Independence Test for Linear Non-Gaussian Data and Applications in Causal Discovery
Authors: Yiqing Li
Xiaofei Wang
Boyang Sun
Yewei Xia
Kun Zhang
Venue: ICLR 2026 Poster (2026)
Keywords: independence tests
Causal discovery
Primary area: causal reasoning

Abstract: Independence testing involves determining whether two variables are independent based on observed samples, which is a fundamental problem in statistics and machine learning. Existing testing methods, such as HSIC, can theoretically detect broad forms of dependence, but may sacrifice statistical power when applied to limited samples with background knowledge of the distribution. In this paper, we focus on the linear non-Gaussian data, a widely supported model in scientific data analysis and causal discovery, where variables are linked linearly with noise terms that are non-Gaussian distributed. We provide a new theoretical characterization of independence in this case, showing that constancy of the conditional mean and variance is sufficient to guarantee independence under linear non-Gaussian models. 
Building on this result, we develop a kernel-based testing framework with provable asymptotic guarantees. Extensive experiments on synthetic and real-world datasets demonstrate that our method achieves higher power than existing approaches and significantly improves downstream causal discovery performance.

Published as a conference paper at ICLR 2026
INDEPENDENCE TEST FOR LINEAR NON-GAUSSIAN DATA AND APPLICATIONS IN CAUSAL DISCOVERY
Yiqing Li" Xiaofei Wang 1.2
Boyang Sun'
Yewei Xia²
Kun Zhang! 1.4
Mohamed bin Zayed University of Artificial Intelligence 3 Fudan University
2Northeast Normal University
*Carnegie Mellon University
ABSTRACT
Independence testing involves determining whether two variables are independent based observed samples, which is fundamental problem in causal discov- ery. Existing testing methods, such as HSIC, can theoretically detect broad forms of dependence, but may sacrifice statistical power when applied to limited sam- ples with background knowledge of the distribution. In this paper, we focus on the linear non-Gaussian data, a widely supported model in scientific data analy- sis and causal discovery, where variables are linked linearly with noise terms that are non- Gau-Sausiinn distributed. We provide new theoretical characterization of independence in this case, showing that constancy of the conditional mean and variance is sufficient to guarantee independence under linear Gau-Gassii mod- els. Building this result, we develop kernel-brrt -bsee testing framework with provable asymptotic guarantees. Extensive experiments on synthetic and real- world datasets demonstrate that method achieves higher power than existing approaches and significantly improves downstream causal discovery performance.
INTRODUCTION
Testing for statistical independence, i.e., deciding whether variables are independent from ob- served samples, is fundamental in machine learning applications. such as in self-supervised repre- sentation learning (Li et al.] (021). feature selection (Candes al.] 2018), and in causal discov- ery (Spirtes et al., 2000p. Over the decades, rich toolbox of independence tests has emerged targeting different scenarios. Classical parametric methods include Pearson's correlation, Spear- man's rank correlation, and Kendall's tau, etc. Recent advance focuses on nonparametric methods like HSIC (Gretton et al. 2005a) and dCor (Székely et al. 2007). In this paper, we aim to test the independence between linear mixtures of independent non-Gaussian components. This test is rather important for causal discovery in linear non-Gaussian models both with and without latent variables.
The assumption of linear model is prevalent in causal discovery algorithms. Without structural as- sumptions on the data generation process, the causal direction is not identifiable from observational data (Spirtes al. 2000) |Peall 2009). Following linear non-Gaussian acyclic model (LiNGAM), the Direct-LiNGAM algorithm (Shimizu et al.) 2011) suggests an independence test scheme on regressor and residual determine causal directions under the no latent confounders assumption. Furthermore, even when latent confounders are present, the causal direction can still be identified through independence testing. Specifically, the recently proposed generalized independence noise (GIN) (Xie et al.) 2020 2022) condition provides an elegant and efficient way to identify the ex- istence of latent variables and recover the causal orders of the latent variables in LiNGAM model. Verifying the GIN condition involves a significant number of independence tests, strengthening the need for reliable independence test in linear non-Gaussian settings.
Given the proven success of linear non-Gaussian models in solving real-world problems across various domains (Dong et al. 2023), surprisingly, to our knowledge there is no independence test specifically designed for the linear, non-Gaussian regime. In practice, researchers often resort to general- general-purpose purpose paramarait independence tests (e.g., HSIC). While these methods control Type I error, their generality can be liability in our specific context. They are designed for arbitrary dependencies and may lack statistical power for the structured relationships generated by linear non- Gaussian models. This creates methodological mismatch: while we employ identifiable models

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that leverage non-Gaussianity, we rely tests that do not exploit this structure- akin using shotgun to shoot a buttterfyy which is inefficient and potentially ineffective.
A natural and pressing question arises: How can design independence test that is tailored to this well-established model class? By incorporating the model assumptions directly into the testing procedure, we develop a method that is not only statistically more efficient but also conceptually simpler. This paper addresses this exact need. We propose a novel independence test specifically designed for the linear non-Gaussian data. We begin by providing new characteri- zation of independence in this setting. We show that, interestingly. for judging the independence of linear non-Gaussian data, conditional variance. Based on this new characterization, we further designed a statistic that can test the conditions simultaneously. With derived asymptotic distributions, our method leverages the model constraints to achieve higher statistical power than generic alternatives, thereby providing more robust foundation for causal discovery algorithms.
enough check the constancy of the conditional mean and the
We summarize our contributions follows.
We propose a novel characterization of independence for linear mixtures of independent non-Gause Gaus-an components using only the conditional mean and conditional variance.
We propose statistic and derive its corresponding asymptotic distributions to test indepen- dence. We also prove the equivalence of the statistic and the independence characterization.
We conduct extensive experiments on both synthetic and real-world data, which demon- strate the efficacy of our method. In addition, we integrate our testing method into an existing causal discovery algorithm and it outperforms other testing methods.
BACKGROUND
Problem Definition. For random variables X X' and E V/ where X' and are their domains, we say X and Y are independent if Pxy = PxPyy, denoted by X IL Y. Given a dataset 'D = ((),)))) where the pairs are independently and identically sampled from the joint distribution Pxy, an independence test constructs test statistic T based on D to test for hypotheses:
Ho
X
IL Y
V.S.
: M.M.
The statistic T' is then compared with a critical value to decide whether to reject the null hypothesis Ho- The quality of an independence test is typically characterized by two quantities: the probability of incorrectly rejecting Ho when it is true (Type error), and the probability of failing to reject Ho when it is false (Type error). An ideal test maintains the Type error at a user-specified significance level a, while achieving high statistical power (1- Type II error rate).
direct way to check for independence based on the definition. That is, estimate the probability densities of the joint distribution Fxy and the marginal distribution Px Py, and then evaluate if Fxr = Px is satisfied almost surely. For example, mutual information measures the depen- dence strength between two variables using the KI. divergence between Pxxy and P Py. However, estimating the probability densities from finite samples is difficult. Some distributions may even have no densities, which may further deteriorate the testing performance. Instead, (Jacod & Protter 2004) provides an alternative characterization of the independence of random variables.
Lemma 2.1. The random variables X and Y are independent if and only if for each pair (f)g) of bounded, continuous functions, i.e. f E Co(X) and g E Cz()).
=
provides a direct test criterion without the need for intermediate density estimator. However, the space of bounded, continuous functions too rich, which will raise the consistency issue. That is, the empirical estimate converges slowly its expectation the sample size increases. Instead, To address this, one may restrict attention to a more manageable yet expressive function class. In particular, kernel-based methods provide principled way address this by restricting to an RKHS, which is both manageable and still rich enough to capture independence.
Lemma 2.1
Reproducing Kernel Hilbert Space. A kernel function k(x x²) is defined as a symmetric, posi- tive definite mapping k: X x X RR which admits representation in terms of an inner product,
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k(x,,') = (d(x), (()))), where ((2) is a feature map in Hilbert space H. Furthermore, say that H. reproducing kernel Hilbert space (RKHS) if H is Hilbert space of functions f - R that satisfies the reproducing property (0(x), fhx f(z). H. A linear operator A : + F, F are separable Hilhert spaces (i.e. the Hilbert space has a countable orthonormal basis), is called Hilbert-Schmidt operator if it has a finite Hilbert-Schmice norm, || Σje, Ag 11 where {gi) W denotes any orthonormal basis of G. In finiteddimensional Euclidean spaces with the linear kernel, this reduces to the Frobenius norm of a matrix. See this link for more details.
where G
Definition 2.2 (Universal Kernel), A continuous kernel k(-,.) on compact metric space (X, d) is called universal if and only if the RKHS JF induced by the kernel is dense in C. (X), i.e. for every function f E C (X) and every 0 there exists function g induced by & with fl olloo S.
Universal kernel provides a smaller space than (x) to consider the functions while keeping the characterization property, used in COCO (Gretton al. 2005b) and HSIC (Gretton et al., (2005a). These methods connect independence with the zero-value of their statistics, when universal kernels used the compact domains X' and y (or more generally, characteristic kernel
Definition 2.3. (Gretton et al. 2007) The Hilbert-Schmide Independence Criterion between X and Y. denoted as HSIC(X,Y) is the HS norm of the covariance operator
Exy Ilus ||Pxx [[tx px) @ (ør py MI
where μx A Ep (X), My - Ep, (Y)). and @ is the tensor product.
When using characteristic kernels, X TL Y if and only if HSIC(X, Y)
Case (1) Nonlinear
Case (II) Nonilnear
Case (II) Linear
Case (IV) Linear
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1.0
0.4 X DTYX)
0.6
P(M))
P(Y)))
Samples
PiYPX) density
-ww: STYX]
(ar(Y|))
Figure 1: Illustration of the motivation. The first shows the scatter plots between X and Y, and the second row gives plots the conditional densities, P(Y X), for each bin (continuous) or value (discrete) of X. For clear comparison. the conditional mean E(Y
and the conditional variance
Var(Y' X) are drawn in red and black dashed lines respectively. Data generation process: (I) X = U, Y T 22, where U, (1-1, 1): (II) ~ W((1, .... 6}). with each value of X Y follows
\/00 1), Laplace 1 - 3, v3). Exp(1) - tw-3. and 0.5 (V(-,, 0.25) + 0.5 0.25) respectively, and Y is normalized so that Var(Y x)= (II) = 3-X + and X, ~ Beta 5) independently; (IV)
Beta(2,5, 5) and Y Laplace(O, 1). Note that for both Case (II) and (IV) we have EYY X) and Var(Y X) are controlled as constants.
MOTIVATION
Recall that classical independence tests designed for Gaussian data, such as Fisher's z test, rely only on the first two moments of the variables. With the assumption of a simple linear model Y = 3X + independnece can be assessed by testing the null hypothesis B = O, where 3 = Cou(X.Y). Var(X)
We give the formal definition of characteristic kernels in the appendix.
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which again only depends the first- and second-order moments. These are possible because for Gaussian distributions, dependence is fully characterized by the mean and covariance structure; for the simple linear model, the relationship is also constrained to these low-order moments. Motivated by this, a natural question arises in the context linear mixtures of non-Gaussian sources: can independence between X and Y also be determined from low-order moment information, e.g. the first and second conditional moments?
Figure 1 presents a preliminary exploration and illustration. Case (I) and Case (II) are nonlinear relationships while Case (III) and Case (IV) contain linear relations only. We constrain the condi- tional mean E(Y X) and conditional variance Var(Y X) be constant in Case (II) and (IV), and leave them free in Case (I) and (III). For nonlinear data, there exist situations in which X and Y are still dependent after enforcing the constancy of the conditional mean and variance. For example, in Case (II) we can clearly see the skewness and kurtosis of TP(Y X) change across different values of X, manifesting the dependence between and Y given the conditions. However, if X and Y are linear mixtures of independent non-Gaussian variables, as shown in Case (IV), it is hard, even not possible to keep X and Y dependent while keeping B(Y X) and Var(Y X) constants.
Imposing tighter model class typically affords higher power. In the linear non-Gaussian setting, the observed constancy of the first two conditional moments provides a precise handle for testing. Building this, develop independence test specialized linear mixtures of independent non-Gaussian components that outperforms generic nonparametric tests when the assumptions hold, leading to more accurate estimation of linear non- non-Gaussian causal graphs.
4 METHOD
Based on the surprising observation in Section 3 we now formally formulate an independence test that utilizes the information in the model class. Specifically, first give new characterization of independence for linear mixtures of independent non-Gaussian components, as shown in The- orem 4.2 Moreover, we build the connection between the conditions in the new characterization and the uncorrelatedness of the first and second order information of Y, which is shown in Theo- rem 4.3 and give the corresponding statistic. Finally, we also derive the asymptotic distributions of the statistic and the estimation of the testing threshold under Ho in Theorem 4.5 4.6 and 4.7
4.1 CHARACTERIZATION OF INDEPENDENCE FOR LINEAR MIXTURES
We provide what appears to be the first explicit characterization of independence for linear mixtures of independent non- -an-aassiian components. We first introduce lemma which would be useful later:
Lemma 4.1. The following two statements for the random vector (X, Y) are equivalent:
E(Y X) = + 3x, Var(Y XX = o² constant,
(2()).t2)
= iip (t1, 0) + do(tt.0)
(ii)
0t
1==0
da1
(I)
8 (2)
(02 + a²) Φ (1,, 0) + 28o do(l).0)
+ B2 B2dl ((t,0)
012
1==0
de
dif
Here o(t1, = Ele X+2)) is the joint characteristic function (c.f.) of (X, Y). The proof of is in appendix. This lemma connects the regression conditional with simple analytic identities of the joint characteristic function. This technical tool allows to derive the following main theorem:
Theorem 4.2. Ler Em independent, non-Gause Gaussian random variables with and let Y >}= U Σø= byē; be two linear mixtures of ,,,..., with coefficients (a and { respectively. Vj [m]] a² + 63 > Then Y 1L if and only if there exist constants e E IR and of 2 0 such that
with finite variances,
(i) (Constancy of regression)
E(Y X)
iii (Homoscedasticity) War(Y x) E|y x Y X])² og.
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Proof sketch. The necessity is immediate. For sufficiency, assume (i)-ii). First decompose
=+ A, L = Σ a,E,, A 4=Σor
> OFE - + B, M = Σ b,E,, B = Σ b,ed, EE
:ES
iES
i&S
where -= {i: agle + 0} are the index of the common components shared by X and Y, and A and B are linear mixtures of the components specific to Y and X. Suppose S * 0. Since the are mutually independent, A, B are independent and also independent of L, M, according to Darmois-Skitovich Theorem Given (i)- and A TL X we obtain constants a, o² such that
E Xx ELL M) = a, Wor(L X)= = War(L M) = =02
Let f denote the c.f. of Ej and 0g = log fj near the origin. Let @t 12) E [e (t ++ be the joint c.f. of (M, L). Applying Lemma 4. with 3 0 and evaluating at l2 = 0 yields, for all t E IR E)ss aj 02 (b,() = ia, Ejes az (b,t) = -o². Integrating the second identity twice and using (0 = 0 and 0) (0) = (e]e gives Ejes By 0,,1, ippl 2 for some constant jz- Exponentiating both sides, EE [1:(,,1)] /3
exp ipul - whose right-hand side is Gaussian c.f. with the Hermitian property. By the deecoopposiiionnn Theorem each fy (j E S) must itself be a Gaussian c.f., hence the corresponding Ey are Gaussian. This contradicts the non- non-Gaussian assumption unless S = Ø. Consequently X and Y share no common source, so X 11 Y.
This theorem guaranties that, in the linear non-Gaussian setting, independence be decided by examining only the first two con- ditional moments, EE(Y | X) and Var(Y' X). even though out- side this setting higher-order information is often needed since only Gaussian distributions have all cumulants] of order 2 3 equals to zero. Recall the comparison in Figure these conditions alone do not exclude more complex forms of dependence in general non- linear Gau-Saan models, showing the importance of linearity.
U.V-UUnfform(-11 n-2000
RXK
We also want to emphasize that under the linear non-Gaussian model our criterion is jointly necessary and sufficient. Although X and Y are linear mixtures, checking only the conditional mean insufficient, which might be counterintuitive. Neither TE(Y | X) being constant nor Var(Y X) being constant alone implies inde- pendence. Figure]jsoww shows an example. We can easily construct two dependent variables (X, Y). X = U + V, Y U - V, where and V are two independent non-Gaussian variables with zero mean (here U. ~ 1() E(Y X) is constant, while Var(Y) X) varies. This also explains why a single linear regres- sion of Y on X cannot serve
An example that the constancy of regression holds while homoscedasticity does not. The conditional mean E(Y X) and variance Var(Y X) are the black and
Figure 2:
red lines in the figure, respec- tively. Clearly X K Y.
valid independence test for linear non- non-Gaassian data. Ho 3 = 0 only considers the conditional mean and along linear alternatives, i..., 3 # O. Dependence can manifest either in the conditional variance or nonlinear mean effects.
4.2 DERIVATION OF THE STATISTICS
Based on this characterization, we reduce the problem of testing independence to testing whether the conditional mean and conditional variance are constant. One may view E(Y X) and War(Y X) as functions of X, and there exist nonparametric procedures for testing whether such functions are constant, e-g., (Bierens) 1990, Fan & Jiang 2007). However, these methods are computationally demanding and, moreover, require separate tests for both Y and y to verify the two conditions in Theorem 4.2] Interestingly, we show that the two conditions can in fact be tested simultaneously with single statistic. To this end, we introduce an intermediate representation of independence for linear mixtures of independent non- Gaussian components. The proof is deferred to Appendix CC.4]
Theorem 4.3. Let El... Em be independent, non-Gaussian random variables with E|s?] < co for all i, Define Y = ETh 045; and X = Em b,s,. Then X 11 Y if and only if for any bounded, continuous function f, Coo(f(X), = 0 and [ox()(X), Y2) =0. O.
*Cumulants are defined as kX) = 1 d" log ox (t) t-a whenever the derivative exists. They are poly- nomial combinations of moments (e.g. K1 = [[))), M2 = Var(X)). Kr 0 for all 3 iff X is Gaussian.
(" der
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Remark 4.4. The implication of the sufficiency relies on the linear Gaussian structure above; without it, constant conditional mean and variance do not imply independence in general.
Based on Theorem 4.3 it suffices to check that, for every bounded continuous function f, Cov(f(XX,,
0 and Cov(f(X), Y2) O. For implementation, we adopt a kernel criterion. Let & be a universal kernel on X (e.g. Gaussian), and let i be the degree-2 polynomial kernel on Y with RKHS H, involving the functions y + and y + y2. Φ, v are their feature maps. We define the first variant of Linear Non-Gaussian Independence Criterion (LiNGIC,) as
LiNGIC, - [[Coo(((X),
- (0)Too(((),,))))))))) (Y)) |[][(((X) - xxx) @ (6(Y) - ply Ill
where u ([)( 14y A EY)(Y), and @ the tensor product. Note this construction co- incides with HSIC equiped with gaussian and polynomial kernels, though polynomial kernels are not characteristic thus seldom used in the literature. In our case, degree-2 polynomial kernel on one side suffices and LINGIC(X, Y) 0 if and only if vf e C,(X), Cov(f(X), 0 and (X), 2²) 0, which is equivalent to X Y for linear non-Gaussian data by Theorem 4.3
Given dataset D (T 91)
biased estimator of (X,
LINGIC(((D)
12
Σ 1.99
1 n²
HLyH
n²
13
n
2...g.r
where k. := : k(,,,)))) b "G My vy). Kx is the n x n matrix with entries k; and Ly with entries f1y. - and 1 is n x 1 vector of ones. This empirical statistic can be derived directly using plug-in estimation method and a sum of three V-statistics (Serfling) 1980).
However, this criterion is asymmetric w.r.t. (X,Y). Since polynomial kernels unbounded on non-compact domains, in practice this can lead to numerical instability when takes extreme val- ues, especially under heavy-tailed distributions. See Appendix E for empirical observation. To symmetrize our statistic, we further design the following feature maps. If we write the feature map in matrix form sp (.), we define the extended feature map op and its kernel matrix K! as
(xx)) O
[x 0
x² (x)
K2 = (x))
(bx))
Lx
[x.,._..,,]] and y [un) to represent vectors of samples. 8² and K² are defined similarly with an exchange of place of ø and v. can be easily veri- fied that ||Cou(sp(( (Y))) (+))) = [C))(o(X), (YY) + +I[Coov((X)) ((Y)) us That is, using these feature maps is equivalent to combining two directions, LXNGIC((X, and (Y, X), directly together. Lastly, use LINGIC(X, Y) A [Cou(p (Y)) i final criterion, which is symmetric to (X, Y). And now the biased estimation of the statistic becomes LiNGIC,(D) = # Tr(Kx Hyy H) + ne Tr(Ky H
where
4.3 ASYMPTOTIC DISTRIBUTION AND ITS APPROXIMATION
We now describe the null distributions of the test statistic. Suppose D = (w; 1n (1)).:)))
We first define symmetric function that satisfies LiNGIC,(D) 1.2 Eii.g,r hajgr
hajor = 4!
1
Σ
(K + K 2 to + KY + KY - 2k X
(2)
(1,06,0,10)
where kY k(,,, 125 I(1;, R.)) the sum represents all ordered quadruples (t, u, U, w) drawn without replacement from (i. q. r). and assume E(h?) < oc.
We use biased estimator instead of an unbiased for the purpose of computation efficiency. The unbiased version of the statistic can be casily obtained by replacing the V-statisties with U-statisties. As mentioned by (Gretton al. 2005h), the biased version converges the unbiased version the rate O(n

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Theorem 4.5 (Null distribution). Under Ho. we have E,hijor 0. In this case, LiNGIC, converges in distribution to weighted sum of X² variables, i.e.
Σ Axix X satisfies )pbi(w;)= = f hesgr var
L.LIGIC,(D) S
(3)
Here Xu are i...d. chi-square variables with freedom one. Since we - (Iis, ye), d are the solutions to the eigenvalue problem integrating over the distribution of variables Wi, Way and Wy-
Next, we give a theorem about the asymptotic distribution when LINGIC, > 0, ic., X.Y. This distribution would be useful in analyzing consistency
Theorem 4.6. When LiNGIC(X, (X, Y) > 0, LINGIC((D) converges in distribution to Gaussian:
(LINGIC,((D) LINGIC(X, Y)) 4 N(0,00
The variance o² (B LINGIC(X, 2)),, where Ej.g.r Eu,, W"
To use LiNGIC as a level-a hypothesis test, need the (1 a) critical value of its null distri- bution. The asymptotic null law Equation (3) is an infinite weighted sum of x² variables and is not tractable to evaluate exactly. One may want use the permutation test (Ernst 2004), which permutes the ordering of the Y sample and keeps X fixed to ensure the independence between X and Y thus estimating the quantile. However, this can be computationally intensive for large As faster alternative, one can approximate the null by a Gamma distribution (Kankainen 1995), fitting
shape and scale via moment matching as in (Gretton et al., 2005a Zhang et 2012):
A²
B
n LiNGIC,(D) Gamma(y, 3), where y
B
A
Here A A E/n LINGIC((D)) and B A Var(n LINGIC(DD).. We estimate them follow.
Theorem 4.7. Under Ho, the estimation of mean with bias of O(n to A can be given by
Ā = w + l1 "llll - in |1mbl
1 11
+ Miittee + [| II Ilabll u ll whtn utt 11:
where ute & Eδ((), U A Edly), uhe Ek(x, Maw A El(x, tye & Ekly, v), and "/ A El(y,v). Also, the estimation of variance with bias of O(n to B can be given by
2(n - () 5) B = (n - 1)2 2) (n= -3)
1F -
Em1
where M; (O denotes the entry-wise matrix product and 2 the entry-wise matrix power)
Mi ((HKx H (HLy H)) M2 = =(HKy H) @ HH)HH)?
M3 2 H) @ (HLxH) o H @ HH
RELATED WORKS
Independence testing has long been an active research direction in statistics. Early approaches include the F-test (Tiku 1967) and the Chi-squared test (Greenwood & Nikulin, 1996) for dis- crete or categorical variables. For continuous variables, the Pearson correlation coefficient (Ben-) esty et al., 2009) is widely used, and in the linear Gaussian setting it fully characterizes indepen- dence. Another category involves nonparametric rank-hased methods, which aim to achieve ro- bustness against distribution shapes by focusing the relative order of observations rather than their precise magnitudes, such as Spearman's rank correlation and Kendall's tau. More recently, CCatterjee 2021) proposed rank-based correlation coefficient that equals zero if and only if two
"Whether the Type II will converge as CO.

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variables are independent and equals one only under functional dependence. (Han al. 2017) developed distribution-free rate-optimal tests for high-dimensional mutual independence based on rank statistics. Nevertheless, the reliance rank-order statistics limits these approaches to de- tecting monotonic dependencies, failing to provide general yet computationally efficient solution. Mutual information (Berrett & Samworth 2019) provides fully general characterization of inde- pendence, yet its practical use is constrained by the difficulty of accurate density estimation. To overcome these limitations, kernel-based independence tests (Bach & Jordan 2002] Gretton al., 20056, 2003 2005a) have been developed. A kernel function implicitly defines an inner product a reproducing kernel Hilbert space (RKHS) (Berlinet & Thomas- Agnan 2011). which induces similarity measure between data points. A widely used example is HSIC (Gretton et al. 2005a).
which quantifies dependence via the squared Hilbert-SShmiddt norm of cross-covariance operators in RKHS. Variants such as random Fourier feature approximations (Zhang et al., 2018) improve computational efficiency. while sometimes at the cost of statistical power. Recent ad advances focus on
improving previous methods (Ren et al. 2024). find tests that suit high-dimension data (Zhang &
Zhu 2024 Zhang et al., (223)) or time series data (Liu el. (023), address rare dependence patterns via adaptive sample reweighting (Li et al.] 2025). etc. Besides, the random dependence coefficient
(RDC) (Lopez-Paz et al.] 2013) achieves marginal invariance through copula transformations and measures dependence by maximizing correlations under random projections, offering computa- tionally efficient solution. However, all these existing methods are not specifically designed for the linear non- Gaussian setting, which is crucial for many causal discovery frameworks.
EXPERIMENT
We test the proposed method on both synthetic and real data to compare its performance with other baselines. We also include the experiments that applying our method to causal discovery methods.
Baselines. All baselines follow their default settings unless otherwise stated. HSIC (rretoon et al., 2007]: the original HSIC test using gamma approximation. dCor (Székely al., 2007): A nor- malized covariance between the centered pairwise Euclidean distance matrices. HSIC-RFF (Zhang et al. 2018): HSIC using finite-dimensional random Fourier feature mappings to approximate ker- nels. LFHSIC (Ren et al. 2024): HSIC test with adaptively learned bandwidth.
For all methods that rely on a characteristic kernel, we adopt a Gaussian kernel whose bandwidth is selected heuristically according to the sample size, with LFHSIC being the only exception, as learns its bandwidths directly from the data. Due to space limit, details about the experiment settings, experiments on more data generation processes, comparisons with more baseline methods, and real-world data experiments, please see Appendix
Laplace(0.1)
Student 3reedom=3
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components
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Figure 3: The experiment results when we change the number of the independent components of the linear mixtures with 500 samples. The number of components d E (2, 4, 5, 6). Fach column shows the results with e; ~ a different distribution. The first row demonstrates Test Power and the second row shows the Type I error. The significance level 0.05 is annotated the black line.
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6.1 SYNTHETIC DATA
Data Generation. We generate n. pairs of two linear mixtures, Y = Σi= oge; and X Σ byeq, where ε , are independent and identically distributed non-Gaussian components. For simplicity, [Laplace 1), 1==3, ((-10, 10), TruncNorm(( (0, 1; -: 2.) For power rate, the weights are ran- domly generated ai, b; (-),11 We also ensure that the dependence between X and Y does not vanish due to randomness by constraining a,,, 2 0.1, Vi. Finally, we whitened the data to ensure zero correlation. For type I error, we make sure X and Y do not share any common components by
restrict them the same distribution, which is chosen from
Di=1 OjE; and X
Σ 2m
Di==+1 biEs, where Ex, I...., 2m, have the same distribution.
L0)1ace(0.1)
Student t 3reedom=3
Uniform(-10.10)
TruncNorm00,1 (0,1:2
Test power (t)
0.4
0.8
0.3
0.4 19
0.0
0.20
Type error
0.15
0.15
0.15
0.15
0.10
0.10
0.10
0.10
0.05
0.05
0.05
0.05
0.00
500
750
750
1000
1000
number of samples
number of samples
number of samples
number of samples
-
HSIC
HSIC-RFF
LHHSIC
K
dCor
- Ours
Figure 4: The experiment results when we change the sample sizes of the linear mixtures of 3 inde- pendent components from different distributions. The sample sizes E (300, 500, 700, 900, 1100).
Results. In Figure 3 we demonstrate that our method consistently controls Type errors and that its power outperforms other baselines in different distributions. More independent and identically distributed components make the standardized linear mixture behaves more like Gaussian and the dependence relation between X and Y becomes method does not suffer from this complexity as much as other baselines do for Laplace 1) and f-=3- Figure the performance when the number of samples varies. All methods benefit from a power gain when more data samples are available. Our method again consistently performs better in testing power, which confirms the need for specific testing method for linear non-Gaussian data.
complex and hard detect. Note that
6.2 APPLICATION IN CAUSAL DISCOVERY METHODS
We then apply our method to classical causal discovery method with linear non-Gaussian assump- tion. Here we use the ground truth causal structure of the flow cytometry dataset, SACHS (Sachs) et al. 2005). and test the algorithm both original real data and synthetic data generated according to the structure. We report the Structural Hamming Distance (SHD) and Fl score. SHD measures how many edge insertions, deletions, or reversals needed to transform the estimated graph into the ground-truth graph. Fl score is the harmonic mean of precision and recall, summarizing how accurately and completely the estimated edges recover the causal edges. More details about the dataset SACHS and results of the real data see Appendix E.4 and E,5
Direct-LiNGAM.
This is a causal discovery algorithm for LiNGAM without latent confounders. It finds the causal order by repeatedly identifying exogenous variables via independence tests with regression residuals. We replace the original HSTC test with other baselines and our method, and run the algorithm the sample size as n = 500 and change the distribution the exogenous variables. We can find the consistently better performance of our LiNGIC in all settings as shown in Table 1
the synthetic data generated according to the structure of SACHS. We fix
With assumption that these components have finite variance.

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Published as a conference paper at ICLR 2026
Table SHD and Fl Score of Direct-LiNGAM algorithm using different testing methods.
SHD (1)
Fl Score (1)
Noise Type
HSIC
HSIC-RFF
dCor
Ours
HSIC
HSIC-RFF
dCor
Ours
Uniform Laplace
2.1
2.8
0.9
0.8
0.93
0.91
0.97
0.98
2.7
4.7
0.2
0.96
0.91
0.84
0.99
Student t
2.6
2.6
2.5
2.3
0,91
0.90
0.91
0,91
TruncNorm
13.2
13.9
17.1
6.3
0.60
0.55
0.45
0.79
CONCLUSION AND DISCUSSION
We studied the problem of testing independence between linear mixtures of independent non- Gaussian components, critical but underexplored task in machine learning and causal discovery. We established a new theoretical characterization showing that independence fully determined by the constancy of the conditional mean and variance under this setting. Building this insight, we proposed a kernel-based testing procedure with provable asymptotic guarantees. Experiments demonstrate clear power gains by leveraging the model constraints. Overall, our findings provide both theoretical and practical advances for causal discovery methods with linear non -auusssi as- sumption and highlight the importance of exploiting structural assumptions in data.
ACKNOWLEDGMENTS
The authors would like to thank the anonymous reviewers for their helpful comments. We would also like acknowledge the support from NSF Award No. 2229881, AI Institute for Societal Deci- sion Making (AI-SDM), the National Institutes of Health (NIH) under Contract R01HL159800 and grants from Quris AI, Florin Court Capital, MBZUAI-WIS Joint Program, and the Al Deira Causal Education project. XW also acknowledges the support from National Natural Science Foundation of China (Grants No. 12171076).
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APPENDIX
Organization of Appendices
Section
A
Table of Symbols and Notations.
Section
Use of the LLM.
Section
The Proof the Main Results.
Section
D
The Derivation of Statistics.
Section
Supplementary Experimental Details and Results.
Section
Discussions.
A NOTATIONS
Table 2: Notation Table
Symbol
Description
X,Y
Random variables (or sets of variables)
N.Y
Domains for random variables
Fx
Reproducing kernel Hilbert spaces (RKHS)
x,y
Sample vectors (or matrices)
Ie yis 24
Specific values of sample vectors (or matrices) Kernel functions on the input spaces X, y Feature maps for X, Kernel matrice samples x, Cross-Covarianss operator Norm in RKHS Expectation of X
(x(xxx)) (o (y.v')
(())
Kx.
ExY
I
[[X]
War[X
Variance of X
Cov [
Covariance X and Y
R20
The of positive real numbers (including 0) Borel o-algebra on R
B(R)
Pxy
Joint distribution of and
PxYIz
Joint distribution of X and Y conditioned on Z
Tr|-] or trace(.)
The trace of a matrix
Tensor product Big notion
O
Number of samples X is independent of Y
XLY
(1)
The of all r-tuples drawn without replacement
(n)
Number of permutations
N(O,1) 1) u(0,1)
Normal distribution with zero mean and standard deviation Uniform distribution in (0, 1)
B THE USE OF LARGE LANGUAGE MODELS
During the preparation of this manuscript, we used ChatGPT to assist with writing by providing the prompt: "Please check whether this part is suitable for paper for submission to international conference." " We applied its suggestions paragraph by paragraph, and outputs were edited by us ensure correctness.
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C PROOF OF THE MAIN RESULTS
C.I PRELIMINARIES AND USEFUT. LEMMAS
Theorem C.I (Cram't er Decomposition Theorem (Cramér 1970g). Let random variable be normally distributed and admit a decomposition as a sum & = E1 + E2 of two independent random variables. Then the summands E and Eg are normally distributed well.
Theorem C.2 SoammissSkkiooich Theorem (DST)). We consider independent scalar random vari- ables X1...., Xn (not necessarily identically distributed) and two linear statistics
L1 = EaiXi,
L2 = B.Xi.
where the B; are constant coefficients. Let Lı and L2 be independent. Then the random variables X, for which 33 # O are all normal.
As mentioned in (Shimizu et al. 2011) In other words, this theorem means that if there exists non-Gaussian x, for which 05S3 # 0, L1 and L2 are dependent.
The following lemmas and corresponding proofs are attributed to (Kagan et al. 1973 Lemma C.3. Let X and Y be random variables and EY exist. Y has constant regression X if and only if the relation
E(Ye EY Ee
holds for all real
Lemma C.A. Lel (X, Y) be a two-dimensional random vector with EX = EY = O. A necessary and sufficient condition for the linearity of the regression of X is the existence of a constant such that, for all real tj.
@b(11
(2)
do (t1,0)
Ot
dt
12=
where à (2) the c.f. of (X
Proof. In view of Lemma 1.1.1, a necessary and sufficient condition for F(Y - 3X X) = 0 for
some constant that
E(Y 3X)e"* itx = 0 for all real
which is easily seen to reduce to the results here.
Lemma C.5. In order for the two-dimensional random vector (X, Y) to satisfy the conditions E(Y X)= = a + 3X,
Var(Y [X) = o² constant,
it is necessary and sufficient that
@o(t) lt)
i (t1, 0) + 3 _d
812
1==
des
(5)
[2]
= (o² + a²) @b (t1, 2ia 3 do(().)))
81g
12=0
di
dif
Proof. The conditions here are equivalent to
E - 3X X) [Y² 8x)² X] 02
which, in view of Lemma are easily seen to be equivalent to the results in the Lemma.
Lemma C.6 (a-decomposition theorem). Let the function o(z) of the complex variable z be regular and nonvanishing on the disc |z| < R and possess the Hermitian property: o(-2) Ø(z). If $1....,0. &. be c.f.s, and aı,, ....... a2 be positive numbers such that for some sequence {t..) of real numbers tending to zero the relation
is satiified, then the functions 6 regular and nonvanishing in |2| R. and relation (6) is valid throughout the disc. If in (6), o is a function of the form exp Qt), where Q(t) is a polynomial with the Hermitian property, then every 6J is normal e.f.
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C.2 A PRELIMINARY RESULTS AND THE CORRECTION ITS PROOF
Theorem C.7 (THEOREM 5.7.1. in (Kagan et al. 1973 Let Xi...., Xn be independent r.v.'
with finite variance. The linear functions L Σ ajx and M b, with a,b, * for j = 1, satisfy the relations
(i) B(L| M) = a + BM, and
(i) Var(L L M) = o = constunt
if and only if the following conditions are satisfied:
(a) the xj for which 0j # Bbg are normal, and
(b) 3 = (E' (E'aaaaa (obboo)))) (300)) of -a
where 03 Var X,, and E" denotes that the summation taken over all j for which aj + Bby.
We claim that the original proof has minor problem with an exchanged and b. Here we give another proof that corrects this problem.
Proof. The sufficiency of the conditions is easily established.
log Φ (41, (2) log E (i(()
log
(5
n II j=1
log
fj (byt) + tz(22)
- j=l
Bg (t1,
{ 0¹ 0'
(1, 12) e² =
Φ² 0" Φ 6'0' = (0" + 02)
We first prove the necessity. Let fj be the c.f. of X,, and 8, log fy (in a neighborhood of the origin where none of the ʃj vanishes).
Ea,0, (b,t) ia + b,,0 Σ (byt) = -o0 + 32 Σ 8700 (b,t).
(7)
(8)
Differentiating (5.7.1) with respect to t, we obtain
aa,bo = 10 (6,1)
From (5.7.2) and (5.7.3), we derive
Σ (a3 - Bbj} 00)
(b,t) (b)))
Σ 1b,)
03 (b,t) = o3-
Integrating (5.7.4), we obtain
II UG (b,0)) exp (112))00)))
where 7; (a, Bby)² Assertion (a) now follows noting that in Lemma C.6 if the above holds with (bt) exp Euut (1/2)a²t²) then every o; is normal c.f. The other results obtained by
setting = 0 in and 7
The sufficiency of the conditions easily established.
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C.3 PROOF OF THEOREM 4.2
Theorem 4.2 Let Et,.... En be independent non-Gaussian rv'' with finite variance. Suppose that Y Y==ays; Lass and X Σ bzez be two linear mixtures of e; Then Y 11 X if afd only if:
(i) B(Y X) =constant a.S., and
(ii) Var X) = o constant a.s.
Proof. D: Suppose that we know X Il Y, then we directly have (i) and (ii).
E: Proof by contradiction. Suppose that we have conditions (i) and (ii). We can rewrite X and Y in
Y + A. L
Σ ajs,, A=
Σ a,E = M Σ b.s, B
Σ
beE
iESc
where S = {i a,d; # 0} and S n.. Since ex, Vi are mutually independent, the linear mixtures of disjoint sets &s are independent, i.e., A 11. B, A, B, and M A, B. Then we have
B x E[L + A X| = E|L x + E|A X] = c E|L X| _ c-ccB E[A],
(9)
Var(Y X)= = Var (L + A X) Var(L L X) + (A),
(10)
+ War(L X) = 0² _ o Var(A).
(11)
We assign L -ド a c2 = Ejss azej. where a/ 0, 0, S Let f, be the characteristic
function (c.f.) of y and 0g = log fy (in neighborhood of the origin where none of the fa vanishes).
The c.f. of (X, L) is @(11, t2) Ele-(taMM+>>) M+ Then through Lemma 4.1 we know / = 0,
01,,t) Σ (b,t) iar jES
(12)
Σ (b,t)) Σ (b,t) -03.
(13)
jES
So we only need to consider j E S
= ii ar,; * 0}. Also we know that for E S, we have
by + 0. We now want to integrate 13 Suppose F(t) = Ejes a3 (b,t)) since by * 0 for
j S. Then Ft( = Ejes (b,t)
-02. Therefore F(t) -2 + C1. Suppose again G(I) = Ejes a) b7 (b,(). Then G'(l) Ejes 12 = F(I) = -02 + C1 which gives
us G(t) - 2 + +Ct Co. Then determine the constants using the initial values.
0,(b,t)(-o log (0) = log E
0,
(14)
ff(b,t)
E jee e8
l,(b,t)).-o
E ii,
(15)
f3(
El
These give us the following points
G(O) = Σ a 0b,
0,; 0) = =0,
(16)
JES
(0) = F(0)
Σ a (b, .0) 0) =)= a
1
E[ie,] A ip.
(17)
bj
jES
jeS
Therefore, the integrated function is G(t) = Ejes 2 nential on both sides,
jes 0
2j (b,tt) = - + ipt. We take expo-
explipt - 0² 2²2) = Hlys 1f, (f,
Since iut - 3 22 has the Hermitian property, through Lemma C.6 all fis are normal c.f. and eys normal variables, which contradicts to the linear non non-Gaussian model. So S = 0.
So now we have Y = A = Eig 0,E; and X = B Eies b,s, Since A n = and E,S are mutually independent, we have X 11 Y.
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C.4 PROOF OF THEOREM 4.3
Lemma C.8. Let (P) be probability space, X - X random element taking values in a metric space X, and let h X + be Borel measurable with TEI((X)] < co. Assume that
I [[J(X)(()) =0 for all J Cr(2).
Then h h(X) = almost surely. Moreover, if h is continuous and the law of X has full support on X (i.e., P(X E U) > 0 for every nonempty open set U), then h(z) for all X.
This result be viewed measure-theoretic analogue of the fundamental lemma of calculus of variations (Gelfand et al., 2000). We provide self-contained proof below for completeness through contradiction. Let μ Po be the law of X.
Proof. Remark:
1) Indicator approximation. By regularity of Borel probability measures on metric spaces and Urysohn's 's Lemma, for every Borel set X' there exists (fn) C Cs(X) with 0 S fn S and fn + la p-a.c., where 1 represents the indicator function of the set A.
2) Extension bounded measurable functions. For simple function g m Gs 1A. define On Σne C.Jn E Ca(A') using the above approximations. Then (X) - g(X) a.s. and 9n (X) & (E, (cil) [x()) with [X)) oc. By the Dominated Convergence Theorem,
B[g(X))(X) = lim (X)((X)((X) 0.
n-
By approximation of bounded measurable funetions by simple functions, the identity (()))) = 0 holds for every bounded Borel measurable Taking g = 1 yields SA h ddu for every Borel A.
3) Conclusion. If (X) 2 e) > for some & > 0, then 0 = S(1ze) hdp 2 e e 2 e)) > 0,
contradiction. Similarly for {h < - Hence h(X) 0 a.s.
If, in addition, is continuous and the law of X has full support, then h(zo) * would imply open ball around zo, which has positive probability. This contradicts h(X)
||| 2 a.s., and therefore h = 0 everywhere.
Theorem 4.3 (Independence for Linear Non- Non- Gan--aussian Data). The linear mixtures of the independent non non-Gaussian variables E Σ 0gE; and Ei= biE,, are independent if and only if Cov(f(X),Y) and Cor(f(X), O, where can be any bounded, continuous function.
Proof. If X IL Y, then from Lemma 2.1 we know Cov(f(X), = 0 for each pair (f,g) g)
of bounded, continuous functions. Clearly the condition satisfied.
= We first consider Cov(S(X), Y) 0. For any bounded and continuous function S.
Cov(J(X), (((x)){)E[[()]])
E(((X)(Y-E))
= =Eif(X) EYT)) = 0.
If we define h(X)= A B[(Y EY) = E(Y X) E| then
B(f(X) = 0.
vf E (X).
Use the results in Lemma C.8 we get that 1(X))
Therefore have the condition EEY
X) = E[y] a.s., which is constant.
With similar discussion, since Cov(f(X), 22) 0 gives us EY E| and it is constant, we can derive that
Var (Y X) -
- X = x - E A constant.
Combine the results and Theorem 4.2 know that X 11 Y
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D DETAILS ABOUT THE UNCONDITIONAL INDEPENDENCE STATISTIC
We first give some preliminaries for later proof and derivation. Definition D.1 (U-statistics). The statistic Un defined as follows called a U-statistic with sym- metric function à of order
Un
11 m
Σ h (X (x.,.....,..... Xi.),
(18)
where E. denotes the summation over the from nn.....n..
(m) combinations of m distinct elements {di..... im 3
Xm)), For every U-statistic Un as an estimator of E [h (X (m)). there a closely related V-statistic defined by
V
Σ h
X
n
km
Proposition D.2. Let Vn be defined by the above function and we have n i.dd samples { drawn from Px.
(i) Assume that B (Xi Xim < 0o for all S S S im S m. Then the bias of
Vn satisfies
(PPx) 0
0 (n
(ii) Assume that E h (Xi,:-.. variance Vn satisfies
Xim. p9] < X0 for all 1 S i S ... S im S m Then the
Var(V.) Var(U,) + O(" 3).
We also define some more statistics for later derivation. For k: 1, ....., m, let
ha ha ... Fk) E [M(X) Xm) Xi Fl,..., Xx xxk]
= E [h(x),..... Ih. X Xm)].
Note that hm h. Further define ÇA A Var (hg (X1...., Xx)). Theorem D.3 (Shao 2008), Theorem 3.16). Let Vn be a V-statistics with E [h(X,,,.... )2] < oo for all 1 < 1 < I lm m.
(i) If A Var(hi 0, then
Vn (V, - -0) $ N(0.
(i) IfCi 0 but Co A Var(ha (X1,X2))
X2)) > 0, then
n (Vn 0) 4 m m 1)
2
where X2, 's are i.i.d. random variables having the chi- chi-square distribution xi and Xy 's are some constants (which may depend on Px satisfying E,=1 C2.
D. CHARACTERIZATION OF UNCONDITIONAL INDEPENDENCE
For (X, Y) E A' x V, the cross-covariance operator Exy Fy + Fx defined by (Fukumizu al. 2004):
VfeFx,geJv (1,Exvo) -xxl[(X)g((}) Ex([(X)) (19)
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and the covariance operator itself can be written
Exy = Ex [(x(X) - 2xx) @ (o(Y) - [rY],, p Ex - Ero(Y),
(20)
where is the tensor product.
This operator is generalization of the cross-covariance matrix be- tween random vectors. HSIC is the squared values) of this operator, mentioned in Def. 2.3
norm (the sum of the squared singular
HSIC Y) Exx
(kx (X, Xy (Y, Y + Exx- Br) Y
(21)
x [xx [xx (X, X J (ky (Y, Y']l-
Assuming the expectations exist, where X' denotes an independent copy of X. An unbiased es- timator of HSIC in sample D {[ v)) drawn from distribution Pxy the sum of three U-statistics: (Grettoon et al. 2007)
HSIC,(D) = (n)2 (1.))
Σ
k
Σ
Σ
(22)
(n)4
(n)3
(fJi..))
(fJ.4)Ely
where 2:x2 = xx (2i,, xz), k = by (yt, Mj), (10m == mn-m)] and the index set i denotes the set all r-tuples drawn without replacement from the set {1..... n). A biased estimator is the one replacing -sstatistics with -statisttcss, as in
1 HSIC&(D) = 2
Tr(KxHKyH) (23)
T
7
n2
1.J.a.
where the summation indices now denote all r-tuples drawn with replacement from (1...., n) and
H
111
D.2 APPROXIMATE THE ASYMPTOTIC NULL DISTRIBUTION
D.2.1 MEAN OF LINGIC, UNDER Ho
An unbiased estimate of L.NGTT((X Y), denoted by LiNGIC. (D), is sum of three U-statisties
LiNGICIa(D)
1
Σ
kgh4 (n)4 (Ks...))
Σ
kiylar (n)a (b.d.))E
Σ
kiglig
(n)2
(KJ)
which has E [LIGGIC,. E[LINGIC(X, Y)] = O under Ho.
The complete proof given in (Gretton et al., 2007). We show only some of the key steps here. The biased estimate of LiNGIC(X, Y). denote as LiNGIC,(D). is sum of three V-statistics
LiNGTC (Z) == n
1
Kiglas +
1 no
Σ kigher S.ggrr
1 2
kislig
n
We can show the difference (LiNGIC,(2) - LINGIC. (2)) similar to Ren et al.
2024
2
Σ
(K + +134))
Σ
(K: 3q
+ k x
n²
3
(i.))
(i.i-))Etz
3
Σ
1 +
Σ
XXIY in
X or
(n)2
(n)s
(n)a (K.gg)))
(L.ig)
2
Σ
(KYL
Σ
(H + kiy(()
n
2
(KJ)
(KJ.A)EiS
3
Σ
1 +
10
Σ
K
Σ
KN + 0 (n-))
(n)2 (1.))))
(n)a (k.1.))
(n)
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when we assumee the kernel is bounded with compact A' and y. Secondly, we take the expectation of the last equation. To simplify, we use the notation Exyy kl = Exyy k( x:) (y, y²) (and so on for the rest), then n (E [LiNGIC,(Z]] E [LINGIC,
-yyzkl 2 (Eyzzi kl + Byo'zkl) kil + Byw'ahkl + E Byyank
yy xx kl 10 kl - GEny kEzzz!
Baykl 2 -2(22 (Beryy kl + Exs't kl) Eay y kl + y kl
3w ysy kl "'lz"''y" kl 6g Kiyy + O (n
Under Ho. x is independent with y, thus we can draw the conclusions that Egy kl Exy'g kl, Bee'skl kl ye and Ene'yy kl = Byx'vy'k kl = E kByyl, Similarly, kl = En kl, E Eyy'skt kl Byg' kl and Byy'sar kl Byy aay' Eyy klEnsi Combining with E [LINGIC v (Z)] = O. we obtain that
E [LN)GIC((2]]
Eary hil + |1ae | 112 Ezk | 11 Eyd || 12)
=
n
1
(Eynkl + || 2 || 11 - Eygh k || 2 E Euttl -E + +0(0 7)
n
where 2z := (x), 14 = (x)) and for ply 1ly are similar. Also note that the estimators of
Ey can be written K,,k Ey x.) - E, A:y which is the same for E.y:: 1
1 Σ kY Egl pla E 1xx Eyy pdy a
Σ, 1/ An empirical estimate can be obtained
n by replacing the term above with
Σ | (n)2 (i,))sgg
k.
|
Mij ||
Keg
(n)) (1.))
(n)2
(K.))
| (n)2 (i.))
The obtained estimate
E m LINGIC(((2)
p || | 1
|
/ 11
+ Miig x + || mot u 11
results in (generally negligible) bias of O (n-1) and can be calculated within the time cost O (n²).
D.2.2 VARIANCE OF LINGIC, (2) UNDER Ho
The complete proof is given in (Gretton et al., 2007). We show only some of the key steps here. According to (Serfinng, 2009, Section 5.2.1), the variance of the U-statistic with the kernel can be calculated by
Var LINGIC,
n2
n 4
So
=
4
n-4) 3
C1
+
6(n=1) 2
Ce + + (n-
where we only need to consider the dominant term
S2 Es, ((ejgr
E (I)
under Na
using degeneracy (51 0) under Ho. Under Ho. using x, y are independent, we have
B hiiggr
6 (ket + +Eprk Ezke - kg)s) (Cy + Eyg' - Ey44 - Egg4))
+ 6
(K) + Eyg k - Eykke Eyk;) (1)) + Egail Egl4 Ezl,)
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-Kkj + Eeze E Exk;) (11) + Eyy' - Eyl, Byli)
+ (ki) + E Egk - Eggi (4i) + Ey,' - Byls Egd1)
(1/ Byyl Byli Eyl)) (Kiy Byy Ey Byghi)
+ (R:g By k - Bygis Ryy 2 (1:) Ezz' Egli -Eal)) 2
c 2 || 2 + =||1 2 | 2 || 1 || 11
where
Chu == [X)x(X) - m+:) @ (vi(X) - pb)] = (X), -B>(())
And
Egg (bi) Eawck KEnk E,kj) )² )²= Ey (x;) - 0 (x²z -m
=Eij ((Φ (2:) ) 2 (0 (x:) /+)) (0 (zz) ) @ (Ø (zzz) -1=tt 11 || 1??
which, following a similar derivation, we have Egy (1) + Eyy Eyli - Egg1) = |
yy
E,j (Ni + Eyy Eyk; - Eykk)) = || 1| and Eig (1,5 + Ezz! - Eyli - Eyl,)² = C 2.1
2
We also have
Eng (ki)) + Errc E, - Egggy) (1)) + Enwl E24 - Enll)) = || !|
Ey (1) + Enge - E.yli - Byy4) (ky + Egy - Eyyki - Enghig) = ||C
Then the variance of the statistic is obtained by
Var (D)] 2(n 4) - 5)
c || I + 2 2 llss ||
(n)
+ || l
| We
||
+ O (n
where || ll the Hilbert-Schmidt norm. An empirical estimate of the product Hilbert-Schminn ll is given by
norms | l
[|
|| and ||C% al ||
B1= 1 (B, (Biig(BB)) with B ((HKg HJ @ H)) = ((HK, (HL
n(n- 1)
respectively, where @ is the entrywise matrix product and 0- is the entrywise matrix power. For Hss we first give the unbiased estimators (where K HKH and i = HHHH): ):
ICAl |
1
K
1 ((Kx @ (x)= diag( K x + iix))1
nnn
1)
n(n- 1)
1
n(n 1) ((RY o 17) diag(R* RY 11)1. RyDYy
HS
ninn
1)
ifj
Therefore the empirical estimate of ||Ck!] || Ils is given by the formula above with
HS
Bg = (HKxH) @ (HLx @ (HKyH) O HHLyH).
The estimate in has a bias of O nn 3) and can be calculated within time cost O (n²). Since the additional terms of the bias vanish faster than the terms in front of it, the result is identical to the case of unbiased.
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E SUPPLEMENTARY EXPERIMENTAL DETAILS AND RESULTS
E.I IMPLEMENTATION DETAILS
Here we provide the implementation details of the methods. In all experiments, we use Gaussian kernels in all kernel-based methods. The significance level is set to 0.05. The results are obtained after averaging the values the 100 tests.
Details about the Baselines. All the baselines follow their default settings unless stated other- wise. HSIC (Gretton et al.] 2007): the original HSIC test using gammaa approximation for p- value. Code from python library causal-learn (Zheng al. 2024); LFHSIC (Ren al. 2024): HSIC test with adaptively learned bandwidth. Code from
LK. RDC (Lopez-Paz et al., 2013): use canonical correlation between finite of random Fourier features. We permute the samples 500 times to compute the empirical p-value. Code from http:///ituub.oooggaayyyrrraaaae
FHSIC (Zhang et al. 2018): HSIC using finite-diments dimensional ran- dom Fourier feature mappings to approximate kernels. Code from http:///ggghu...mmoooo////////e/
E.2
MORE RESULTS ON SYNTHETIC DATA
E.2.1
EXPERIMENT RESULTS ON MORE BASELINES
We also conducted the same synthetic experiments
more baselines, RDC (Lopez-Par al.
2013 and MI (Berrett & Samworth] 2019). The results are shown in the Figure 6
and
Laplace(0.1)
Student_t 3
Triangular(-1.0.11
TruncNorm001::22
Test power (f)
0.3
0.7
0.2
0.8
0.6
0.1
0.7
0.20
0.20
0.20
0.20
Type error
0.15
0.10
0.10
0.10
0.10
0.05
0.05
0.05
0.00
0.00
0.00
0.00
components
components
components
components
C
HSIC
*
RDC
-
Ours
Figure 5: The experiment results when we change the number of the independent components of the linear mixtures with 500 samples. The number of components E 3, 4, 5, 6}. Each column shows the results with E; ~ a different distribution. The first row demonstrates Test Power and the second row shows the Type error. The significance level 0.05 is annotated the black line.
SCIT is designed for conditional independence testing under linear non-Gaussian SEMs by regress- ing out conditioning variables and applying generic nonparametric kernel-based independence check residuals. When setting Z 0. SCIT reduces testing whether the coordinate-wise RBF similarity between (a. is significantly larger than that between (z, r), where is permuted copy of y. This generic independence test does not leverage the linear non-Gaussian structure our method does. We added SCIT with Z = 0 as one of our baselines in the Table
E.2.2 EXPERIMENT RESULTS ON PERMUTATION TESTING
To show the performance of our derived asymptotic distribution, also run experiments HSIC and LiNGIC with a permutation test. The number of permutations equals to 500. The results are shown in the Figure and the null distribution approximated by the sum of chi-square variables as mentioned in (Zhang et al, 2012).
and 10 Here we also included HSIC_chi one of the baselines, which means
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Laplace(0.1)
Student_ freedom=
Triangular(-1.0.1)
TruncNorm00.1-22
Test power (f)
0.9
0.3
0.8
0.2
0.8
0.7
0.1
0.20
0.20
0.20
0.20
Type error
0.15
0.15
0.15
0.15
0.10
0.10
0.10
0.00
500
750
1000
0.00
500
750
1000
0.00
500
750
1000
0.00
500
750
1000
number of samples
number samples
number samples
number samples
-
Ours
Figure 6: The experiment results when change the sample sizes of the linear mixtures of 3 inde- pendent components from different distributions. The sample sizes n E {300, 500, 700, 900, 1100).
(aplace(0,1)
Student freedom=3
Uniform(-10,100
TruncNor(0,,;:22
Test power (f)
0.8
0.6
0.4
0.4
0.1
0.2
0.2
0.20
0.20
0.20
0.20
Type error
0.15
0.10
0.10
0.10
0.10
0.05
0.05
0.05
0.05
0.00
0.00
0.00
0.00
components
components
components
components
-
HSIC
-
MI
-
Ours
Figure 7: The experiment results when we change the number of the independent components of the linear mixtures with 500 samples. The number of components E (2, 6). Each column shows the results with e; ~ a different distribution. The first row demonstrates Test Power and the second row shows the Type error. The significance level 0.05 is annotated the black line.
E.2.3 SENSITIVITY TO DEPENDENCE STRENGTHI
We conducted additional experiments where all parameters are fixed and only the dependence strength between and is varied. Specifically, our data generation process is defined as:
Y = = Sy + c2. = Sx + V c Nx:
(24)
where Sx; Sy are generated from shared latent sources, and Nx, Ny are generated from indepen- dent sources. Thus, O corresponds to independence (X 1 Y), while c 1 yields the strongest dependence (entirely shared source).
We report results (with d 3 independent components drawn from Student-/ distribution and sample size n 500) in Table 4
E.3 COMPUTATIONAL COMPLEXITY AND RUNTIME ANALYSIS
The computational cost of independence test, LiNGIC, is comparable to standard kernel-based independence tests such as HSIC, as it primarily involves estimating the Hilbert-Schmidt norm of a covariance operator in the Reproducing Kernel Hilbert Space (RKHS).
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Laplace(0,1)
Student freedom=
Triangular(-1.0.1)
TruncNor(00.1:22
Test power (f)
0.8
0.0
0.6
0.2
0.2
0.1
0.20
0.20
0.20
0.20
Type error
0.15
0.15
0.15
0.15
0.10
0.10
0.10
0.00
500
750
1000
0.00
500
750
1000
0.00
500
750
1000
0.00
500
750
1000
number of samples
number of samples
number samples
number samples
-
Ours
Figure 8: The experiment results when change the sample sizes of the linear mixtures of 3 inde-
pendent components from different distributions. The sample sizes n E {300, 500, 700, 900, 1100).
Table 3: Comparison of Type I error and Power across different distributions and dimensions (d).
HSIC
SCIT
LiNGIC
Distribution
d
Type
Power
Type
Power
Type
Power
2
0.04
0.84
0.07
0.57
0.07
0.84
3
0.01
0.77
0.01
0.61
0.04
0.80
Laplace
4
0.06
0.61
0.01
0.60
0.05
0.67
5
0.04
0.56
0.06
0.51
0.07
0.70
0.09 6
0.49
0.05
0.39
0.06
0.66
2 0.07
0.79
0.02
0.54
0.08
0.85
3
0.05
0.85
0.06
0.69
0.06
0.97
Student
4
0.05
0.83
0.03
0.62
0.06
0.95
5 0.07
0.75
0.06
0.60
0.04
0.93
0.04 6
0.70
0.07
0.57
0.03
0.96
2 0.07
0.87
0.05
0.00
0.07
0.88
0.04
0.76
0.07
0.01
0.05
0.83
Uniform
4
0.02
0.47
0.06
0.00
0.06
0.54
0.05
0.37
0.03
0.00
0.06
0.45
6
0.03
0.24
0.04
0.00
0.03
0.31
2 0.06
0.19
0.12
0.01
0.08
0.41
3
0.02
0.11
0.01
0.01
0.02
0.25
Truncnorm
0.04
0.05
0.06
0.01
0.05
0.12
5
0.04
0.03
0.04
0.00
0.05
0.04
6 0.06
0.07
0.05
0.00
0.06
0.03
Specifically, given samples, computing the nxx Gram matrices requires (n=2) kernel evaluations. discussed in Gretton al. (2005a), the centering and evaluation of the V-statistic Tr(K HL, H) can be implemented using three double sums, each of which is O(z²). Consequently, single LiNGIC evaluation has an overall time complexity of (nn2) (quadratic in sample size), matching the complexity of standard kernel-based independence tests.
Importantly, our main contribution is the improved statistical power under the linear non- -a--Gussiia setting rather than computational acceleration. Nevertheless, LiNGIC operates comparable runtime scale to the widely used HSIC. Designing faster variant of LiNGIC (c.g., via random Fourier features Nyström approximation) remains potential direction for future work.
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Laplace(0.1)
Student_ freedom=
Uniform(-10,10)
TruncNorm00.:222.
Test power (1)
0.6
0.6
0.3
0.6
0.8
0.4
0.7
0.20
0.23
0.20
0.23
Type error
0.15
0.15
0.15
0.15
0.10
0.10
0.10
0.05
0.05
0.05
0.05
0.00
0.00
components
components
components
components
ISIC
chi
HSIC_perm
-E-
Ours_perm
Figure 9: The experiment results when change the number of the independent components of the linear mixtures with 500 samples. The number of components E (2, 4, 5, 6). Each column shows the results with e; different distribution. The first demonstrates Test Power and the second row shows the Type error. The significance level 0.05 is annotated as the black line.
Laplace(0.1)
Student_ 3
Uniform(-10.10)
TruncNorm0.11-2..0
Test power (8)
0.4
0.9
0.9
0.9
0.2
0.7
Type error
0.15
0.15
0.15
0.15
0.10
0.10
0.10
0.10
0.05
0.05
0.05
0.05
19
500
750
0.00
500
750
1000
500
750
1000
500
750
1000
number of samples HSIC
number of samples
number samples
number samples
HSIC_cth chi
-K--
HSIC_perm
-B> Ours
Ours_ Qurm_perm
Figure 10: The experiment results when we change the sample sizes of the linear mixtures of 3 inde-
pendent components from different distributions. The sample sizes n e {300, 500, 700, 900, 1100).
For empirical evaluation, compared the runtime (in seconds) of our testing method against base- lines, as well as their corresponding applications causal discovery (specifically using Direct- LiNGAM). The results detailed below.
As shown in Table and T we observe that LiNGIC operates on similar timescale as classical Kernel-based tests.
E.4 SACHS DATASET AND RESULTS OF THE REAL DATA
Sachs Dataset. The the flow-cytometry data published by (Sachs et al. 2005) is a popular real- world data set for causal discovery methods, which gives expression levels of proteins under various experimental conditions. Among all data, we use n = 853 observational measurements proteins. We take the popular reconstruction of the signaling network in original SAHCS dataset paper (the right graph in Fig. II] as the ground-truth causal graph, which contains 11 nodes and 17 directed edges.
11
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Table Power comparison (higher is better) under varying dependence strength The last column reports the Type error at с 0 (lower is better, target a 0.05).
Dependence Strength (c)
Type
Method
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
(c 0)
HSIC
0.01
0.04
0.05
0.10
0.17
0.28
0.37
0.53
0.72
0.83
0.05
LiNGIC
0.08
0.06
0.12
0.23
0.39
0.53
0.69
0.86
0.94
0.95
0.05
Table Runtime comparison (in seconds) of different independence tests with varying sample sizes (ni). "Perm" denotes permutation-basee tests (shuffles=100).
Sample Size (n)
HSIC
HSIC perm
LiNGIC
LiNGIC perm
RDC
500
0.011
2.693
0.036
7.159
2.190
1000
0.051
16.267
0.150
36.427
2.478
2000
0.205
66.178
0.499
102.564
3.131
3000
0,411
166.481
1.154
233.445
3.778
4000
0.678
301,443
2.016
405.439
4.352
5000
0.995
483.046
3.126
638.219
4.773
E.5
REAL DATA EXPERIMENT RESULTS
E.6
MORE RESULTS ON DIRECT-LINGAM
E.7
ADDITIONAL SIMULATION SETTINGS AND RESULTS
We conducted new simulations similar to the settings described in Section 7.1 of SSimizu et al. (2006). We evaluated our method against the baseline (DirectLiNGAM with HSIC) under varying graph structures (sparse vs. dense), dimensionalities, and non- non-Gaussian noise distributions.
Data Generation. We simulated both Sparse (S) graphs (with average degree of 2) and Dense (D) graphs (with an edge probability p 0.7). We also varied the dimension of the dataset (number of nodes) within d {5, 10, 15). We used strictly lower-triangular matrices B with weights sampled uniformly from 1.5, -0.5 (05, 1.5). The disturbance terms were generated from Gau-Sian distributions (e.g., Uniform, Student-l, and Truncnorm), followed by random permutation of the causal order.
Experimental Results. We report the performance mean over 100 trials datasets with N 200 samples. Our method demonstrates strong generalizability and consistently maintains higher performance with respect to the FI score (f) and SHD (L).
PIP2
PLCS
PIP2
PICE
PIPS
PIP3
Mek
Mek
Akt
Akt
JNK
JNK
PKA
PKA
738
p08
PKC
PRC
Figure 11: The first two figures in Figure 5 in Mooij & Heskes
Left: Consensus network, according to (Sachs et al., 2005); Right: Reconstruction network, reconstruction of the signaling network by (Sachs et al. 2005), in comparison with the consensus network. Black edges: expected. Blue edges: unexpected, novel findings. Red dashed edges: missing.
2013
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Table 6: Runtime comparison (in seconds) of Direct-LiNGAM using HSIC
LiNGIC as the
independence measure (N = 200).
Settings
Direct-LiNGAM (HSIC)
Direct-LiNGAM (LiNGIC)
d 20
4.75
7.05
50
62.46
99.52
d 100
500.17
802.77
Table 7: SHD and F1 Score of Direct-LiNGAM algorithm using different testing methods the ground-truth data of the SACHS dataset.
Sachs
MI
HSIC
RDC
HSIC-RFF
dCor
LiNGIC
FI (t)
0.18
0.10
0.18
0.10
0.29
0.22
SHD (J)
15
16
15
16
14
15
E.8 PERFORMANCE ON HIGHT-DIMENSIONAL GRAPIS
We conducted additional simulations evaluate method on high-dimensional causal graphs with the number of nodes d E {20, 50, 100)-
Data Generation. We simulated graphs with average degree of 2. We used strictly lower- triangular matrices B with weights sampled uniformly from 1-1,5, -0.5 u [0.5, 1.5]. The dis- turbance terms were generated from Uniform or Truncnorm distributions, followed by random permutation of the causal order.
Experimental Results. We report the performance mean over 100 trials on datasets with n 200 samples. Our method demonstrates strong generalizability and consistently maintains higher performance with respect the score (f) and SHD (L).
F DISCUSSIONS
F. THE PREVALANCE OF NON-GAUSSIAN DISTRIBTUTIONS
do not care about the non-Gaussianity requirement in causal discovery, Gaussian data are far more prevalent than Gaussian ones in the real world, as mentioned in (Spirtes) & Zhang 2016). According the Cramér's decomposition theorem (Cramér) 1970), if any of the variables with non-zero coefficient in the linear composition is non-Gaussian, then the composition must be non- non-Gaussian. This implies the rareness of the Gaussian distribution linear data since easily "polluted" by non-Gaussian ones.
In fact,
F,2 JUSTIHICATION FOR PAIRWISE INDEPENDENCE TESTING
In our linear non-Gaussian setting, joint independence can indeed be reduced to pairwise indepen- dence. This relies a fundamental result in Independent Component Analysis (ICA). As stated in
Comon [1994):
Proposition F.1. Let s be a random vector with mutually independent components, and x = Gs. Then, the mutual independence of the entries is equivalent their pairwise independence.
In our framework, the variables are linear mixtures of independent noise terms. Specifically, we can write:
== Σ a.; a====a =Te x= ==>mn bis; =b_
Z Σ Cg5s = T
(25)
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Table 8: Comparison Fl Score (higher is better) and SHD (lower is better) on different settings.
Fl Score (1)
SHID (L)
Settings
Nodes (d)
HSIC
Ours
HSIC
Ours
0.906
0.909
0.93
0.86
Student t (D)
10
0.717
0.754
13.11
11.43
13
0.614
0.652
43.3
39.21
5
0.913
0.935
0.58
0.43
Student (S)
10
0.833
0.847
2.82
2.52
15
0.756
0.773
7.36
6.53
5
0.905
0.927
0.92
0.72
Uniform (D)
10
0.620
0.700
17.91
14.21
15
0.508
0.566
53.95
48.21
5
0.945
0.968
0.49
0.36
Uniform (S)
10
0.916
0.927
1.72
1.51
15
0.841
0.875
4.52
3.49
0.474
0.572
4.78
3.95
Truncnorm (D)
10
0.435
0.434
25,92
25.75
15
0.412
0.450
63.31
59.84
5
0.506
0.613
3.52
2.81
Truncnorm (S)
10
0.442
0.556
10.84
8.98
15
0.402
0.432
19.18
18.93
Table 9: Comparison of F1 Score and SHD on high-dimensional graphs (d {20, 50, 100)).
Fl Score (f)
SHD (L)
Distribution
Nodes (d)
HSIC
Ours
HSIC
Ours
20
0.828
0.835
6.9
6.5
Uniform
50
0.550
0.613
58.9
51.1
100
0.363
0.454
191.0
159.0
20
0.422
0.489
24.3
22.3
Truncnorm
50
0,296
0.367
119.3
101.9
100
0.283
0.358
249.7
208.6
where a, b, c are coefficient vectors and is a vector of mutually independent non-Gaussian com- ponents. This system can be expressed in matrix form as:
X Z
b
e A Ge.
(26)
Since has mutually independent components, the condition of the proposition is satisfied. Hence, the mutual (joint) independence (Y, XY, Z) is equivalent to their pairwise independence under our model.
Therefore, to test the joint independence (Y, X, Z). it is sufficient to apply LiNGIC to the pairs XY. (Y, Z), and (X, Z) and combine the results accordingly. Extending LiNGIC to direct multivariate test in this setting (in the spirit of dHSIC) remains an interesting direction for future work.
28

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Published as a conference paper at ICLR 2026
F.3 LIMITATIONS
We acknowledge that our theoretical guaranties rely on the assumption of linear mixtures of non- Gaussian variables. Extending the setting more general nonlinear mixing models would be non-trivial task and represents a promising avenue for future research.
29


Reviews:
[{"review_id": "y2E8JAUEWj", "reviewer": "Reviewer_FrNB", "summary": "This paper proposes a new independence test, LiNGIC, specifically for linear non-Gaussian (LiNGAM) data. It argues that general tests like HSIC lack statistical power in this structured setting. The core contribution is Theorem 4.2, which proves that for LiNGAM data, independence is equivalent to the conditional mean and conditional variance both being constant. The LiNGIC statistic is a kernel-based test designed to check these two conditions simultaneously. Experiments demonstrate that LiNGIC achieves higher statistical power than baselines and improves downstream causal discovery performance.", "questions": "1. The authors suggest that in certain applications, real-world data follows a linear non-Gaussian setting. Can they provide a concrete example? In particular, the proposed method is evaluated on the Sachs dataset, yet there is no evidence supporting a linear non-Gaussian assumption. In fact, methods designed for nonlinear settings often achieve better performance, indicating that the Sachs dataset may not be an ideal benchmark for this paper.\n\n2. While the paper emphasizes applications in causal discovery, the evaluation in this context is quite limited. I recommend providing detailed causal discovery results under various graph structures and data settings. Additionally, the description of the Sachs dataset in the appendix is insufficient. Which type of data was evaluated: observational, interventional, or both?\n\n3. To make the evaluation more comprehensive, the authors should consider including results from general nonparametric constraint-based conditional independence tests or causal discovery methods, such as:\n\n> [1] Strobl, Eric V., Kun Zhang, and Shyam Visweswaran. \"Approximate kernel-based conditional independence tests for fast non-parametric causal discovery.\" Journal of Causal Inference 7.1 (2019): 20180017.\n\n> [2] Zarebavani, Behrooz, et al. \"cuPC: CUDA-based parallel PC algorithm for causal structure learning on GPU.\" IEEE Transactions on Parallel and Distributed Systems 31.3 (2019): 530-542.\n\n> [3] Zhang, Hao, et al. \"Residual similarity based conditional independence test and its application in causal discovery.\" Proceedings of the AAAI conference on artificial intelligence. Vol. 36. No. 5. 2022.\n\n4. The \"Application in Causal Discovery\" (Section 6.2) shows mixed results. Although the proposed method performs best on data with \"Uniform\" noise, it fails on \"Laplace,\" \"Student t,\" and \"TruncNorm\" noise, achieving F1 scores near 0.0—comparable to other methods. This is a significant weakness, as it contradicts Figures 3 and 4, which show that LiNGIC has superior statistical power for these distributions. The paper does not address this discrepancy.", "limitations": "", "rating": 4, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "- The paper provide strong theoretical results. \n- In general, the paper is well-written.", "weaknesses": "- Although the authors have made efforts to justify developing conditional independence tests under the strong linear non-Gaussian assumption, I remain unconvinced. In practice, it is difficult to verify that real-world data strictly satisfies this assumption. Moreover, it is not entirely fair to compare their method with approaches that assume general nonparametric settings, as the learning problem is substantially simpler under the linear non-Gaussian constraint.\n\n- The evaluation of causal discovery performance is limited, with insufficient analysis across different graph structures, noise types, and data settings. Key dataset details (e.g., observational vs. interventional) are also unclear.\n\n- The paper does not include comparisons with general nonparametric conditional independence tests or established causal discovery methods, limiting the empirical context of the results.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "daiuBjnBZH", "reviewer": "Reviewer_cGuL", "summary": "The paper targets independence testing tailored to linear non-Gaussian settings. Its main theoretical observation is a characterization: for linear mixtures of independent non-Gaussian components, $X \\mathrel{\\unicode{x2AEB}} Y$ holds iff both the conditional mean and the conditional variance are constant. Building on this, the authors design a test statistic (LiNGIC) that couples a universal kernel on $X$ with a quadratic feature/polynomial kernel on $Y$, making it sensitive precisely to departures in those two conditional moments. They derive an asymptotic null and use a Gamma approximation for fast calibration. Empirically, the paper (i) reports Type-I control and power under several non-Gaussian families and increasing mixture complexity, and (ii) demonstrates downstream causal discovery by swapping this test into Direct-LiNGAM. Across synthetic settings, the method shows higher power than standard unconditional tests in the target regime, while keeping computation low thanks to the closed-form calibration.", "questions": "See weaknesses.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The paper targets the dependence between linear non-Gaussian mixtures and proves a specific and clean equivalence between independence and constancy of the first two conditional moments. \n2. The theoretical work of the test statistic seems to be solid and complete, although standard results have been given in other papers.\n3. The empirical results on pairwise dependence clearly show the advantage of the proposed method, as it generally has higher powers across all regimes.", "weaknesses": "1. The proposed method can only be applied to pairwise independence testing. A joint independence test (e.g. dHSIC) between a number of variables would have more potential applications, especially in causal discovery. I wonder whether such an extension could be possible. \n2. The computational advantage of the proposed method would be strengthened if the author could provide runtime comparisons of the method on pairwise independence testing and application to causal discovery. \n3. In the experiments, the proposed method and HSIC both used the Gamma approximation. Although it alleviates the computational burden, its performance might be reduced. A usual alternative is to use the permutation test, and the author did not discuss this part. Therefore,  results for HSIC and LiNGIC with permutation calibration should be included to see whether LiNGIC + Gamma approximation remains superior.\n4. In general, the power issue is more severe in high-dimensional causal discovery problems, i.e., many (joint) independence tests may simply reject dependence if multiple variables are involved. Therefore, I would like to see more experiments on LiNGAM with $d = 20, 50, 100$ with SHD, F1 score comparison to evaluate whether the proposed method brings a significant advantage to this scenario.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "qux6nUxTM8", "reviewer": "Reviewer_EbqJ", "summary": "The paper is related to causal discovery and proposes a novel independence test for data with linear dependencies and non-Gaussian noise. The kernel-based testing framework comes with asymptotic guarantees. Several numerical experiments for synthetic as well as real-world data are performed and results are compared to different baseline approaches.", "questions": "What are the required computational costs?\n\nDoes the approach scale in the number of components d and the sample size n?\n\nWhich model quantity could become a bottleneck?\n\nCould the assumption of linear mixtures be relaxed to more general ones?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "S1 The considered problem is interesting and relevant.\nS2 The paper is technically sound and overall well-written.\nS3 The methods used are suitable.\nS4 The evaluation and comparison against baselines is convincing.", "weaknesses": "W1 The required computational costs and scalability remain somewhat unclear.\nW2 Limitations of the paper’s methods could be better discussed.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "BUcHeGngKi", "reviewer": "Reviewer_GaAN", "summary": "This paper investigates the problem of independence testing under linear non-Gaussian data, which is an important problem in both statistics and machine learning. Under the framework where variables exhibit linear relationships and the noise terms follow non-Gaussian distributions, the authors provide a new theoretical characterization of independence, proving that the constancy of conditional mean and conditional variance is sufficient to guarantee independence in this specific model. Based on this insight, they develop a test statistic and derive its asymptotic distribution. Finally, extensive experiments on both synthetic and real-world datasets are conducted to validate the effectiveness of the proposed method.", "questions": "1. As stated in line 336 of the paper, I am very interested in seeing how the proposed method performs when using a permutation approach. Specifically, how do the Type I error rate, testing power, and computational efficiency compare with the current approach that uses a Gamma distribution to approximate the null distribution?\n2. When evaluating the test power in independence testing, could you fix the parameters $a_i$ and $b_i$, and examine how the testing power of the proposed method changes as they vary from small values (e.g., 0.1) to large values (e.g., 1)?\n3. Could you provide more simulation experiments when applying the proposed method to causal discovery, similar to the experimental setups presented in the “Simulations” section of reference [1]? Using only the causal graph corresponding to real-world data does not demonstrate the generalizability of the proposed method to different graph structures.\n4. Line 278 of the paper states that both Gaussian and polynomial kernels are used, but the simulations emphasize that only the Gaussian kernel is adopted, which is quite confusing.\n\nRef: [1] A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research, 2006.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "Originality: This paper provides a new theoretical characterization of the independence testing problem under the framework of linear non-Gaussian models. It introduces an innovative perspective by proposing that the constancy of conditional mean and conditional variance serves as a sufficient condition for independence, which represents a clear conceptual advancement.\n\nQuality: The theoretical derivations in the paper are rigorous and logically coherent. The authors not only define the test statistic and analyze its consistency but also derive its asymptotic distribution, thereby providing a solid theoretical foundation for significance evaluation.\n\nClarity: The paper is clearly written and well-structured, presenting a coherent flow from problem motivation and theoretical formulation to methodological development and experimental validation. The notation is well-defined, and the main theorems and lemmas are easy to follow.\n\nSignificance: The proposed work offers a new theoretical tool for independence inference under non-Gaussian conditions, providing a strong complement to existing independence testing methods that rely on Gaussian assumptions.", "weaknesses": "1. The paper lacks an empirical comparison with RDC [1] and MI [2], two methods capable of detecting a wide range of dependency types.\n2. Some relevant literature on independence testing and dependence measurement is missing, such as references [3] and [4].\n3. The paper claims to have conducted experiments on the SACHS real-world dataset, but the corresponding results are not reported. In addition, three figures are provided in the Appendix E.3, yet it is unclear which one is used as the ground truth. If only the first figure in Figure 7 is used, the other two seem unrelated to the paper.\n4. There are no empirical results for cases where $X$ and $Y$ are high-dimensional. To my knowledge, the main challenges in independence testing arise from nonlinearity and high dimensionality, but the paper presents no results concerning the high-dimensional setting.\n5. The definitions of the two evaluation metrics in Table 1 are not formally provided.\n\nRefs: [1] The randomized dependence coefficient. Advances in neural information processing systems, 2013.\n[2] Nonparametric independence testing via mutual information. Biometrika, 2019.\n[3] Distribution-free tests of independence in high dimensions. 2017, Biometrika.\n[4] A new coefficient of correlation. 2021, Journal of the American Statistical Association.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)