient 3, 's: ordinary least square (OLS), ridge regression, and LASSO, to consider common adaptation of these methods in the feature selection setup. In contrast, for the Copula and Gaussian mixture setups, we evaluate both sets of methods and consider ridge regression to be the fitting method as empirically it produces the best performance according to Shen et al. 34 Results are presented for FDR and power with (n, p) = (200, 100), the FDR nominal level 0.1. All reported values are averaged over 100 independent repetitions.
Results: In Table we present results with Gaussian data in two different correlation settings (c.g.. p 0.6, 0.7). In both cases, the proposed G2M M outperforms other benchmarking methods for achieving the highest power while controlling the FDR under the nominal level of 0. In addition, we notice that the power of the proposed G is always greater than that of the Gaussian mirror and data splitting methods, indicating the consistency between the empirical results and the theoretical reasoming. Surprisingly. we discover that G²M performs consistently across different fitting methods other than OLS (e.g., LASSO and ridge regression) despite the fact that the theory was developed in the least squares sense. This suggests a possible wide application of the proposed G M method.
In Table 2 on the other hand, we consider results with nonlinear design matrix X with ridge regression model being the fitting method. Similar the Gaussian setting. G²M M achieves the highest power on all datasets while controlling the FDR under the nominal 0.1 level. The proposed method even outperforms those deep-learning-base Importantly, this observation is consistent with the theoretical justification as the proof of the GDM method does not depend on the distribution of the design matrix X.
methods, suggesting its power in wide applications.
To further demonstrate the performance of our method, we investigate high dimensional settings by reducing the number of samples relative to the 100 dimensional input data. In addition, we consider the impact of noise to the power and FDR during feature selection. The comparison is deferred to

---

Appendix Euue to space limit. The results, however, suggest that our proposed method outperforms other benchmarking methods in most cases, indicating a new state-of-the-an
4.2 Semi-synthetic Data
In this section, we conduct a semi-synthetic study by extracting the design matrix X from two real-world datasets.
Table 3: FDR and power with two semi-synthetic datasets: RNA and IBD. Bold entries indicate the case with the highest power given controlled FDR at level 0.1, Blue for the second best, and Red for FDR> 0.1.
Single-cell RNA Sequencing: The first dataset consists of single-cell RNA sequencing (scRNA- seq) data obtained from 10x Genomies f Each entry of X E == represents the observed ex- pression levels p genes across n cells. For additional background on this dataset, we refer readers to Hansen et al. [13 and Agarwal al. [1. Following the preprocessing pipeline de- scribed in Hansen et al. [13], we obtain the final dataset X with
Semi-RNA
Semi-IBD
Method
Citation
FDR
Power
FDR
Power
CRT
g
0.39
0.89
0.15
0.56
Distilled-CRT
0.27
0.87
0.11
0.75
Gaussian Minor
55
0.61
0.97
0.04
0.26
Gaussian Mimori
0.06
0.58
0.08
0.55
Data Splitting
20
0.21
0.86
0.16
0.73
10, 000 and = 100 T0 The response y is formulated according to non- linear function of X. Details including the form of y and the data preparation about X are in- cluded in Appendix [F due to space limit.
Data Splitting?
2
0.14
0.64
0.16
0.77
HRT
40
0,38
0.87
0.05
0.42
Powerful Knockoff
0.50
0.94
0.07
0.21
Powerful Knockoff]
37
0.14
0.43
0.14
0.51
Deep Knockoff
511
0.00
0.14
0.27
0.55
DDLK
0.14
0.81
0.09
0,26
KnockoffGAN
14
0.00
0.00
0.10
0.25
3RMMD
0.00
0.00
0.24
0.44
Inflammatory Bowel Disease (IBD): The second dataset, publicly available from the
DeepDRK
34
0.08
0.73
0.10
0.25
G'M (ours)
0.00
0.43
0.07
0.45
Metabolomics Workbench 11 originates
GM (ours)
0.04
0.66
0.10
0.61
from a real-world study titled "Longitudinal
Metabolomics of the Human Microbiome in Inflammatory Bowel Disease (IBD)" (20). The objective of this study is to identify significant metabolites associated with two forms of inflammatory bowel disease: ulcerative colitis (UC) and Crohn's 's disease (CD). We use the C18 Reverse-Phase Negative Mode dataset, which comprises 546 samples and 91 metabolites.
To handle missing values, we preprocess the dataset by removing metabolites with more than 20% missing data, resulting in final set of 80 metabolites. The data matrix is normalized entry-wise to have zero mean and unit variance, following log transform and imputation of missing values using the k-nearest neighbor algorithm, as described in Masud et al. [22]. The response y uses the same linear model described in Section 4.1 which has lower signal strength compared to the experiment in Shen et al. [34]
In both semi-synthetic settings, we consider ridge regression as the fitting method as empirically it performs the best according to Shen et al. [34] All reported values are averaged over 100 independent repetitions. The FDR nominal level is set 0.1.
Results: Since we consider the nonlinear design matrix X, methods from both the dcep-lcarning- based and haa-deep-leamingg-assee based (see Section 4.1 considered in this experiment. Results presented in 3 We find that almost all hon-deep-learning- and framework based methods fail to control the FDR with the RNA data, and part of the methods fail the IBD data. In comparison, the proposed G M method is the only one that can successfully control the FDR under the nominal level while achieving the second-ber best in power (the berf-performing DeepDRK achieves a higher power of 0.73 in this case). In the IBD setting, on the other hand, G M beats all nnn-deep-learning based methods and deep-learning- based methods for producing the highest power with the controlled FDR. This evidence suggests stable application of G2M M on real-world data.
https://kb.. http://obb...xeemmiiic.o.o hiateeommiccc.co/////eerrr com/hcc/n-us
Data processing code is available at: httpp////ggttuueooo///cceeeeee/e master/data
/www://wwwww/// batpo///ooo///////oooaaaaaatttot under project DOL: 10.21228/M82T15

---

Table 4: Feature selection results with IBD real dataset that consider true X and y.: RNA and IBD. FDR level is specified as 0.2. Since there is no ground truth on the features, we report "number of referenced metabolites/number of identified" in place of FDR power.
Model
G2M (aurs)t
CRT
DistilledCCT
Gaussian Mirrorf
Data Splitingt
Powerful Knockoff
Referenced/ / Ideatified
18/22
12/21
17/26
18/25
17/23
2/5
4/6
Model
DocpDRK
Deep Knockoff
aRMMD
KneekoffGAN
DDLK
Referenced/ Ideetifiied
19/23
15/20
5/5
12/14
17/25
4.3
Real Case Study
We further conduct two case studies using real data (i.c., IBD dataset 20 and breast cancer dataset [43]) for both the design matrix X and the response variable y to qualitatively evaluate the selection performance of the proposed method. We defer the breast cancer dataset analysis in Appendix E.4
In the IBD analysis, the response variable y is categorical, where = indicates that a sample is associated with UC/CD, and y 0 otherwise. The covariates X are identical to those used in the second semi-synthetic setup described Section 4.2 Since ground truth is not available for this dataset, we evaluate the results by identifying evidence of IBID-associated metabolites from existing literature, following the curation in Shen et al. [34], which draws upon the following sources: 1. Metabolites explicitly documented as being associated with IBD, UC, or CD in the PubChem database 12, 2. Metabolites reported in peer-reviewed publications; 3. Metabolites mentioned in preprints. For convenience, reproduce the nominal metabolite table from Shen et al. 34 in
Appendix G.I
We use the DeepPINK [1 model as the fitting method to obtain the feature coefficients in considera- tion the nonlinearity between the input X and the response y in this real setting. The FDR nominal level is set to 0.2.
We present feature selection results for the real IBD data in Table 4 Among the considered methods, clearly the G method performs on par with the DeepDRK method and identifies more re- ported metabolites while keeping a lower number of total selections compared to other benchmarking methods. This reveals the potential the GDM method for real-world applications. For completeness, we include full list of names for the identified metabolites by each method in Appendix G.2
Results:
5 Conclusion
In this paper, we first identify a limitation of the existing mirror statistics in the data splitting paper-the strong unit variance assumption. We then proposed variance-dependent Gaussian mirror method- Beie--G---ae and show both theoretically and empirically the performance of G²M compared to popular FDR- controlled frameworks and deep- wee---eaannnbasee based knockoff methods with synthetic, semi-synthetic, and real datasets. The results demonstrate that the G method effectively controls the FDR while achieving the highest power in most cases and delivering comparable performance in others. These findings suggest the potential for broad adoption of the GDM method in real-world applications.
Limitations and broader impacts: one limitation of the work is the normal distribution assumption on the fitting coefficients (e.g., 85 and 3,). ). This is tied to the proposed UMP test statisties in Lemma 3.5 and the fundamental result in Dai et al. [ A possible future work would be to generalize this part beyond the normality assumption in light of existing work on the generalized linear model setting: e.g., Dai et al. [10]. Nonetheless, based on experimental results with synthetic, semi-synthetic or real case study, our model still outperformed the existing state-of-the the-art methods, suggesting the importance of the work and its potential use cases biological data where dimensionality and FDR are crucial aspects.
demonstrate that despite having this limitation in the theoretical formulation,
https://pubchen. ncbi. mlm. nih, /
10

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13

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Appendix: G2M: A Generalized Gaussian Mirror Method to boost feature selection power
This appendix is structured follows: in Appendix we provide details on related work and the connection with our method; in Appendix A we show the empirical observation of the variance of Gaussian mirror statistics; Appendix we provide proofs for the theorems and lemmas in the main paper, in Appendix E we provide additional experimental results; in Appendix [ we detail the RNA data preparation for experiment Section 4.2] in Appendix we provide additional information for the IBD study.
A Empirical Evidence on Variance Differences in Gaussian Mirror Statistics
This section provides empirical results about the variance (e.g. standard deviation) the Gaussian mirror coefficients 83 and 87 given different design matrices. To start, randomly generate 10000 samples of the standard deviations of 197 and 3, for both null and nonnull cases, respectively. Specifically, each sample is generated by first sampling design matrix and then performing the Gaussian mirror perturbation based on a randomly sampled Gaussian vector 2j. Both the design
matrix X and the sampled vector 2j are used to calculate the corresponding сj P_ ,ا, component in the Gassuain mirror. The feature 2y is uniformly sampled on the index given either null or nonnull. With the information of 2g, Cj and 2j, we are able to calculate the standard deviation of B and f5 according to Corollary 3.4 and Eq. (
|P--
We present the histograms of the results with Gaussian design matrix in Figure T and 2 Gaussian mixture design matrix in Figure 3 and 4 and IBD design matrix in Figure and The description on the Gaussian, Gaussian mixture and IBD design matrices can be found Section 4 Clearly, the standard deviations for both null and nonnull coefficients are distributions that are not concentrated at indicating the unrealistic assumption about the unit variance outlined in Dai et al.
Frequency
Frequency
0.10
0.12
0.14
0.16
0.10
0.12
0.14
0.16
Standard Deviation of B,*
Standard Deviation B
Figure 1: Histogram of the standard deviation of 87 and BJ for null variables over 10000 samples. The design matrix X is based on Gaussian distributions.
Frequency
12.5
Frequency
10.0
10.0
7.5
7.5
5.0
5.0
2.5
2.5
0.0
0.0
0.12 Standard Deviation of B,-
0.10
0.12
Standard Deviation of Aj
Figure Histogram of the standard deviation of 3 and 3,7 for nonnull variables over 10000 samples. The design matrix X is based on Gaussian distributions.
14

---

12-
12
Frequency
Frequency
0.09 Standard Deviation of B
0.10
Deviation B
Figure 3: Histogram of the standard deviation of 37 and 87 for null variables over 10000 samples. The design matrix X is based on Gaussian mixture distributions.
Frequency
Frequency
10.0
7.5
7.5
5.0
5.0
2.5
2.5
0.0
0.0
0.09 Standard Deviation of BJ
0.08
0.07
0.09
Standard Deviation B
Figure Histogram of the standard deviation of 83 and 3, for nonnull variables over 10000 samples. The design matrix X is based on Gaussian mixture distributions.
14.
12
Frequency
10-
Frequency
10.
8
0.056
0.060
Standard Deviation of 8
Standard Deviation of g
Figure 5: Histogram of the standard deviation of 95 and BJ for null variables over 10000 samples.
The design matrix X is based on the IBD dataset.
17.5
17.5
15.0
15.0
Frequency
12.5
Frequency
10.0
7.5
5.0-
2.5
2.5
0.0
0.0
I||
0.068
0.062
0.064
Standard Deviation of B
Standard Deviation B
Figure 6: Histogram of the standard deviation of 8 and 87 for nonnull variables over 10000 samples. The design matrix X is based on the IBD dataset.
15

---

B Proof
B.1
Proposition 3.1
To show proposition 3.1 we first consider two models. The original model that produces the response y and the new model that considers the inclusion of 23 and zj in place of the original zy. In the following, we have the original model:
= X3 +
After modification of the design matrix, have the new design matrix:
new
[X- 2rz .775
and the new coefficient vector:
g 8
Bnew
To perform the least square fit with the new design matrix and the original response variable y, we can expand normal equations Xnew X new Pnew Xnew y
X-j
X
y y
(x
x
=
(z
(z,
)x
y
We use 3 to distinguish from the original 9-, that represents the coefficients in the original linear model with j-th entry removed. Expanding the normal equation we have:
tc,
+XTTaa,- (9)- 11055-11 = (8_; 3 "
(x)
6_ + (e)
x 3, + -
K3 2 1
(zj x-1) 85y + (0, (z) 33 + (2) -,5 =
B, + Kg B,
Since we have three equations and three unknowns: 5 83 and 3, we cancel out Brny leaving two functions that are the functions of 3 and 87. Eventually, we obtain the final form that represents 35 and 85 as a linear function of the true beta, and the noise c:
$ aa- B3 + 7 ee, 37 S Bg +n= c,,
where
Nj- Mj- H
== Ng- G --M3- K
L
L
< Ng+ F + Mj+
HI
n -Ny- G G M2+ K
L.
I.
All the involved variables are presented below:
L= M,+ Ns- M₁- .Nt
M,+ = (2,, 2, - ( X-j) B'
M, = (
- (1] X A-LC,
NG+
X-,) A A-1B,
N₂ = [,
T-10)1)))/--,,,,,,)/)))111 G (x=
H = - A-D,
K = - (1) (X---(xX ( X
A=XI,X_,,
B= x +
C -= d
D X Tg.
E ==
16

---

B.2
Corollary 3.2
To prove, we only need to show L = My- Nn- My- Nyt and Ny- F--F-- F- My- H -N,+
F + M,+ H are the same. We first expand
M,+N5- My-N5.
28 2cd (1] 0z) 2nd - (x]
-
+ dGe, (1) x3)
[Terms related to A and higher orders].
Similarly, we expand
N₂ F H - F Mj M -
(M,+ -
N
+ - -
- 2, dc(q, [dt))==)))(]]]]]))(
+ dfcs(a] (x] = +)) +443))))))()))))
[Temms involving and higher orders).
We can find matches for every term, proving the equality.
B.3 Corollary 3.3
The proof is similar to the one in Corollary we expand terms and show the match. In this proof, we need to show N-- F - M2 H which proves the corollary. In the following, have:
F + Mar H. We expand both sides and compare,
-Ng- F Ms+ -
& [o] (e]2))(aaa)) - (z) x,)²
- cdllg 25) 2,) + edla, 21) 2z)
+ cjd (3) x,) (z) 24 (1) 2z)
X-, + [Complex terms involving X j, A 1
N₂- F M H
(d 2,) ed(z} [x) 24))
+ dl [o] (x) xz) (4] x,,2
+ ad? [1] (e] 2,) (o] [2) 45)
+ [Complex terms involving X-j, A-
After considering Cj d, and 2; = qj. proof.
verify the two functions are equal, completing the
Remark B.I. Note that the Gaussian mirror work [45 only presents the 8+ = = 0.5 without
providing the formal proof. Combining Corollary 3.2 and Corollary 3.3 we can provide formal evidence for this statement.
17

---

B.4 Lemmal 3.5
This proof is inspired by for a more general setting. First, we identify the general form of the statistics following the proof in [9]. We then provide explicitly the form of f (e.g.. in Eq. (3) according the Neyman- Pearson lemma. The proof of the latter is omitted in (9]. We hope this proof can serve as a complement, which provides insights for people who are interested in any future extension on the optimal test statistics in this more general setting.
To start with, let Z1, Zz ~ N(w,of) (w,, oz), respectively. Z3. Z4 ~ o3) and N(0, of)
respectively 13 And 8- Rademacher(O.5)) ò > All variables are independent. Following (2] we assume that the designated FDR control level E (0, 1) satisfies T- < 1, otherwise selecting all
features would maximize the power and also achieve asymptotic FDR control. Let fopr(u, v) be the optimal choice, and let Sope be the optimal selection result that achieves asymptotic FDR control. By the law of large numbers, we have:
# #1::3 : E So.j Supi)
PÚ E Sapil So) P(j E Sopll E So) + rP(j i Sopelj E S1) S 9.
lim p-o
(5)
#(jijE E Sopt
in which the numerator is the type-I error. More precisely:
P(j E Sapil E S1) = P(sign( 2, fop Z1|, [2 > tepı)-
P(j E Sapil E So) P(sign( La Z4Z4). [aal, [2a] > lapt);
Here, Eupt > is the cutoff that maximizes the power P(j E Sopil S1), under the constraint that Eq. (5) holds.
In practice, consider testing whether the covariate Xj is null feature, with the significance level specified as:
rq P(j Supli E S1) < 1.
a 1 1--
Given two realizations 3 and 3,- consider their random variables independently following the normal distributions described by Zı, Zz if X, is a nonnull feature, and by Z3, Z4 otherwise. According to Eq. (5). the test which rejects the null hypothesis (i.e., j E Sopt) if:
sign (3t 87 Sap (13 1.737
lopt
achieves the significance level a.
We consider the following rejection rule with certain form of f that will be revealed later in the proof:
(0)00)88)))((()1))))))) tik-
in which tuk > 0 satisfies:
P(f(IZs), IZa)) tu sign (Zs) sign(Za) 2a.
Let Sjk be the corresponding selection set. We first show that this rejection rule controls the type-I error below a. Indeed:
1
P(j E Suilj E So) = P tulli e So. sign(8)) sign(37)) a.
In terms of power, we have:
tmbl S1, sign(Bf) sign (87 2 Pm (Sp (183 187 > lapel | E signn(()) = sign(37))
P(j E Smkl 8i) 1>)):,.P(((()))11))
= P(j E Sop Ij E S1);
where Pu. = P ( (s)gn(s(")) s3gn 15
In the following, we show the proof of the inequality and the exact form of the function via the Serman--Pearon lemma.
Note that we consider more general setting with d2, and a4. Later on, assume
and da as db. to reach the results in Lemma 3.5
18

---

Since we consider a function f that takes the realizations inputs (e.g., Eq. (6). we first need to study the distribution of |55 |87] conditioning on the equal sign constraint: sign( sign(37) (37).
We first define the event to represent sign(83) = sign(37). Then the events would have two cases: 1. both terms are positive; 2. both terms are negative. Under Ho, since SJ and 3 are independent and symmetric around zero. Therefore:
EEL PHo (3 > 0,8J > 0) + PHo (3 0, < 0) = 0.25 + 0.25 0.5.
Under H1, the other hand, we have:
РE 0.5 Pf + 0.5 Pe|==-6:
where
[ 1- + (-)
PEpp== = Pai
+
1 ①
02
with & being the cumulative density function of the standard normal distribution. Combining the two equations above, we obtain:
01 [1 の2
PEIH =
+
After characterizing the probability of the equal sign event, consider two conditional density functions under Ho and H1, respectively. Specifically under Ho:
fiet 1,87 HI, (a, (3 0, >
0) SHo(a, b) PHo (85 < <0,55 < -HH (-a; -b)
PEZHO
where Jno ( = foz (3:: ( (8,
Similarly under H1, we have:
1 1071117111 (a,b) P++ S++ (4,a))) P S (a,
PEWH,
where:
f+, = foz(0;
(a;
for (b; 6, o22
f (a, fot (a; -5,
foz (b; 5, 0,0
1
8
+ Overall, the likelihood ratio between H, and Ho can be represented as:
P4+ - P_
Φ
+
2
02
A(a, b) $18 1185 HE. (a,
f/s, 1118, 11, (a,
U. [P+ exp(- SS) + P_ exp( exp(-Sa S+)
where
1
0304
U
PEJH
0102
S_ (a (a-0)a 8)2
(b 8)2
6
207
20
20
20
S+ (a + 8)3
(b 8)2
d
62
20
207
20
20
P+ +
0
02
P_ =
19

---

In addition, we let 01 = 03 and 02 = 01 leading to:
1
U =
PEIB
S_ 8(6 2a)
8(6 26)
20
20
8(5 + 2a)
5(5 + 2b)
S+
20
20
P+ = +
0a
8
[
6
P_
+
0a
Letting A(a, b) be f(a,b) b) completes the proof, as A(a, b) the UMP test statistic according to the Neyman-Pearson lemma.
B.5 Theorem 3.6
To show that the proposed test statistic sign(ab) (ab) (a, b) (Eq. (3) is better than the Gaussian mirror 44 and data splitting [9] counterparts, we prove with two parts. In the first part, we show that the mirror statisties can vary for different j's. Secondly, we show that the proposed test statistic (Eq. (3) is UMP given j.
B.5.1
Part I
Differing from existing works like Gaussian mirror or data splitting that considers generic distribu- tion for all By under Ho and Hi, we interpret the FDR under Hg from another angle that considers the changes in distribution to 3, across different j's. Namely, given an arbitrary mirror test statis- tic sign(33 (8) 37 (3) ,37)) that depends on the index j, and a general index- cadex-agnostic counterpart (85 (B8 ,97) ), the FDR can be defined as:
(f 3-) B-) < ty)
Σ JE(L....
P())((iin((8) (,1,11 177) <&g Ho.
where P(j) refers to the probability of the presence of the index j. In practice, we do not need to know the specification of this distribution as the proof relies only on the symmetry of the conditional distribution: P(sign(3 B7 (3 87 Ho. Apparently Σye(1....p P(j) = holds. ty is threshold chosen such that the FDR can be controlled by the nominal level q. t adds the dependence on the index j. In previous works, is common to assume identical By in Ho and H₃ to have the same distribution across all j's, leading to the fact that sign(3) ( .37 < by the same as sign( 8 ff
< tg. However, one considers the selection rule for these approaches (e.g.. Gaussian mirror/data splitting), it is clear that such a universal assumption need not to held. This because to control the FDR, the only necessary property the symmetry of the null distribution. According to Eq. 7 we realize that even in the case where s8gn (5) .87) varies across j's, the overall FDR can still controlled if all of these distributions are symmetric about zero. The only difference is that for each sign(8) 3, ( (,7) the threshold N is different given different distributions. And the proof higher power is straightforward given this distributional flcxibility.
B.5.2 Part II
According to Lemma
3.5
we already showed that the proposed test statistic sign( (7) .7)
Eq. 3 UMP given the setting considered this paper. This means that given any arbitrary if for some FDR level (not necessarily / refers to the general selection rule across all B, 's), we can achieve the highest power compared the test statistics in Gaussian mirror and data splitting. This leads to the conclusion that the overall power is the highest, hence completing the proof.
20

---

Remark B.2. This proof simply reveals fact that under the controlled nominal FDR level q. to find a better test statistic, we only need to maximize the power of individual test statistic for every 33 (or, equivalently, x)), rather than general form of test statistic across all [P....,P).
To the best of our knowledge, we are the first to provide this reasoning in the proof and we hope this brings insights into developing the UMP test statistics, which is beyond the scope of this paper.
C Complexity Discussion between Benchmarking Methods
The complexity of the Gaussian Mirror method is Onpp² + p²) (for n > p). Essentially it runs ordinary least square fit (OLS), each of which has (mpp + p²) complexity (n p). The computational complexity of Algorithm 2 is O (npp² + y² + ppk) (for > p), where the (ppii) part is introduced by the k-means (following Lloyd's algorithm). k is the number of clusters and i stands for the number of iterations until convergence. In comparison, the data splitting runs two OLS, resulting in O(np² + p²) complexity. The knockoff framework, according to Askari et al. [3], requires at least (pp*s) 5 to solve for the knockoff variable a semi-definite programming setting. Later on, it requires one OLS for 2p dimension, resulting in Ogp + 4np² + 8y²) complexity. Other deep-learning baaseing-based knockoff variables require additional deep learning model fitting to obtain the knockoff statistics, making the complexity amalysis hard given the choice of optimizer and the model architecture, hence ignored. According to Zhong et al. [46], the computational complexity for CRT, assuming = n is Opp" log" p). We did not find any rigorous complexity analysis about HRT and dCRT, however, based on the results reported in Table 2 of the paper (26). we believe the complexity is between model-X knockoff and CRT.
D Design Matrix Setup for Synthetic Experiment
This section presents the model applied to the design matrix X in the synthetic dataset setup in Section 4.1
Gaussian: We replicate the multivariate normal benchmark described in Romano et al. [30]. Specifically, we sample x ~ _NO0,)), E). where Σ is a d-dimensional covariance matrix with en- tries Σn = --i-il, This autoregressive Gaussian data structure captures strong correlations between adjacent features, with diminishing correlations as the distance between features increases. For our experiments, we consider E (0.5, 0.6, 0.7} to provide additional context on how the change of p affects the feature selection performance, in addition to the chosen 0.6 from Romano et 33 Due to limited space in the main paper, we present results for p 0.5 in Appendix E.I
X Ei T&N Σx), where the proportions of the three Gaussian components are given by (TI, T2, R3) = (0.4, 0.4). The mean vectors μ4 E I are defined as μı 1p 20 (k - 1). where is a a p-dimensional vector of ones. The covariance matrices Σg E px have entries (i,
Gaussian mixture: We utilize Gaussian mixture model for X, represented
k-0.1 and Paase 0.6. Both Gaussian and Gaussian mixture computed as Pk |r- where Pk Prase
implementation can be found in
Copula Models: To evaluate data structures with more complex correlation, we incorporate copula models (32). Specifically, we consider two copula families: Clayton and Joc, each parameterized with a consistent copula parameter of 2. Marginal distributions are selected as either a uniform distribution (via identity transformation) or an exponential distribution with a rate of 1. These copulas are implemented using the PyCop library [5 Essentially, copulas are statistical tools designed to model and simulate complex dependencies among random variables, independently from the shapes of their marginal distributions. Unlike traditional multivariate models that assume linear or Gaussian relationships, copulas allow to construct datasets where variables exhibit non-linear or asymmetric dependencies, better reflecting patterns seen in real-world data. In our study, we use two widely-studied copula families: the Clayton copula, which models strong lower-tail dependence (i.e., variables tend to move together when their values are low), and the Joe copula, which captures strong upper-tail dependence (i.c., variables tend to move together when their values are high). By
21

---

specifying a copula parameter of 2, we control the strength of these dependencies in consistent way across scenarios. For each simulated dataset. the individual variable distributions (marginals) are chosen to be either uniform (via identity transformation) or exponential (rate=1), allowing to assess the robustness of our methods under different data distributions. This setup enables comprehensive evaluation of how our proposed methods perform under diverse and realistic correlation structures.
E
Additional Experiment
This section extends the experiments in the main paper to cover e.g., the high dimensional setting and various noise levels, to further demonstrate the performance of the proposed method. In particular, we consider synthetic experiments and focus on the ridge regression model for the fitting coefficients (see Sec. 4.1 for details).
E.1 Additional Gaussian Synthetic Dataset Result
This section presents benchmarking results for the Gaussian setting complementing Table T with rho 0.5. Results are presented in Table
Table FDR and power with Gaussian design matrix X for = 0.3. Bold entries indicate the case with the highest power given controlled FDR level: 0.1. Blue entries indicate the second best. Red entries indicate FDR> 0.1.
OLS
Ridge
LASSO
Method
FDR
Power
FDR
Power
FDR
Power
CRT 8
0.38
0.93
0.27
0.95
0.17
0.99
Distilled-CHI CRT
19
0.54
0.99
0.34
0.99
0.15
0.98
Data Splitting
9
0.00
0.00
0.13
0.82
0.08
0.70
Data Splittingt
0.06
0.75
0.13
0.82
0.13
0.76
Gaussian Mirror 45
0.10
0.79
0.05
0.77
0.07
0.84
Gaussian Mirrori
45
0.10
0.82
0.05
0.80
0.07
0.87
HRT [40
0,00
0.34
0.00
0.36
0.01
0.36
Powerful Knockoff [37
0.00
0.00
0.07
0.49
0.05
0.77
Powerful Knockoff [37
0.06
0.80
0.09
0.60
0.05
0.80
G (ours)
0.10
0.88
0.07
0.90
0.07
0.94
G G2M (ours)
0.10
0.88
0.07
0.90
0.07
0.94
E.2
High Dimensional Setting
We extend Table 1 and Table with two additional low sample-size settings (i.e., n/p 30/100 and 90/100), considering ridge regression for its better performance compared LASSO.
Notably, given that we already consider a low signal strength setting in both synthetic and semi- synthetic experiments, following the experimental setup in Shen al. (34) (see Sec.44ir the main paper), to balance the reduced sample size with signal strength, we apply a multiplier to boost the signal strength. This is common practice in statistical analysis. Specifically, we use the multiplier /200/ vP. where p 30 100, to ensure fair comparison between the cases and the n/p 11pp=00 20pp200/1000 case in Sec.
Additionally, for Data Splitting, Gaussian Mirror, and the proposed G,M, we consider the non- dagger versions, as we observed that the dagger versions cannot properly control FDR when p > n. The results for the Gaussian setting and the non- non-Gaussian setting are included in Table 6 and 7 respectively.
From Table we observe that the proposed GdM outperforms all other methods by achieving the highest power with controlled FDR. In addition, Table 7 suggests that G²M is generally good feature selection algorithm for having the best or the second best power with controlled FDR. More importantly, in this case, G2 never exceeds the FDR nominal level of 0.1, suggesting its soundness in the proposed theoretical framework.
22

---

Table 6: Extension of Table with high-dimensional settings: FDR / power under two different n/p ratios and two correlation strengths. Red entries indicate FDR failures (> 0.1), Bold entries indicate the highest power among methods controlling FDR I 0.1, and Blue entries indicate the second highest.
n/p = 90/100
n/p = 30/100
Method
p = 0.6
p= p= 0.7
p= 0.6
p= 0.7
CRT
8
0.00 0.15
0.00 / 0.09
0.00
0.00 /.011 0.01
Distilled- Distilled-CRT
19
0.59 0.95
0.59 / 0.92
0.04/0.00
0.09 / 0.09
Gaussian Mirror [45]
0.11 0.555
0.08 0.54
0.01 /0.03
0.00 0.000 0.00
Data Splitting 9
0.08 / 0.53
0.07 / 00.52
0.02 /0.05
0.06/0.00
HRT 40
0.00 / 0.08
/
0.00
0.00/0.00
Powerful Knockoff (3
0.02 / 0.11
0.03 / 0.13
0.00
0.00 0.00)
G²M (ours)
0.06 / 0.58
0.08 / 0.55
0.09 0.09 0..13
0.05 0.11
E.3 Noise Level Variation
To study robustness to noise, we follow Shen et al. [34] and vary the signal strength by scaling 3 as -
p p_ (already in the paper), and _ Below, we report results for the additional Toy7 and T5yn P
10,n'
20y/ settings. All other experimental parameters remain the same. Similarly, ridge regression is considered for all methods. And results for the Gaussian setting and the non-Gaussian setting included Table 3 and respectively.
observe that G beats other methods by having the highest power with controlled FDR-a a case that is similarly revealed in Table (6] Besides, Table allo also suggests a good performance of the proposed G for having out of 10 best performing results (i.e., consider the highest power with controlled FDR). This indicate the outperformance the proposed G²M both Gaussian and non-Gaussian settings.
From Table 8
E.4 Another Real Case Muddy-Breast Cancer Dataset
We conduct an additional real-world case study using the Breast Cancer Wisconsin (Diagnostic) dataset [43], which consists of 569 patient samples. Each sample is labeled with binary diagnostic outcome (malignant vs. benign). accompanied by 30 quantitative features extracted from digitized images of fine-needle aspirate (FNA) of breast mass tissue. Following the same evaluation protocol as our previous case study, we reviewed the biomedical literature to identify features that have been consistently reported as highly indicative of malignant tissue. These serve as the ground truth for our analysis. Based on this reference set, we report in Table X the number of "referenced features are recognized as clinically relevant. Results are in Table I0
features
identified" features selected by each competing method. In total, 22
F Preparation of the RNA Data
We first normalize the raw data X to value range [0, 1] and then standardize it to have zero mean and unit variance. Y is synthesized according to X. We consider the response Y is generated following the expression:
E [m/4]
6 9k
(1)
(1,, (4)
(3)
(5) 122
m/4
(8)
Y
x
9k
X4k-3 + 4/ X4R-2
(4) Lanh
(40) X4R-1 + 5x XAk
(8)
23

---

Table Extension Table 2 with nigh-dimensional settings: FDR and power for each dataset under two sample-size regimes. Red entries indicate FDR> 0.1, Bold for the highest power among methods controlling FDR < 0.1, and Blue for the second highest.
FDR Power
Dataset
Method
n/p 90/100
n/p 30/100
CRT
0.00
0.00 0.00
Distilled-CRI
19
0.10/0.13
0.00/0.00
Gaussian Mirror [45
0.06 / 0.52
0.01 0.001
Data Splitting b
0.05 0.332
0.06/0.07
HRT [40
0,00 0.001
0.00
AR
Deep Knockoff 30
0.77 / 1.00
0.79
DDLK 38
0,25 0.221
0
KnockoffGAN 14
0.02 0.10
0.00
sRMMD 22
0.75 1.00
0.79
DeepDRK
0.01 /.220
0.00/0.00
G (ours)
0.04 0.39
0.04/0.04
CRT S
0.00/0.00
0.00 0.00
Distilled-CRI
19
0.12/
0.08 0.04
Gaussian Mirror
0.03 0.100
0.11
Data Splitting 9
0.20
0.02
HRT (40
0.00
0.00
Copula: Clayton & Exp.
Deep Knockoff 30
0.47
0.77
DDLK (38
0,00/0.00
0.00
KnockoffGANN 14
0,00
0.00
sRMMD 22
0.42 0.81
0.75 0.81
DeeplDRK 34
0.08 / 0.15
0.17 / 0.35
G²M (ours)
/ 0.20
0.02 / 0.03
CRT X
0.00/ 0.00
0.00 0.00
Distilled-CRT
19
0.13 / 0.19
0.07 / 0,07
Gaussian Mirror 45
0.02/ 0.18
0.12 0.18
Data Splitting 9
0.07 0.28
0.04 0.02
HRT 40
0.00
0.00
Copula: Clayton & Gamma
Deep Knockoff 30
0.50
0.78 /
DDLKK (38
0.00/0.00
0.00
KnockofIGAN
14
0.00/0.03
0.00
sRMMD 22
0,52 0.995
/
DeepDRK R4
0.01 0.008
0.13
G (ours)
0.03/ 0.188
0.10/ 0.10
CRT 8
0,00 0.00
0.00 0.00
Distilled-CRF CRT 19
0.03 0.09
0.00
Gaussian Mirror 45
0.05 0.28 0.28
0.01 / 0.02
Data Splitting 9
0.06
0.02 0.01
HRT 40
0.00/0.00
0.00
Copula: Joe & Exp.
Deep Knockoff
0,53
0.77 0.888
DDLK (38
0.00/0.00
0.00
KnockoffGAN 14
0.00
0.00
sRMMID 22
0.42/0.79
0
DeepDRK 34
0.09 / 0.21
0.08 / 0.00
GPM (ours)
0.06 / 0.27
0.02 0.01
CRT S
0.00 0.00
0
Distilled-CRT
19
0.04 0.07
0.00
Gaussian Mirror [55
0.09 0.52
0.01 0.02
Data Splitting 9
0.07 0.28
0.02
HRT 40
0.00/0.00
0.00
Copula: Joe & Gamma
Deep Knockoff 30
0.50 0.50/0.97
0.00
DDLK 38
0.01 /..00
0.00
KnockoftGAN 14
0.06/0.27
0.04
sRMMD
0.48 0.96
0.79
DeepDRK
34
0.05 / 0.30
0.03/ 0.05
G (ours)
0.09 / 0.62
0.05/ 0.05

---

Table 8: Extension of Table with with different noise levels: FDR / power under two additional different P and P 20 Red entries indicate FDR> 0.1; among those with
signal strength parameters 10.7 FDR v 0.1, Bold entries indicate the highest power and Blue entries indicate the second highest.
P
Method
10
20
= 0.6
0.7
0.6
0.7
CRT 8
0.04 / 0.96
0.05 0.88
0.07 0,42
0.09 0.25
Distille-CRRT
19
0.40 1.00
0.40 1.00
0.26 0.89
0.21 0.66
Gaussian Mirror 445
0.09 / 1.00
0.06 / 0.88
1..52
0.06 0.40
Data Splitting 9
0.11 0.811
0.07 0.777
0.03 0..5/
0.08 / 0.45
HRT 40
0.01 0.71
0.02 0.46
0.02/0.13
0.00/0.00
Powerful Knockoff 37
0.08 / 0.78
0.02 0.53
/.05/0.20
0.02 / 0.06
G (ours)
0.07/1.00
0.05 0.93
0.02 0.58
0.02 0.45
where follows the standard normal distribution and the 20 for 20 covariates that sampled uniformly.
G Supplementary Material for the Case Study
G.1
Nominal Metabolite List
In Table II
we include the list of the nominal metabolites curated by 34 for the IBD dataset.
G.2
Additional Results for the IBD Study
Here provide the supplementary information for the experimental results described in Section 4.3 In TTble|12|and 13 we provide the list of identified metabolites by each of the considered models. This table provides additional information for Table 4 in the main paper which only includes metabolite counts due to limited space.
25

---

Table 9: Extension of Table with different noise levels: FDR and power under two signal strength and 20 Red entries indicate FDR> 0.1; Bold for the highest power among methods
regimes 10
controlling FDR & 0.1; and Blue for the second highest.
Dataset
Method
10 (FDR / Power)
20 (FDR Power)
CRT
[
0.31 / 0.40
0.30 / 0.40
Gaussian Mirror 45
0.06 / 1.00
0.07 / 0.62
Data Splitting 9
0.09 / 0.94
0,06 / 0.64
HRT 40
0,01 / 0.39
0.02 / 0.09
Deep Knockoff 30
0.99
0.76/ 1.00
Gaussian Mixture
DDLLK 38
0.79 1.00
0
KnockofIGAN 14
0,21 1.00
0
sRMMD 22
0.60/ 1.00
0.77 / 1.00
DeepDRK 34
0.06 / 0.94
0.15
G²M (ours)
0.06 1.00
0.05 / 0.78
CRT [8]
0.01 0.26
0.02 0.12
Gaussian Mirror
0.05 0.95
0.08 0.46
Data Splitting 9
0.86
0.13 0
HRT (40
0.02 0.588
0.00/0.06
Deep Knockoff 30
0
0,38 0.82
Copula: Clayton & Exp.
DDLK 38
0.49/0.88
0.02 / 0.05
KnockoffGAN 14
0,06
0.05 0.12
sRMMD 22
0.10/
0.37 / 0.81
DeepDRRK 34
0.07 / 0.85
0.17 / 0.51
G²M (ours)
/
0.12
CRT X
0.00 / 0.19
0.00 0.04
Gaussian Mirror 45
0.09 / 1.00
0.05 / 0.69
Data Splitting 9
0.10/0.96
0.09 0.72
HRT [40
0.02/
0.02 0.29
Deep Knockoff
0.07 0.97
0.37 0.994
Copula: Clayton & Gamma
DDLK 38
0.95
0.30
KnockoffGANN 14
0.08 0.91
0.09
sRMMD 22
0.06 / 0.97
0,35 0.94
DeepDRK 34
0.06 0.92 0.92
0.12/
G (ours)
0.07 / 1.00
0.10 0.86
CRT [8]
0.11 / 0.57
0.08 / 0.23
Gaussian Mirror 45
0.05 / 0.76
0.15 0.35
Data Splitting 9
0.09 0.788
0.20 / 0.36
HRT 440
0.01 0.39
0.00
Deep Knockoff 30
0.18/0.91
/
Copula: Joe & Exponential
DDLKK 38
0.25
0.02 0.02
KnockoffGAN
14
0.05 0.05/0.35
0.03 / 0.06
sRMMD 22
0.16/ 0.88
0.40 0.68
DeepDRK
0.74
0,20 /
G (ours)
0.08 / 0.91
0.19 0.43
CRT 8
0.11 0.56
0.10
Gaussian Mirror [45
0.05 0.98
0.08 0,56
Data Splitting g
0.07 0.92
0.10 0.58
HRT 400
0.01 0.911
0.01 0.166
Deep Knockoff 30
0.96/0.96
0.35
Copula: Joe & Gamma
DDLK 38
0.48 0.98/0.9
0.11
KnockoffGAAN 14
0.05 0.77
0.07 0.30
sRMMD 22
0.06 0.95
0.95
0.38 0.92
DeePDRK
34
0.05 0.86
G (ours)
0.09 / 1.00
0.08 / 0.68
26

---

Table 10: Feature selection performance the Breast Cancer Wisconsin (Diagnostic) dataset 43 Since the true support is unknown, we report "number of literature-referenced features number of identified features" instead of FDR power.
GMM (ours)t
Distilld-CCRT
Mirrorf
Splitingt
Knockoff
Referenced/ Identified
18/21
5/6
12/15
18/26
11/15
2/4
21/26
Model
DocDRR
Deep Knockoff
aRMMD
KneckoffGAN
DDLK
Referenced/ Idletifiied
15/20
11/15
13/18
17/22
8/13
Table 11: IBD-associated metabolites that are supported by the literature. This table includes 47 referenced metabolites for the IBD case study. Each metabolite is supported by one of the three sources: PubChem, peer-reviewed publications, or preprints. For PubChem case, we report the PubChem reference ID (CID), and for the other two cases, report the publication references.
Reference Type
Metabolite
Source
Metabolite
Source
palmitate
CID: 985
taurccholate
CID: 6675
cholate
CID: 221493
p-hydroosyhhenllacctte
CID: 127
linoleate
5280450
deoxycholate
222528
murrchenodeexycchlate
CID: 387316
12.13-diHOME
dodocanedioate
arachidonate
eicosatrienoate
eicosadienoate
docosapentaenonte
3
taurolithocholate
salicylate
saccharin
oleate
arachidate
glycocholate
chenodeoxychollte
phenyllactate
22]
10
glycolithocholate
5
Publications
urobilin
223
caproate
hydrocinnamate
22:
myristate
adrenate
23
20
olmesartan
letradecanedioate
hexaddeaanddionte
oxypurinol
porphobilinoeen
caprate
36
undecanedionate
stearate
oleanate
elyccchendeoxycchlate
33
sebacate
nervonic
lichocholate
aipha-murieholate
Preprints
17-methylstearnte
25
myristoleute
25
taurodeoxycholate
ketodeoxycholate
27

---

Table 12: A list of literature- supported metabolites out of a total of 80 candidates. indicates the important metabolites marked by the corresponding algorithms.
Metabolite
Distiled-CRT
Gaussian Mirrei
Data Spliiting
HRT
Powerful Knockolf
12. 1213--H
9. 4.10-HHOME
capmoate
hydrocinnamate
mandelate
3-dydraxyactanoate
indoleacetate
dodecamestz
undecanetionate
myristoleare
dodecanedioate
pentadocamoute
hydroxymyristiate
palmitoleate
palmitati
10 hehheaaccoenrtte
alpha
linoleate
oleate
hexddecanedioate
10 monadcemoate
nonallccnnniic
17 7--methylstearal
elcosaeentenooee
arachidonate
cicosarriencate
cicosadioncate
arachidate
phytanate
decossbesseerotte
dncosspentacnonnie
adrenate
13-docosssoate
eicosanedioate
olcanate
masilinnt
lithcholate
chenodooxychulate
deoxycholiic
ketodeoxychoolate
alpha courchoolate
cholate
elycolihhocholtte
elyeodeosychholle
elycoursdeeuscchoosee
elycocholate
aurolithwchollat
murrcchnnsdossychhe
taurodeonychollaee
tauro alpha muricholate/fauro beta muricholate
taurocholate
salicylatic
azclate
sebacate
carbxyyitugnofen
d.3-dicarboxylleee
d-hydroxystyrene
acety-yrosiie
alpha-CEHHC
oxypurinol
palnittylllaam/amiiee
pheny Ractate
j-hydrosyphenglaeetee
porphabilinogen
urobilin
nervonic acid
oxymstacoline
28

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Table 13: A continued list literature- supportte metabolites out of total of 80 candidates to "*" indicates the important metabolites marked by the corresponding algorithms.
Table 12
Metabolite
DeepDRK
Deep Knockoff
3RMD
KnockoRGAN
DDLK
12.13-diHOMM
9.10-iHHMME
cuproate
hydrocinnumate
mandelate
b-bydroxyoctannute
capratc
indoleacetate
3-by --yydroyydeaaneee
dodecamoute
undecanedionate
myristolestc
myristate
dodecanedionte
pentadecamoatee
by yydrixynyiisttte
palmitoleate
paliiitte
cetradccasdiiotte
2-bydroxy hexadleannotle
alphu limolenate
linoleate
oleate
sicarate
bexadleanedicate
10 nonadacnontc
nonadecanoaee
methy Isteurate
dicosupentaetoate
arachidonate
cicosatricnonte
cicusadienoate
cicosenoate
arachidate
phytanate
docosupentaenoute
adrenate
13-decosencate
dicosanedioate
oleamate
masilinate
litocchlltte
chenodeoxycholate
dcoxycholate
kctodcoxycholaat
alphu maricholate
cholate
elycolithocholate
hynnchenadeoxyccolltte
dycodeoxycbolate
glycocholate
aarooiihochhllaee
taurochenodeouycclll
taurodeosy yccholllccclttte
tauro-alphat huu---llpae-nuiccchttettttr---e- hat.aaaccctttuo---t-teencchhhhee
taurocholate
salicylate
saceharin
azelate
carboxyiboprofon
olmesartan
1.2.3.4
wewteeayyrrobbbttaceeoo 1iw.rroooiiin.--iiiarrrrllla dicarboxylate
4-bydroxystyrene
acelytyrosine
alpha-CEHC
carnosal
oxypurinol
pulmiloyeethanlamidee
phenyllisctate
porphohilinogen
urobilio
oxymctazoline
29

---

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36


Reviews:
[{"review_id": "4U2GOExs93", "reviewer": "Reviewer_h9tx", "summary": "The paper identifies the limitation of existing mirror statistics and presents a Generalized Gaussian Mirror method to enhance feature selection. The method is compared with multiple FDR related methods across different datasets (synthetic and real world), which shows efficacy in controlling FDR while achieving high power.", "questions": "1.It would be good to add rationale on why k-means is selected in estimation algorithm. \n\n2.The paper references paper 'Shen et al [32]' quite a lot, regarding problem set up, method, experiment setting, and result discussion. It would be good to add more discussion and clarifications on what's the novelty/originality of this paper compared with existing 'Shen et al [32]'\n\n3.some discussion on the method complexity would be good and on how it compares with other methods in terms of computational complexity.", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:   \n+ The related work covered sufficient works with proper discussion linking all related works, and motivating the proposed method.\n+ Method is evaluated on multiple datasets, both synthetic and real world data, and compared with state-of-the-art methods. \n+ Problem is well motivated with clear background and rationale. \n+ The paper includes detailed proofs for each part of the method.\n\nWeaknesses:   \n- The paper address the limitation of existing mirror statistics, novelty is not that significant. \n- The experiment part can be enhanced with some significance test on the result comparing with other methods. \n- It's good that the paper points out the limitation of assuming normal distribution on the fitting coefficients, however, the limitation is still there. It would be good to discuss more on how this normal distribution can be lifted/resolved.", "was_revised": true, "final_justification": "I appreciate the authors' response on the clarifications on the questions, and provided additional comments justifying paper novelty. Based on the rebuttal comments, i have updated my score in clarity and originality section. \nthanks.", "rebuttal": ""}, {"review_id": "elwVe5jD6h", "reviewer": "Reviewer_tMF4", "summary": "This paper addresses the task of generating realistic and diverse human motions from natural language descriptions, with a focus on improving generalization across different datasets and motion styles. The authors propose a novel method called MotionAuto, a diffusion-based model that separates the motion generation process into two stages:\n\n1. Text-to-Motion Representation (T2MRep) Generation: Given a text input, the model generates a latent motion representation using a transformer-based architecture with diffusion sampling. This stage learns a distribution over possible motion intents, grounded in the textual description.\n\n2. Motion Decoding: The generated T2MRep is then passed to a motion decoder that reconstructs a full motion sequence using a generalizable motion prior.\n\nA key contribution of the paper is the Generalizable Motion Prior (GMP)—a modular and transferable motion representation trained to model the manifold of realistic human motions independent of specific text annotations or datasets. By disentangling motion prior learning from language grounding, GMP enables zero-shot and few-shot transfer to new datasets and tasks.\n\nThe model is evaluated across multiple datasets (HumanML3D, KIT-ML, and BABEL) and shows superior performance in both standard and cross-dataset generalization settings, with improved diversity, realism, and text-motion alignment compared to strong baselines. Qualitative examples and ablations further support the effectiveness of the modular design and latent representation.", "questions": "1. What is the structure and semantics of the T2MRep latent space?\n\nQuestion: Can the authors clarify what kind of information the Text-to-Motion Representation (T2MRep) encodes? Visualizations (e.g., t-SNE or PCA plots) or clustering analyses could help reveal whether semantically similar motions occupy nearby regions in this space. A better understanding of T2MRep's structure would support its interpretability and clarify why diffusion in this space works well.\n\n2. How transferable is the Generalizable Motion Prior (GMP) across domains or tasks?\n\nQuestion: Beyond cross-dataset generalization, have the authors tested GMP in new tasks (e.g., action classification or robotic control) or motion styles (e.g., stylized, multi-agent)? Even preliminary results or discussion about potential use cases would strengthen the claim of “generalizability.” Demonstrating use of GMP outside the training regime would increase confidence in its utility and modularity.\n\n3. What are the failure modes of the proposed system?\n\nQuestion: In what scenarios does MotionAuto fail to generate meaningful or aligned motions (e.g., under ambiguous, long, or compound text prompts)? Including a qualitative analysis or examples of such failures can help guide future improvements and clarify the method’s limits. If failure cases are rare or easily explained, this would reinforce the system's robustness.\n\n4. How does the method perform under low-resource settings?\n\nQuestion: Have the authors measured how MotionAuto performs when fine-tuned or trained on smaller datasets, especially when GMP is reused? A few-shot or low-data experiment would be helpful to demonstrate whether GMP meaningfully reduces the data burden. Positive results here would further validate one of the paper's key motivations: generalization with limited supervision.\n\n5. What is the computational efficiency compared to baselines?\n\nQuestion: What are the training and inference time comparisons between MotionAuto and other leading methods like MDM or MotionDiffuse? Including runtime or memory usage benchmarks (even briefly in the appendix) would give a more complete picture of the model’s practicality. If the proposed method is competitive in terms of resources, it would increase its appeal for broader deployment.", "limitations": "While the technical contributions are well-developed, the paper does not sufficiently address potential limitations or negative societal impacts of text-to-motion generation. Here are some suggestions for improvement:\n\n- Ethical Concerns: Consider discussing risks such as misuse in surveillance, deepfake-style animations, or biased motion generation (e.g., due to dataset imbalances).\n\n- Representation Bias: Acknowledge if datasets like HumanML3D or KIT-ML might overrepresent specific motion types, body types, or cultures, which could affect generalizability and fairness.\n\n- Physical Safety: For applications in robotics or avatar control, incorrect motion generation could cause physical harm or user confusion. This should be mentioned as a risk.\n\n- Failure Mode Transparency: Limitations in generating rare, multi-agent, or highly dynamic motion types are not explicitly addressed and would be important for understanding the boundaries of the method’s reliability.\n\nIncluding even a brief discussion in the main paper or appendix would enhance transparency and responsibility.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "-Strengths\n\n--Quality\n\nSound technical approach: The proposed two-stage architecture, combining text-guided latent representation generation and motion decoding via a generalizable prior, is methodologically well-motivated and executed.\n\nStrong empirical performance: The model shows consistent improvements across standard and cross-dataset evaluations (HumanML3D, KIT-ML, BABEL). Metrics indicate better realism, diversity, and text-motion alignment.\n\nComprehensive evaluation: The authors provide detailed ablation studies and visualizations to justify the design choices. Both quantitative and qualitative results support the claims.\n\n--Clarity\n\nClear presentation: The paper is well-written and logically structured. Figures and diagrams (especially the model overview on page 3) effectively illustrate the pipeline and components.\n\nDefined terminology: Concepts such as T2MRep and GMP are clearly introduced and consistently used.\n\n--Significance\n\nAddresses a key limitation: The work tackles generalization in text-to-motion models—a major hurdle in practical deployment and cross-domain usage.\n\nPractical utility: By decoupling motion learning from text supervision, the proposed framework lowers the annotation burden and increases reusability of pretrained components.\n\n--Originality\n\nNovel modular design: The introduction of GMP as a reusable, disentangled motion prior is a creative and impactful contribution.\n\nDiffusion in latent space: Applying diffusion to a motion-specific latent space (rather than directly to poses or joint angles) adds a fresh perspective on modeling uncertainty in motion generation.\n\n-Weaknesses\n\n--Quality\n\nLimited analysis of failure cases: While success is well-demonstrated, the paper would benefit from a deeper exploration of when or why the model fails (e.g., unusual text prompts or out-of-distribution motion types).\n\nFew details on computational cost: The runtime or training/inference efficiency is not thoroughly reported or compared to baselines, which may matter for real-world adoption.\n\n--Clarity\n\nLatent space interpretability: The nature and structure of the T2MRep and GMP representations could be more explicitly explained or visualized. What semantics, if any, do these latent variables capture?\n\n--Significance\n\nLimited modality diversity: All experiments are in the domain of skeleton-based 3D human motion. It remains unclear how well the approach generalizes to other data modalities (e.g., 2D video motion, gestures, or robotic trajectories).\n\n--Originality\n\nBuilding on recent trends: While the combination is novel, many components—diffusion models, latent space modeling, and modular priors—build upon recent work in motion synthesis and text-to-motion generation. The originality lies in the effective integration and the use of GMP, rather than in new fundamental techniques.", "was_revised": false, "final_justification": "", "rebuttal": ""}, {"review_id": "pRXPTZBRLv", "reviewer": "Reviewer_Y37A", "summary": "This paper tackles the problem of variable selection under false discovery rate (FDR) control for linear regression models. The authors propose a generalized Gaussian mirror ($\\text{G}^2$M) test statistics to study the feature selection problem. They established rigorous theoretical guarantees for the $\\text{G}^2$M statistics, including the entry-wise UMP properties and the advantage over existing statistics. Algorithms are proposed to handle both the case where one has access to true nonnull $\\beta_j$'s and the estimation in practice. Substantial empirical simulations are then conducted based on the estimation algorithm, which shows the advantages of the $\\text{G}^2$M statistics in terms of testing power and FDR control.", "questions": "1.  In section 3.1, the selection of $\\rho=\\{0.6,0.7\\}$ needs some further justification.\n  \n2. The simulation using the Copula model on page 6 needs additional clarification, given that the reader may not possess prior knowledge of this methodology.\n\nTypos: \n\n1. Page 7, line 214: Eq. (3) cannot be found in the main file (assuming it is the equation between line 215 and 216). \n2. Page 9, line 323: 'An possible' should be 'A possible'", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths: \n\n1.  The paper provided a good motivation for why one might care to consider using the $\\text{G}^2$M statistics for the variable selection problem under false discovery rate (FDR) control. \n \n2. The theoretical foundation is strong, with clear lemmas and theorems providing a closed-form $\\text{G}^2$M statistics that is UMP and is more powerful over existing statistics including the Gaussian mirror and the data-splitting test statistics.\n  \n3. The methodology provides greater empirical power than multiple benchmarking methods in both the synthetic and the semi-synthetic simulation settings.\n\n\n\n\n\nWeaknesses: \n\n1. The paper does not discuss whether the proposed $\\text{G}^2$M test statistics have an intuitive interpretation, which may be of interest in the case of real-world data.\n \n2. There is still some ambiguity concerning the difference between $\\text{G}^2$M and $\\text{G}^2\\text{M}\\dagger$. Given that the highest statistical power is predominantly achieved using $\\text{G}^2\\text{M}\\dagger$ instead of $\\text{G}^2$M, further justification is required to substantiate the selection of the $\\tau_q$ in Eq.3 (assuming it is the equation between line 215 and 216) as the optimal threshold.\n \n3. It would be desirable to have at least one more real case study in section 3.3 as the current example does not sufficiently illustrate the comparative advantages of the proposed method over existing approaches.", "was_revised": true, "final_justification": "I would like to thank the review for the carefully-written rebuttal, and also more numerical results. I am satisfied with the response, and I will keep my rating.", "rebuttal": "Thank you for your time and feedback, and for assessing that the paper has a strong theoretical foundation and the methodology provides greater empirical power compared to benchmarks. \n\n\n***QA1: The paper does not discuss whether the proposed test statistics have an intuitive interpretation, which may be of interest in the case of real-world data.***\n> Essentially, the proposed test statistics is mainly the result of a variance extension of the mirror statistics. We have included in Appendix B why it is necessary to consider a variance-generalized setting. Primarily, this is because we observed that even with the simplest Gaussian simulation setting (see Appendix B), the variance of the mirror statistics is not 1. When data is more complex (e.g., the real-world data), it is expected that the unit-variance setting will be violated. In this case, the original Gaussian Mirror/ Data splitting setting that depends on the unit-variance assumption is likely to fail. In QA3 below, we have also included another real-world dataset and demonstrate the gains of the proposed statistics from the feature selection perspective.\n\n***QA2: There is still some ambiguity concerning the difference between G$^2$M and G$^2$M$^\\dagger$. Given that the highest statistical power is predominantly achieved using G$^2$M$^\\dagger$instead of G$^2$M, further justification is required to substantiate the selection of the $\\tau_q$ in Eq.3 (assuming it is the equation between lines 215 and 216) as the optimal threshold.***\n> Essentially, the $\\dagger$ version and the regular version differ only by the selection rule of $\\tau_q$. The regular G$^2$M utilizes Eq. (3), and the G$^2$M$^\\dagger$ considers the equation between lines 215 and 216. And according to [1], the two equations differ only when\n$\n\\min_{t > 0} \\lbrace t : \\frac{1 + \\vert \\lbrace j : w_j \\le -t \\rbrace \\vert}{\\max(1, \\vert \\lbrace j : w_j \\ge t \\rbrace \\vert)} \\le q \\rbrace > \\min_{t > 0} \\lbrace t : \\sum_{j \\in [p]} \\mathbb{1} \\lbrace j : w_j \\ge t \\rbrace < \\frac{1}{q} \\rbrace$,\nwhere $\\mathbb{1}$ denotes the indicator function.\n\n>This means when $q$ is small or the fraction of non-nulls is small, then the first term usually finds a larger $\\tau_q$, in favor of the FDR control. In this case, the power is pretty low, as when $\\tau_q$ is large, there are only limited non-nulls selected. Instead of choosing $\\tau_q$ with the first term (e.g., Eq. 3), the second term chooses a smaller $\\tau_q$, which results in higher power. We will elaborate on the above in the revised manuscript near lines 215 and 216, where the new selection rule is introduced. In the revised manuscript, we will add an equation number to $\\tau_q$ between lines 215 and 216. \n\n***QA3: It would be desirable to have at least one more real case study in section 3.3, as the current example does not sufficiently illustrate the comparative advantages of the proposed method over existing approaches.***\n> We have prepared another real-world case study that focuses on a breast cancer dataset [2] that has 569 samples. It contains the binary label (i.e., cancer v.s. normal) as the dependent variable and 30 features as the independent variable. We have detailed the description of the data in the revised manuscript in Section 3.3.\n\n>Following a similar setting for the real case study, we investigated the literature and identified the features that have been proven to have high relevance to the cancer tissue. Using this as the ground truth, we report the “referenced/identified” features in the following table. In total, there are 22 relevant features.\n\n| Model                 | G²M (ours)† | CRT    | Distilled-CRT | Gaussian Mirror† | Data Splitting† | HRT   | Powerful Knockoff |\n|-----------------------|:-----------:|:------:|:-------------:|:----------------:|:---------------:|:-----:|:-----------------:|\n| Referenced / Identified | 18/21       | 5/6  | 12/15         | 18/26             | 11/15          |  2/4   | 21/26               |\n\n| Model                 | DeepDRK     | Deep Knockoff | sRMMD  | KnockoffGAN  | DDLK  |\n|-----------------------|:-----------:|:-------------:|:------:|:------------:|:-----:|\n| Referenced / Identified | 15/20       | 11/15         | 13/18    | 17/22        | 8/13 |\n\n>It is clear that G$^2$M maintains the lowest false discovery rate, as there are only 3 missing. Meanwhile, it obtains 18 correctly identified features that are referenced, suggesting its good performance in real-world data.\n\n>The experimental setup is identical to the existing real case study; the only change is the dataset. And such information, including the results, is reflected in the revised manuscript. The selected features for each of the models are also explicitly displayed in the Appendix in the revised manuscript. We will add the results in the revised manuscript. \n\n\n***QA4: In section 3.1, the selection of  $\\rho$ needs some further justification.***\n> We follow [3] to consider the $\\rho=0.6$ setting. However, we also want to provide additional context on how the change of $\\rho$ affects the feature selection performance. Hence, the $\\rho=0.7$ setting is considered. We will add the above explanation in line 198 about this choice. \n\n***QA5: The simulation using the Copula model on page 6 needs additional clarification, given that the reader may not possess prior knowledge of this methodology.***\n>  We will include a new section in the Appendix in the revised manuscript to include the description of the Copulas. Essentially, Copulas are statistical tools designed to model and simulate complex dependencies among random variables, independently from the shapes of their marginal distributions. Unlike traditional multivariate models that assume linear or Gaussian relationships, copulas allow us to construct datasets where variables exhibit non-linear or asymmetric dependencies, better reflecting patterns seen in real-world data. In our study, we use two widely-studied copula families: the Clayton copula, which models strong lower-tail dependence (i.e., variables tend to move together when their values are low), and the Joe copula, which captures strong upper-tail dependence (i.e., variables tend to move together when their values are high). By specifying a copula parameter of 2, we control the strength of these dependencies in a consistent way across scenarios. For each simulated dataset, the individual variable distributions (marginals) are chosen to be either uniform (via identity transformation) or exponential (rate=1), allowing us to assess the robustness of our methods under different data distributions. This setup, implemented using the PyCop library, enables a comprehensive evaluation of how our proposed methods perform under diverse and realistic correlation structures.\n\n\n***QA6: Typos***\n> Thanks for pointing out the typos. We will correct them in the revised manuscript. \n\n**References** \n\n[1] Ren, Z., & Barber, R. F. (2024). Derandomised knockoffs: leveraging e-values for false discovery rate control. Journal of the Royal Statistical Society Series B: Statistical Methodology, 86(1), 122-154.\n\n[2] https://archive.ics.uci.edu/dataset/17/breast+cancer+wisconsin+diagnostic\n\n[3] Sudarshan, M., Tansey, W., & Ranganath, R. (2020). Deep direct likelihood knockoffs. Advances in neural information processing systems, 33, 5036-5046."}, {"review_id": "nWuf91UGFe", "reviewer": "Reviewer_TuyV", "summary": "This paper proposes a generalized gaussian mirror method for feature selection, which addresses the unit variance constraint on the mirror test statistics. Two algorithms are proposed, one for known coefficient scale and the other for unknown. The authors show theoretically that the proposed G2M method is more powerful than existing methods. Extensive numerical validates their theoretical claims and show superiority of the G2M method.", "questions": "1. Since the authors assume a linear model between X and Y, is G2M valid for general regression models? Or is it possible to extend G2M to general models?\n2. The coefficients $\\beta^+$ and $\\beta^-$ are used without clear definitions. I have noticed a rough definition in the appendix but they are frequently used in the methodology part. It would be better to move their definitions to the main text before their usage.\n3. Although the method is based on existing works that ensure FDR control, it would be better to explicitly formulate this property in a short theorem. It might be a straightforward result or even only a proposition, but showing this issue is worthy since there are also many intuitive methods for feature selection which has no error rate control guarantee.\n\n\nI will be willing to raise my score if the paper is reformulated appropriately.", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 1, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strength:\n\n1. The theoretical claims are concise and formulated in an easy-to-follow way.\n2. The authors have conducted extensive and detailed numerical experiments to validate their theoretical claims and the superiority of their proposed G2M.\n\nWeakness:\n\n1. With a rich background of the feature selection problem and extensive existing methods mentioned by the authors, the paper does not formulate or describe any existing method in the main text. For readers who are not familiar with the field, it could be quite difficult to follow and even more difficult to figure out the contribution made by the authors.\n2. In comparison, the details for simulation settings and implementation take too much space. I believe it is the main reason for weakness 1 since the page limit is 9. I strongly recommend the authors to delay unnecessay details of the experiments to the appendix and save more space for the introduction part.", "was_revised": true, "final_justification": "Most of my questions are about the formulation and writing of the paper. The authors have discussed the related parts and provided plans of how to reformulate. The paper will be acceptable if all these plans are put into effect.", "rebuttal": "Thank you for your time and feedback, and for assessing that the theoretical claims are easy to follow and numerical experiments are extensive and detailed. \n\n***QB1: With a rich background of the feature selection problem and extensive existing methods mentioned by the authors, the paper does not formulate or describe any existing method in the main text. For readers who are not familiar with the field, it could be quite difficult to follow and even more difficult to figure out the contribution made by the authors.***\n> We put the two related works– the Gaussian mirror and data splitting methods in Appendix A. And this was primarily due to the page limit. To improve clarity, we will move Appendix A to the main paper in the revised manuscript..\n\n***QB2: In comparison, the details for simulation settings and implementation take up too much space. I believe it is the main reason for weakness 1 since the page limit is 9. I strongly recommend that the authors delay unnecessary details of the experiments to the appendix and save more space for the introduction part.***\n> Thank you for the suggestion. We will shorten the description of each of the datasets used in the experiment section in the revised manuscript and also move the full description to a separate section in the Appendix to balance the introduction of the related work section according to QB1.\n\n***QB3: Since the authors assume a linear model between X and Y, is G$^2$M valid for general regression models? Or is it possible to extend G$^2$M to general models?***\n> It is possible to extend G$^2$M to the generalized linear model setting. For example, Reference [1] discusses the generalized linear model setting with Gaussian mirror statistics. We believe there is a similar extension for G$^2$M. However, given that the form of test statistics in our paper is significantly different from the original Gaussian mirror, the extension of G$^2$M to this case is non-trivial. And this is deferred to future work. We will add the above discussion in Section 4 of the revised manuscript.\n\n***QB4: The coefficients  $\\beta^+$ and $\\beta^-$ are used without clear definitions. I have noticed a rough definition in the appendix, but they are frequently used in the methodology part. It would be better to move their definitions to the main text before their usage.***\n> According to the response to QB1, we moved the related work section from Appendix A to the main paper in the revised manuscript. This should provide enough context on how the coefficients $\\beta^+$ and $\\beta^-$ are generated from the data splitting and the Gaussian mirror context. Essentially, the $\\beta^+$ and $\\beta^-$ used in G$^2$M follow the same generation process in a variance generalized setting. We will explicitly mention this connection in the updated related work section for the Gaussian mirror in the revised manuscript. \n\n***QB5: Although the method is based on existing works that ensure FDR control, it would be better to explicitly formulate this property in a short theorem. It might be a straightforward result or even only a proposition, but showing this issue is worthwhile since there are also many intuitive methods for feature selection that have no error rate control guarantee.***\n> Thanks for the suggestion. In the revised manuscript, we will state the theorem on FDR control for data splitting in the revised manuscript. Specifically. We state:\n>\n>**Theorem 1**: *For a given set of mirror statistics $w_j$, $j \\in [p]$, if mirror statistics satisfy the following property:*\n> *1. If $j \\in S_0$ is a null feature index, then given any real number $t \\ge 0$, $p(w_j \\ge t) = p(w_j \\le -t) = 0.5$.*\n> *2. If $j \\in S_1$ is a non-null feature index, then $p(sign(w_j) = 1) = 1$.*\n>\n>*Then given a nominal FDR level $q$, the feature selection set $\\lbrace j \\vert w_j \\ge \\tau_q\\rbrace$ controls FDR, where $\\tau_q$ is chosen via Eq.(3).*\n\n>The proof idea is already informally presented in lines 469-473, and we will provide the following proof for this Theorem in the revised manuscript.  \n\n>*Proof*:\n>Given the above two properties for the mirror statistics, we realize that $\\vert \\lbrace j: w_j \\le -t, j\\in S_0\\rbrace \\vert = \\vert \\lbrace j: w_j \\ge -t, j\\in S_0\\rbrace \\vert$ due to the symmetric-about-zero property for the mirror statistics $w_j$ under the null distribution. As a result, the selection rule $\\hat{S}_1 = \\\\lbrace w_j \\ge \\tau_q\\rbrace$ (e.g., the denominator of Eq. (3)) selects features based on the level-$q$ quantile $\\tau_q$ on the right side of the null distribution, ensuring the FDR level to be at most $q$.\n\n**References**\n\n[1] Dai, C., Lin, B., Xing, X., & Liu, J. S. (2023). A scale-free approach for false discovery rate control in generalized linear models. Journal of the American Statistical Association, 118(543), 1551-1565."}]

Author rebuttal:

Meta-review: The paper proposes a generalized Gaussian mirror test statistics for feature selection under false discovery rate (FDR) control for linear regression models.
The experimental results of the proposed method demonstrate significant improvements in power while still maintaining the nominal FDR level compared to the baselines.
Through discussions with the reviewers, the authors clarified their novelty, provided the results on more datasets, and promised to improve the clarity by moving some of the appendices to the main body.
Overall, the reviewers agreed that the paper made a solid contribution for feature selection under FDR control.
Decision: Accept (poster)