On the identifiability of causal graphs with multiple environments
Authors: Francesco Montagna
Venue: ICLR 2026 Poster (2026)
Keywords: causal discovery; heterogeneous data; multiple environment; nonlinear independent component analysis
Primary area: causal reasoning
TL;DR: We show that the causal graphs of structural causal models with arbitrary mechanisms are uniquely identifiable from the auxiliary information of only two environments. This bridges the gap with ICA identifiability results with multiple environments.

Abstract: Causal discovery from i.i.d. observational data is known to be generally ill-posed. We demonstrate that if we have access to the distribution induced by a structural causal model, and additional data from (in the best case)  *only two* environments that sufficiently differ in the noise statistics, the unique causal graph is identifiable. Notably, this is the first result in the literature that guarantees the entire causal graph recovery with a constant number of environments and arbitrary nonlinear mechanisms. Our only constraint is the Gaussianity of the noise terms; however, we propose potential ways to relax this requirement. Of interest on its own, we expand on the well-known duality between independent component analysis (ICA) and causal discovery; recent advancements have shown that nonlinear ICA can be solved from multiple environments, at least as many as the number of sources: we show that the same can be achieved for causal discovery while having access to much less auxiliary information.

Published as a conference paper at ICLR 2026
ON THE IDENTIFIABILITY OF CAUSAL GRAPHS WITH MULTIPLE ENVIRONMENTS
Francesco Montagna
Institute of Science and Technology Austria, Chan Zuckerberg Initiative francesco monnagna@ ist ac
ABSTRACT
Causal discovery from i.i.d. observational data is known to be generally ill-posed. We demonstrate that if we have access to the distribution induced by structural causal model, and additional data from (in the best case) only two environments that sufficiently differ in the noise statistics, the unique causal graph is identifiable. Notably, this is the first result in the literature that guarantees the entire causal graph recovery with constant number of environments and arbitrary nonlinear mechanisms. Our only constraint is the Gaussianity of the noise terms; however, we propose potential ways to relax this requirement. Of interest on its own, we expand on the well-known duality between independent component analysis (ICA) and causal discovery; recent advancements have shown that nonlinear ICA can be solved from multiple environments, at least as many as the number of sources: we show that the same can be achieved for causal discovery while having access to much less auxiliary information.
INTRODUCTION
Causal discovery seeks to recover cause effect structure from data, which allows counterfactual reasoning and prediction under interventions (Pearl) 2009 Peters et 2017: Spirtes) 2010 Spirtes et al. 2000). However, learning causal structure from purely observational i.i.d. data is, in general, ill- posed: multiple directed acyclic graphs (DAGs) are distributionally equivalent, i.e., indistinguishable from the data distribution.
In the interventional causal discovery literature, hard interventions-dicee modifying the causal structure-are known to unlock identifiability of the underlying graph. Classic results from Eberhardt jet al. (2005) show that the number of sufficient hard interventions to identify the causal order scales logarithmically with the number of nodes. A large corpus of intervention-based causal discovery
research has largely built on these findings (Ebberhardt) 2008 He & Geng
2008 Hauser & Bühlmann
2012 [Shanmugam et al. 2015 Kocaoglu al. 2017 7: Wang al. 2017 Lindgren 2018 Eaton
& Murphy [0077] Triantafilouu & Tsamardinos 2015 Lorch et al. [0022] Ke et al. (023b).
Recent work has explored the problem of causal graph identifiability from multiple environments and soft interventions (i.e., in the setting where non i.i.d. data might naturally occur and does not stem from changes in the causal structure) (Perry al. 2022 Huang al. 20200 Heeizee-eeml et al.
2018, Peters et al. 2015 Ghassami et al. 2017 Jaber Jaber et al. [2200] Jalaldoust et al. 2025 Brouillard et al., 2020 Heurtebise et al. 2025): however, from an identifiability perspective, these results do not provide guarantees of recovery of the unique causal graph with a limited number of environments under generic assumptions.
Our research overcomes this limitation. We prove that, for structural causal models (SCMs) with arbi- trary nonlinear mechanisms, auxiliary information from only two sufficiently distinct environments is enough to identify the unique causal graph. Our only constraint is the Gaussianity of the noise terms; however, we outline potential ways to relax this requirement. To our knowledge, this is the first proof of identifiability for full graphs of arbitrary size and generic functional mechanisms from constant number of environments. Strengthening our findings is the contrast with where the number of experiments needs to scale with the number of nodes.
regimes,

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Our work is also of independent methodological interest. In particular, key to theory is the duality between causal discovery and independent component analysis (ICA). Reizinger et al. (2023) recently formalized that nonlinear ICA identifiability results naturally extend to structure learning (well known in the linear case since [Shimizu et al. (2006)). This is of great relevance in light of the late advancements in multi-environment ICA identifiability pioneered by HHvärinen & Morioka (2016); however, directly bootstrapping these findings to causal discovery doesn't I carry great promise, being ICA the harder problem of the two: we show that where ICA identifiability requires a number of environments that scales linearly with the number of variables, causal graph identifiability can be achieved with data from just two extra domains. This calls for causality-only identifiability results in the multi-environment setting, as developed in our work. Inspired by the recent success of ICA with multiple environments, we are hopeful that our approach paves the way to novel causality theory that weakens the requirements in terms of heterogeneity of the data and parametric assumptions.
Our main contributions are summarized as follows:
We show that the causal graph underlying an arbitrary invertible causal model with Gaussian noise terms is identifiable from only two auxiliary environments, when they sufficiently vary. Moreover, we outline potential avenues to relax the Gaussianity assumption.
A methodological contribution consisting of proof techniques that are novel for causal discovery and leverage the (well-known) duality between structural causal models and independent component analysis; to the best of our knowledge, these are the first causality- only identifiability results for nonlinear SCMs that stem from this connection.
We empirically validate our theory. Our synthetic experiments reflect that when the assumptions of our theory met, we can infer the causal direction, even for cases that were previously known to be non non-identifiable. The code to reproduce our results can be found at bttps://githnee wtt.a//ww........eeeeeeeoooooooe
hatp///aussss wwwammmmlllieeeveeee git
RELATED WORKS
Soft interventions and multiple environments for causal discovery. Several works in the liter- ature have addressed causal discovery identifiability and estimation via non i.i.d. data (stemming from soft interventions and multiple environments). Peters et al. (2015) and Heinze- Demll et al. (2018) identify the parents of a designated target node via invariance across environments, yielding partial identifiability of causal directions. They assume linear and nonlinear additive noise models, respectively. Huang et al. (2000) use nonstationarity to recover the skeleton and orient some edges. Perry et al. (202)) leverage sparse mechanism shifts, proving high-probability graph recovery with bounds that improve
the number of environments grows. [Rothenhäusler al. (2015) is the
closest to our work, but their results are limited to linear models. Ghassami et al. (201 7 2018) and Heurtebise et al. (2025), similarly to our work, study identifiability of structural causal models from multiple environments, but their identifiability results are specialized to the linear case. Recently, Jalaldoust al. (2025) formulated statistical test that can find a superset of the parents of target node. Yang et al. Brouillad Brouillard et al.) 2020); Jaber et al.) (2020) characterize equivalence classes identifiailiiy from interventions. From a methodological perspective, Brouillard et al.) (2020); Ke et al. (023a) introduce differentiable approaches to causal discovery with interventions; Mooij et al. (2020) propose a unifying framework for causal discovery from observational and multi calli--enirromment data. All of these results are complementary to our work, which is, to the best of our knowledge, the first to provide guarantees of identifiability of the causal graph from a finite number of auxiliary additional environments, potentially only two.
ICA and causal discovery. The seminal work of Shimizu et al. (2006) shows that if an SCM can be expressed as a linear non-Gaussian ICA model, the underlying causal graph is identifiable, Reizinger et al. (2023) generalize this to the nonlinear case. Monti et al. (2020) show that time contrastive ICA (Hyvärinen & Moriokal 2016) can identify bivariate causal graphs with arbitrary nonlinear mechanism. The common ground of these findings is that they adapt the existing ICA identifiability theory to the problem of causal discovery. This approach is clearly important, especially in the light of the recent advancement in multi- environmeet ICA identifiability (Hyvärinen et al. [2199] 2020a Gresele et al. 2019; Hälvä & Hyvärinen 2020; Hyvärinen & Morioka 2021) however, in the nonlinear setting, it fails capture the gap between the
Khemakhem et al.
2017: Hälvä al.
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two problems: while ICA attempts recover the mixing function and the independent sources each point, causal discovery concerns the much simpler problem of structure identifiability. Our work shows that this difference is key to demonstrating causal discovery identifiability from constant number of sufficiently different environments, where ICA requires at least as many as the number of sources (see e.g. Theorem in Hyvärinen & Morioka (2016)).
PRELIMINARIES
First, define structural causal models, independent component analysis, and how they relate. Then, we describe the problem of causal discovery from multiple environments and define identifiability of causal graphs in this context.
3.1 STRUCTURAL CAUSAL MODELS AND ICA
Let us consider set of causal variables X, with components generated according to a structural causal model
X, ((XPA..:)) Vi Ii.,,
(1)
where XPA, are the causes of X,, specified by directed acyclic graph (DAG) 9 with nodes X. PA; C (1....dd} d} denotes the indices of the parents of X; in the graph (see Appendix G.1 for precise definitions on graphs). The functions F; are the causal mechanisms that map causes to effects. We assume mutually independent noise terms S (Si,..., Sa) with density pa. where is set of parameters defining the density function. Further, we restrict to structural causal models where there are no latent common causes.
Notational remarks. We use [d] (1,... d}. We use uppercase letters for random variables (or vectors), lowercase letters for their realizations. Vectors are denoted in bold, so that we have v (1x)==1 where v; are the scalar vector's components. Probability density functions are differentiated by their argument, where the distinction clear from the context: for example, for random vector Z we only write p(z) to specify its density at certain value z. Further, we define the support to keep track the nonzero entries in matrices: for M E Rmx" supp(M) := j(i,j]l e [m],j E [n] and Mij * 0}; for matrix valued function M Rdd - Rmx the support is defined supp(M)
{ [m], E [n] and there is Rd s.t. M,s(x) + 0). Given vector V = (Vi))<aa and subset [d]. we define V, := (Vihez. For indexing, we reserve superscripts as in V to distinguish between environments (Definition 3)
is well known that the SCM of Equation ( can be expressed in the form of an ICA model
x = f(S),
where f is the ICA mixing function, uniquely specified by the SCM (we show how to construct in Appendix G.2). Definition (ICA model). We define a pair (f. pe) as an ICA model, where is a diffeomorphism in RC, and pa is factorized density parameterized by 0.
A more detailed introduction to independent component analysis is presented in Appendix D
known (Reizinger al. 2023) that, under some faithfulness assumption, the support of the Jacobian of the mixing function completely identifies the causal structure.
Definition (Faithfulness). Consider x = f(s). We say that Ję-1 (x) is faithful for each i, E [d] Jr 1 (x), <>> S; is constant in X; on the entire domain. In other words:
supp(Jr- (x)) = supp(JJ-1).
(3)
Proposition 1 (Proposition 1 in Reizinger et al. (2023)) ). Let 1 faithful. Then, for each j:
Jr-1 (x) = - j PA
This formulation of faithfulness is well known and at the of the LiNGAM algorithm for linear SCMs (Shimizu et al. 2006). and is satisfied almost everywhere under some regularity conditions on f. When this is the case, the above proposition means that for causal discovery we interested in
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the support of the inverse Jacobian, and, by Equation 3 this can be recovered by having access to the support at a single point where faithfulness is satisfied.
Next, we introduce the notion of environment and define the causal discovery problem when multiple environments are available.
3.2 DEFINITION OF IDENTIFIABILITY FROM MULTIPLE ENVIRONMENTS
Intuitively, causal discovery is the inference problem of finding the causal graph underlying structural causal model from the data. We are interested in causal discovery from multiple environments, i.e., when data are collected from different but related structural causal models (which we express as ICA models).
Definition (Environment), Let x f(S) be ICA model. Consider the random variable Sè p'. For i = 1,..., k, we call the ICA model X f(St) an auxiliaryy environment. We adopt the the base environment, p' denotes the probability
convention that X S := X,S, and call density of the sources defined by the 8h environment.
The key part our definition that the mixing function is invariant across environments (real- world examples where this is satisified can be found in Appendix G.5), while allow for changes in the sources distribution:
is obtained from structural causal model (as it assumed across all of our paper), all auxiliary environments share the same causal mechanisms and causal graph the base model X f(S)).
Next, formalize what mean by identifiability in the context of causal discovery with multiple environments. Intuitively, identifiability is achieved when the graph underlying the structural causal model is uniquely specified by the causal variables' distribution. In the definition, we denote the pushforward of a density p by with f.p.
Definition& 4 (Identifiability of the causal graph). Consider the auxiliary environments (f. pb) obtained from the base causal model of Equation (T). = L....h k. Let F be the space of diffeomorphisms in H and family of factorized densities. We say that the causal graph underlying the SCM is identifiable if, given (F.P') E x P, 1,... k, then:
f.Pb p.p. Vi [k] supp(Jr-1) = supp(J
The above definition of identifiability, based on the support of the Jacobian inverse of the mixing function, may be bit unfamiliar, but it's equivalent to what is commonly meant when asking that causal DAG is identifiable: any alternative causal model that matches the distribution of the data is compatible only with the ground truth causal graph (represented with the inverse Jacobian's support).
Relation with ICA identifiability. Compare Definition of identifiability of the causal graph with the notion of identifiability in ICA of Definition 5 in the appendix: for causal discovery, all care about is the support of Jr-1, which can be identified from any point where the Jacobian is faithful; for independent component analysis, we need to guarantee that the exact values of the Jacobian can be recovered over each point of the domain, up to trivial indeterminacies. This phrasing clarifies that, in the nonlinear setting (where the Jacobian varies with x), causal discovery a much simpler problem than ICA: it only requires identifying the support at a single point, rather than the value at any point. This is reflected in our main identifiability result (Theorem [[: we will show that the causal graph of nonlinear SCM can be identified with the information from only two auxiliary environments; this in stark contrast with ICA identifiability results for general mixing functions, that usually require number of environments that scales linearly (O(d) with the number of sources.
Problem definition. We aim to characterize the conditions under which the causal graph G is identifiable from the fewest possible environments.
THEORY
To develop theory, we rely on the following assumptions the ICA model of Equation
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Assumption 1. f is global diffeomorphism and twice differentiable.
Assumption 2. Each environment is obtained as a rescaling of S. namely S is distributionally equivalent L,S for each E [e]. with L; diag(X),.... X4)- We ask that for each j E [d]] there is at least one value of i E [e] such that X3 + O.
Assumption 3. For f (x) where μs.. the mean of the vector of sources, the Jacobian is faithful (Definition 2
Assumption 4. S has Gaussian density pe with 0 mean and covariance matrix parameters.
Discussion on the Assumptions 14 Assumption T is standard when proving identifiability: the results in Hoyer et al. (2008); Zhang & Hyvärinen (2009); [mmer al. (022) based higher- order derivatives, and have strong requirements that guarantee diffeomorphic causal mechanisms (Corollary 3.5 in Foominguez-Olmede et al. 2023)). Also Assumption 2 is mild and somewhat necessary: it simply asks that the interventions are meaningful, i.e. that they affect the variance; interventions on the mean, intuitively, are not informative they shift the density graph by a constant, without affecting its shape (the gradient and the Hessian of the density, where information about the causal graph lies). Assumption requires that the Jacobian of the inverse of the mixing function is informative about the causal structure at the mean of S (and it's almost surely verified over X samples, under some generic regularity conditions on f). The reason behind it is that we probe the identifiability of the Jacobian's s support at the mean. The only real simplifying constraint is Assumption 4 of the Gaussianity of the sources, which is, however, not new in the literature (see, e.g., Rolland et al. 2022)). Later, we discuss why this assumption is needed in the paper and potential ways to relax it (Section
In the remainder of the paper demonstrate that, under these assumptions, leveraging the ICA formalism we can prove the identifiability of causal graphs, potentially with as few as two auxiliary environments. Our starting point is the invertibility so that write the density of X with the change of variable for each value x f(s) as:
p(x) = po (s) J--
Consider an alternative invertible ICA model (Definition I
(f, Pb) such that:
p(x) Pg (/)
(5)
We define the indeterminacy funccttio
h
(6)
which "quantifies" how different the two ICA solutions are. By the multivariate chain rule, the following relation among Jacobian matrices holds:
Jr J..
We show that (under Assumptions 1 Po) there is at least one point x = f(s) f()) such that the Jacobian (s) is scaled permutation, meaning that Jr- support is identifiable up to column per- mutation. Given that for acyelic causal models permutations are easily removed (Shimizu al. (2006), this is equivalent to identifiability of the causal graph in the sense of Definition as we discuss next.
4.1 IDENTIFIABILITY FROM SECOND ORDER DERIVATIVES OF THE LOG-LIKELIHOOD
In this section, we present our main theoretical result and the intuitions behind Our argument for identifiability relies on the analysis of the Hessian of the log-likeliine likelihoohd of X for all environments. We consider the case where f f- f-(() = s μs (by construction, there is a unique corresponding s = fx))).. We partition the set of e auxiliary environments into two groups I₁ = {1...., e } and To {er + 1, 1...., 61 + e2), where 61 62. Then, define the following quantities:
Σ
D po(s) DÎ log (s)
ich
(8)
Ω2 Σ
D log D log (s),
where D2 denotes the differential operator that returns the Hessian matrix. Similarly, define
fi by replacing with Ò. The introduction of Ω4 Gg 1 1, 2, is instrumental for the next result.
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Lemma 1. Letx = f(s) f()), where ps. Let Assumptions 1 and satiisfed. Then:
Σ
D p(x) - Dx log pxxx) xx- .2 1p - xx) 1x (9)
Σ iE/s
D logg(x) D (x) Jp- (x (x) 11 (x)
(10)
The proof is derived by direct computation and can be found in Appendix C.2 We point Lemma in Varici et al. (2025) for related results that analyze the difference of first-order derivatives of the log-likelihood, the context of causal representation learning with soft interventions.
We can intuitively illustrate how the identifiability of the Jacobian's support follows from our
Lemma 1 A first remark is that the 94 S4 matrices are diagonal. That is because, for a vector of mutually independent random variables, the Hessian of the log-density diagonal (see Appendix G..3
G.3
for details about it). Second, by the chain rule, Equations (9 and 10) imply Jn(s) (a,s)) = S4 for I = 1, from which
- ( =2,
12, '222.
(11)
This means that (s) maps one diagonal matrix to another: if the eigenvalues of θ 1 f22 are distinct, that enough to force (s) to a scaled permutation, which is exactly our goal. This sketched argument is key to understanding how Equations (9) and (TO) provide enough constraints to identify the support of Jr-1. Clearly, this discussion implicitly requires that f2 and f42 are full rank. This can be achieved under the following conditions over the rescaling matrices L; .iag(A).....AAe that define the multiple environments.
Assumption 5 (Sufficient variability). For each j [d]
e1+t2
C1 and Σ
(25)2 # 62.
(2
The assumption basically requires that there is sufficient variability between the different environ- ments. Similar requirements of sufficient variability are ubiquitous in the nonlinear ICA literature (e.g. Hyvärinen Morioka (2016); Khemakhem et al. (2020b); Lachapelle et al. (2022). Intuitively speaking, Assumption 5 is satisfied when, for each of the two groups of environments (1 ..., E1 and e1 + 1.... e + e2), each source S, is subject to rescaling. To see that, consider the LHS the first equation: X; = for each i = I,.,. é corresponds to the case when the variable S,, is never subject to rescaling in any of the environments, and indeed yields violation of the assumption. Note that even if Sj is subject to rescaling for some index i, the values of (A) (X))ae(e]] can always be tuned such that the assumption is violated; however, this corresponds to pathological choices of the rescaling coefficients, which never occur in general (shown in Proposition in the appendix).
Next, are ready to state our main identifiability result.
Theorem 1. Consider the groundtruth ICA model (f., po) of Equation and the alternative (f. Pg)- Let Assumptions 5 be satisfied. and assume that the elements in the set {(9) (2)) are pairwise distinct. Let x f(s) f($) and jsg then, the indeterminacy function h == f of satisfies (s) = D, meaning that the causal graph G is identifiabe.
Theorem assumes that the elements in the set ment excludes pathological choices the coefficients of the rescaling matrices T. that define the multiple environments, and it is generically satisfied (Proposition in the appendix).
are pairwise distinct. This require-
Proof sketch (full proof in Appendix C.4
By Lemma we have
M = Ω. 1= 1,2,
(12)
where M := Jn-1 (s). Define A: θ2 ②2 and B = 1 From Equation 12 we can show that A = M-'BM i.c. that A and B are similar. Moreover, being ( Q2 )a i==1 elements pairwise distinct, the diagonal elements of A and B are never repeated. Note that the eigenvectors of

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diagonal matrix with all distinct eigenvalues are aligned with the standard basis: given that M, by definition of similarity, maps the eigenvectors of A to eigenvectors of B. we conclude that it is a scaled permutation. The permutation is removed leveraging the acyclicity of the causal model, according to Lemma 1 in Reizinger et al. (2023). Assumption implies that the causal graph is identified. IT
Identifiability from two auxiliary environments. The theorem tells that, given that we have access to two groups of auxiliary environments, both inducing changes in the variance of all sources, at the mean of the sources the ground truth and the alternative models are equivalent up to rescaling. This constrains the support of J/ 1 of the alternative model to be equal that of Jr which is cnough to guarantee identifiability of the causal graph. It is interesting to discuss the theorem when e C2 2, showing that the above result demonstrates identifiability with as few as two additional environments. In this setting, if L1 = ding(>})ş and L2 = diag(>} (})) with 21, >3 * for each j E [d], then we have two extra environments where the variance of all the sources is affected by rescaling. This is sufficient to guarantee that the assumptions of Theorem [ are met. An important consequence is that the number of required environments does not scale with the number of nodes in the graph, in contrast with similar findings for nonlinear ICA identifiability, As long as there is sufficient variability in the sources of two environments (relative to the base model), we are always guaranteed that the causal graph can be recovered.
Theorem I beyond Gaussianity, Theorem [ inherits the assumption of Gaussianity from Lemmal I here, we briefly discuss potential ways to relax it. At a general point x = f(s) the Hessian of the log-likelihood is equal
JF-1(x) log p' (s) D& log (x)) + Ds, log p' (s)) (x).
(x).
The log-determinant term cancels by taking the difference between environments. To recover note that the summation of second-order derivatives vanishes Equations (9) and (T)) in [e
when V po (s) = O, namely at the mean of the Gaussian sources. However, this can hold for any source distribution that has at least one point where the gradient is zero, a remark that naturally extends Lemma [ (and hence, Theorem I to larger class of causal models. Moreover, from a practical perspective, even if the gradient of the log- likeelioooood of the sources does not vanish, Lemma [ is approximately true when the gradient is sufficiently small. This can occur, e.g., for heavy-tailed distributions. This analysis should convince that Gaussianity is a sufficient but not necessary requirement, and hopefully inspire future research to extend our identifiability results. Mathematical details on the steps in this paragraph, as well as an expanded discussion on the generalization of our theory for more general classes of distributions, are found in Appendix E.4
Next, we validate the conclusions of Theorem T with experiments.
5 EMPIRICAL RESULTS
In this section, we report and analyse empirical results that validate our theory. Our experiments on synthetic data show that if the assumptions of Theorem @aneem[hooee holdd the causal direction can be recovered from the data. In the main paper, we focus on bivariate graphs, commonly adopted as the casiest yet non-trivial setting for testing identifiability (e.g., Hoyer et al. (2008): /Zhan & Hyvärinen (2009)
Immer et al. (202)). Additional experiments on multivariate causal graphs are in Appendix E.5
5.1 SYNTHETIC DATA GENERATION
We generate synthetic data from bivariate causal models with independent noise terms, sampled from a normal distribution with unit mean and covariance entries uniformly drawn between [1, 1.5). Given the variables 11, 22 and the graph ø
+ 22 we consider the following causal mechanisms that comply (i) x2 of arctan(s2) + s (ii) x2 := sîs2 + arctan($2) (x2) (iii) D2 : of + arctan( $1 )82 + 81 2.. Note that any of these models can not be reparametrized to post nonlinear or location scale noise model, which are the most general SCMs identifiable from pure observations (Zhang & Hyvärinen 2009: Immer et al. 2022). Additionally, we consider data from linear Gaussian model, notably non-identifiable. We experiments on datasets with (3, 6, 9)
with the assumptions of Theorem I
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SHD with Gaussian noise
1.0
environments
km3
0.8
Mean SHD (# ssdd
km9
0.6
0.4
0.2
0.0
monlinear anm
pnd
Inm
linerr anm
abbirary nonlinear (1)
arbitrary nonlinear (i)
arbitrary nonfintta (11))
Figure 1: Average SHD (0 is best, 1 is worst) achieved by Algorithm | over 50 seeds on binary graphs. When the assumptions of Theorem T are satisfied, the method can appropriately infer the causal direction, both in the observationally identifiable setting (nonlinear ANM, PNL., LSNM) and the observationally non-identifiable one (linear Gaussian model and the three SCMs with arbitrary nonlinearity). The number of environments does not have a notable effect the accuracy.
environments. For each environment, we generate 2000 observations. In Appendix E.4 we discuss experiments with non-Gaussian independent sources. Interestingly, these additional results seem to support our hypothesis that Theorem could be extended to other source distributions.
5.2 ANALYSIS OF THE EXPERIMENTAL RESULTS
In this section, we analyse the empirical results. First, we introduce an algorithm for inferring the Jacobian support that leverages theory.
Algorithm. The simplified pseudocode found in Algorithm (a detailed version is presented in Appendix E.3 The steps in our procedure closely follow the proof of Theorem [ this approach to algorithmic design is not necessarily the best, which is why we highlight that our method is not within our main contributions. For single inference, the input is the data tensor X E IR &xxxx for each environment from to & it consists of a dataset with n observations of d causal variables. Additionally, we are given the sets I1, 12 [k] of indices that split the auxiliary environments into two groups, as required by our theory. The first environment taken as the base one. We have two steps where statistical estimation is involved: (i) For each environment, the gradient and the Hessian of the log-likelihoud are approximated via the Stein gradient estimator, introduced in Li Turner (2018) and popularized in causal discovery by RRolland al. (2022) Montagna et al. (2023b); ; (ii) For each environment
[k), we need to find the observation j E [n] such that f-1( (.))) R uss that is, the data point generated mixing the source vector at the mean. Fortunately, this can be consistently estimated from the score V Voog Px, as we demonstrate
Proposition 2 in the appendix. These two steps are achieved by [ the end of the first for loop. At this stage, all statistical quantities have been estimated: we note that, being the Stein estimator consistent, the algorithm is correct. in the infinite sample limit. In the second for loop, we take the points at the estimated mean that we previously found, and compute the difference of the Hessians between the base and auxiliary environments: this exactly mirrors the first equality in Equations (9) and (10) of Lemma Next, in the algorithm's 's notation, we compute
M = fa [1] v JeST -2JE-11
(13)
Then, we solve the linear system Amm. [1]M = Fa [2] to find M. In the infinite samples limit Equation 13) is a precise equality, such that M and S2 L2 are similar: diagonalizing M we find Jr-1 up to scaled permutation. The permutation indeterminacy is removed leveraging the assumption that the causal graph is acyclic via standard arguments (see Shimizu et al. (2006) and Reizznger et al.) (2023)). Finally, the algorithm returns the estimated support of the inverse Jacobian.
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Algorithm I: Estimating supp Jp-1 from the data (algorithm sketch)
Data: X E Rxxxxxd
V env: n dimensional obeervations,
11 [k]
11 Set of indices splitting the envircnments in two groups
Result: Estimate of supp Jp-
+ score_estimatte (X) Rkxuxd
H f dess_estimater X) E pxxxxdxd
// each environment e, find the aample ccrresponding the of the source for e 1,..,, do
me + i.t. r- (X(e,))
end
// Difference of Ressions at the mean (i.e. Equations
and EG
Ham + 0 0 R2xdxd
for l = 1,2 do
for e E I8 do
AH H 11 [0. mil
11 me]
11 ma is the index For the base env irnnent
Hams [ + Hams [8] Ah.
end end
M [1 Hu [2] J 1 D2-4-1
11 Hamld] xo d7-1 Redg911-11 by Equations
and
10
Jr + diagonalize ~ Jp DPP
return supp (Jr. P-1) 11 P. can be found using the acyal lofty of the causal graph.
Analysis of the experiments. In Figure D we illustrate the empirical performance of our method on several synthetic datasets generated from bivariate causal model. We consider SCMs with the arbitrary nonlinear mechanisms (i). (ii), (iii) described in Section[5.1] and linear Gaussian models; as a sanity check, we also experiment on nonlinear additive noise models (ANM), post-nonlinear models (PNL), and location scale noise models (LSNM), which are all the nonlinear SCMs where identifiability can be achieved from observational data (see Appendix E.2 for details). All datasets are generated under the assumption that causal effect exists (i.e., the ground truth graphs always have one arrow). We measure the errors through the structural hamming distance (SHD). This equivalent to the number of edge additions, removals, or direction flips that required to recover the ground truth graph from the estimated one: SHD=0 corresponds to correct inference, SHD=1 to an eтror. For each experimental configuration, consisting of function type and number of environments, consider 50 seeds over which we compute the empirical mean and deviation of the SHD. The results are in line with our theory: we see that for the three models with arbitrary mechanisms, and the linear Gaussian SCM (all non-identifiable from pure observations), the average SHD is close to O, which is especially evident when we do inference with only 3 environments. Interestingly, environments doesn't always have beneficial effect. This is not surprising, we showed that two auxiliary environments are sufficient for inference. The method can also infer the causal direction for the ANM, PNI,, and LSNM. We conclude that the empirical outcomes support our theory.
that adding
Remark on multivariate graphs. Multivariate experiments are delayed the Appendix E.5 linear Gaussian SCMs, we find that our method can infer the causal order with only 3 environments for graphs up to nodes, which is strong evidence in support of our theory. In the nonlinear setting, our method struggles to scale to high dimensions, and we limit our experiments to 5 nodes. A detailed discussion on the scalability of our approach is provided the Limitations section B.2 in practice, scaling causal discovery with multiple environments beyond the bivariate setting is well-known. unaddressed challenge, already found in Reizinger et al. (2023) and Monti et al. (2020). Given that algorithmic contributions fall beyond the scope of our paper, we leave this open problem for the future.

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CONCLUSION
We demonstrated that the causal graph of a structural causal model with arbitrary nonlinear mech- anisms is identifiable; surprisingly, this can be achieved given the auxiliary information of only two (sufficiently different) environments. Our main assumption is the Gaussianity of the noise terms, for which, however, we discuss potential relaxations. Our findings extend on the well-known duality between ICA and causal discovery: the first problem concerns the identifiability of the independent sources at each point, whereas causality only needs to access the support of the Jacobian mixing func- tion at single point, when faithfulness is satisfied. 'he exciting consequence of this asymmetry is that while ICA identifiability requires a number of environments that grows linearly with the number of sources, for causal discovery, constant number is sufficient: this makes our thcoretical results appeal- ing even in high dimensions. We hope that our work inspires novel identifiability theory beyond the Gaussianity constraint. Moreover, in light of our results, finding efficient and effective algorithm for causal discovery with multiple environments and in high dimensions is a promising research direction.
Reproducibility statement. Section 5.1 describes the specifics for generating the synthetic data of our experiments. Appendix E. discusses the computational resources that were required for their execution. As supplementary material, we provide a zip folder that allows reproducing our empirical analysis. Particularly, it contains the Python code for: Algorithm 2 the synthetic data generation, the experiments execution, and the visualizations of the figures of this paper. For theoretical results, we explicitly state and discuss in detail all the assumptions (Assumptions (our main contribution). A proof sketch and a detailed demonstration are included in the main text and the appendix. respectively (Appendix C.4)
5) required in Theorem [
ACKNOWLEDGMENTS
FM acknowledges the Chan Zuckerberg Initiative for the financial support.
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CONTENTS
Introduction
2
Related works
Preliminaries
Structural causal models and ICA
3.2
Definition of identifiability from multiple environments
4
Theory
4.1
Identifiability from second order derivatives of the log-likelllne legliiklihhod
5
Empirical results
5.1
Synthetic data generation
5.2
Analysis of the experimental results
6
Conclusion
10
LLM usage statement
17
Limitations
17
B.1 Theory
17
B.2 Experiments
18
B.2.1
Synthetic data
18
B.2.2
High dimensional graphs
18
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Proof of the theorefical results
18
Preliminary theoretical results
18
Proof of Lemmal]
19
Identifiability of the mean of the sources
19
4 Proof of Theorem
21
D Independent component analysis
22
Experiments appendix
22
Computational resources
22
Structural causal model identifiability from observational data
23
Detailed pseudocode of Algorithm
................................
23
B,4
Experiments beyond Gaussianity
23
E.5
Experiments on higher dimensional graphs
25
E.5.1
Experiments on linear SCMs
26
E.5. 22
Experiments nonlinear SCMs
27
Assumptions deepdive
28
Beyond Gaussianity
28
F2
Beyond causal sufficiency
29
G
Additional content
30
G.I
Graph theory
30
6.2
From SCM to ICA models
30
G.3
Hessian of the log-density independent random variables
................................
30
G.4
Measure theoretic arguments in support of the assumptions
31
G.5
Fixed mechanisms environments in real-world data
32
A
LLM USAGE STATEMENT
In this work, LLMs were occasionally used for polishing and improving the writing. All research contributions in terms of theory and experiments' analysis were carried by the authors.
B
LIMITATIONS
In this section, we discuss the limitations our work and the open problems it leaves.
B. THEORY
The main constraint our theory is the requirement Gaussian noise terms. In the main text (cf. Section 4.1 the paragraph Theorem 7 beyond Gaussianity), we discuss how this assumption sufficient but might not be necessary. In fact, our theory can be extended to a structural causal model where the distribution of the sources has vanishing gradient at point. Our work does not address how to extend these result to arbitrary continuous distributions, which remains an open problem.
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B.2 EXPERIMENTS
B.2.1 SYNTHETIC DATA
One limitation in our work is that experiments run synthetic data. This is common in the causal discovery literature due to the challenge of accessing data with a reliable ground truth causal graph. Moreover, data collection often happens under the i.i.d. assumption: this hinders the application of our algorithm on common benchmarks such as, e.g., the Sachs dataset (Sachs et al. 2005), which doesn't 't dispose of multiple environments.
B.2.2 HIGH DIMENSIONAL GRAPHS
In Appendix E.5 we analyse experiments over graphs with more than 2 nodes. We find that, for linear Gaussian SCMs, our method can accurately infer the causal order of 50 nodes with few three environments. However, for nonlinear structural causal models, performance quickly deteriorates with the number of dimensions. In general, we find that in the nonlinear setting, developing an effective algorithm for multivariate causal discovery with multiple environments is challenging problem. This doesn't It come as surprise, being already well reported in the recent literature: Reizinger et al.) (2023) (Table 1) show that for graphs with 5 nodes, neural-basee views fails even converge to a causal order on 40% of the test runs; on 10 nodes, convergence occurs with 27% rate. Perhaps even more remarkable are the findings of Monti et al. (2000) (Figure 2) showing that, as the causal mechanisms become nonlinear, contrastive-bant to recover the causal order better than random baseline even for just two nodes. This highlights that algorithmic multi-environment causal discovery, even for small graphs, is an open and challenging problem that requires intensive research of its own-which is not in the scope of our paper.
contrastive learning from multiple
nonlinear ICA fails
Despite the clear limitation, it is important to keep in mind that the goal of our experiments is to demonstrate that the assumptions of Theorem -ou main contribution-arr sufficient identify the causal direction, and not to present novel algorithmic contributions. To this end, bivariate models are well-known be the easiest yet non-trivial setting: in fact, our experimental setup is reminiscent of that of Hoyer et al. (208);; Zhang & Hyvärinen (2009), two seminal papers in the identifiability theory of causality which limit their theoretical and empirical studies to bivariate causal graphs. This also aligns with several empirical and theoretical identifiability studies in causal discovery (e.g.. Mooij al. (201T): Ghassami et al. (2017); Montagna al. (2024); [mmer al, (022)); Xi et al. (2025); Monti et al. (2020): Strobl & Lasko (2033), which makes our choice to focus on two-variable graphs well-justified. We leave the challenge of developing an algorithm suitable for multi-environment causal discovery in higher dimensions open problem.
C PROOF OF THE THEORETICAL RESULTS
C. PRELIMINARY THEORETICAL RESULTS
In this section, we collect the theoretical results useful for the proof of Theorem !
Lemma (Full rank of Q under rescalings). Assume Gaussian sources S with independent coordi- nates, and environments generated by rescalings St = L.S with L, diag(A; .o.g(,....,,.,, and X3 & O. For 1 E {1,2} define the index sets I f1,.... e1} and 12 fer + 1.....e e1 + e2), and recall
2
Σ RENI
D3 lo)))
D2 lo
evaluated at the same s. Then each Q4 is diagonal with entries
(O4)
[4
of
iich
and therefore
S4 is full rank
Vj
|d]
Σ
# 14].
(2:)²
iET.
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Proof. For univariate Gaussian, D2 log p(sp) -1/03. In environment have S} 2j.SJ. so Sj has variance (A}o))² hence D2 logp'ss) =1/(1)00,) Thus
(D", log p(g;();) D2 log
(ogp((s)) (+,))
= o3 (2)) -1
Summing over E It gives the stated diagonal form. A diagonal matrix is full rank iff none of its diagonal entries is zero, which yields the equivalence.
Lemma 3. S4 is invertible implies Qu invertible.
Proof. By Lemma] I for = 1, we have:
Jr (x) (x)xxxX4 1 = (xp
Under Assumption 5 by Lemma 2 the LHS a product full rank matrices, and so is full rank; so must be the RHS. Given that rank (AB) I min (aank(A), rankk()) (for generic matrices A, B) conclude that Q also full rank.
C.2 PROOF LEMMA
We report the content of Lemmal 1 followed by its proof.
Lemma 1, Let x = f(s) = f($). where s - us.. Let Assumptions
2 and satisfied. Then:
Σ D log (x) D log p' (x) (x)² 82 Jx J/- (x)* J7
(9)
iEh
Σ D logp D logp' (x) Jp-1 (1)xxx)))1gg1g
[g J1p (xx) 1xx)
(10)
ieh
Proof. By direct computation, can be verified that for each 0, ..., ei + ez, we have:
D logp'(x)
= D log Jp- (x) + Jp- (x) DZ log
(14)
+ a. log (5x) (x).
Given Ug. Assumption of Gaussianity, together with the fact that Si is distributionally
equivalent to L,S for some diagonal L, (Assumption 2 imply 0z. log p/ (sk) = 0 for all k. Then, the summation vanishes. It follows that, for all environments i 1, 61 + e2:
D& log p(x) D log p¹ (x) = Jr (x) (D' log pà (s) D2 log ()))) Jr (x).
The same results hold we replace with and with Å. Then, Equation 9 follows summing the above over all 1,... C1, and Equation (IO) follows summing over e1 61 + 62-
C.3 IDENTIFIABILITY OF MEAN THE SOURCES
In this section, we show that under the assumptions of Theorem 1 the mean μs of the sources is identifiable.
Proposition 2 (Identifiability of the sources mean). For each i 1.., C22 suppose the diagonal entries of the rescaling matrices L; generating the environments are randomly drawn a joint distribution that absolutely continuous with respect the Lebesgue measure on (RR O)d(er Then, the following is verified with probability one over the samples E1te1
k
logg((x) Vlogp'(x)= =0 - s=f -14.
(15)
We introduce two lemmas instrumental to the proof of the proposition.
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Lemma 4. Consider the base ICA model of Equation (2) and let i -...kk k be the index denoting an auxiliary environment (Definition)) 3). Let Assumptions to be satisfied. Given x = f(s) such that Jp (x) is full rank, for each k C1 e2
=
k v llg)(ss)
v Vlog((x)- - V Vlogr'(x)= = =
V llogp's) =
(16)
Proof. By the change of variable formula for densities, we obtain the score of x for a generic environment i = 0, k (as usual, p pº):
V log (x) JJ- (xx² 7 V log + logg/Jr-1
Then, for each 1,...,:: k:
logp(x loog'' (x) 1x-1(x) V logp(s) v log]
Taking the summation:
V Vogg((x) - V Vooyy(xx) = (x-( V logp(s)) - Vlogp'
1=1
From the above equation, the right-to-left implication trivially holds. Considering the other direction we have:
Σ v Vlgg((x) logr'(x)
Jr (x) V logp(s) V log (s) 0.
Being the Jacobian of the inverse mixing function full rank matrix, its null space is the zero vector, which implies:
V l(g - V log = O.
Lemma 5. Consider the base ICA model X = f(S) of Equation 2 Let i 1, .... k be the index
of the auxiliary environment X f(S),, with S' S'= L,S. L, = diag X Xa), and X3 + 0. Let Assumptions ! and 4 be satisfied. Assume the joint law of (1} absolutely continuous with respect to Lebesgue measure on (IR \ {0)) (0))ak, Then, for each ls e1 + 62. the following holds with probability one over {Li)e samples:
1,...,d. d, 1, 1,... k} is
logp(s) Vlogp'(s) = s=f f= = Ps-
(17)
Proof. The backward direction is immediate, due to the Gaussianity assumption. Let's focus on the forward implication.
V Vlog((s) - V lo) =0 Σ Dn log (ogp(s;) 0a, logp'(s;) 0, Vj 1,..., d.
We denote with Kjj0ž of respectively the mean and variance of Sy, and define 2 = 1. For each i 0, ..... k have:
0. logg'' (83) -- 8q
(2)0,)2
Then:
Σ 0n logp(s,) - 0. logy' (8,) Mj -j
(x:)²
Therefore, the sum vanishes if and only if for every j. either 8j J A
(2) =k By
Proposition 3 (X))- = occurs with probability zero, and thus the claim is verified.
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We are ready to prove the proposition.
Proof of Proposition By Lemma
4 we have that for each k S E1 + E2:
logp(x) V log
v log
V Vlogp'(s) =0
1=1
Then, the result follows by application of Lemma
L
C.4 PROOF OF THEOREM
We repropose the statement of Theorem followed by a detailed proof.
Theorem 1. Consider the groundtruth ICA model (f, po) of Equation ② and the alternative (f. På)
Let Assumptions be satisfied. and assume that the elements in the set ( 92))))== are
pairwise distinct. Let x = f(s) = f($) and s = usg then, the indeterminacy function = f-1 satisfies (n) D, meaning that the causal graph G is identifiable.
Proof. By Lemma 1 for = 1,2 we have:
Jr-1 ( T = (x)
which implies
M πg 1 1,
(18)
where M = Jn (S). By Lemma 2 Q is invertible, which also implies i invertibility (by Lemma 3 Then, define A := 27 f22 and B = ?2 Ω2. From Equation 18) it follows:
A M BM,
(19)
which implies that A and B similar, implying that they have the same set of eigenvalues. Take X, v eigenvectors of A. Then, the following chain implication holds:
Av Av >
MAv = XMv > BMv = AMv,
(20)
So, M is mapping from eigenvectors of to cigenvectors of B. The next step is showing that each eigenspace of A and B is always spanned by one vector in the standard basis. As preliminary step, we show that the diagonal elements of are pairwise distinct: first, by similarity, we have that A and B have the same eigenvalues. Being both matrices diagonal, the eigenvalues are the diagonal elements. Then:
where the last step follows from Equation 99
(54 )ss
(aa) (O2)ss
A
Byj- i 1,.,..
(21)
(22)
By assumption, we have that the elements the set { (12a)e heeld are pairwise distinct. The above equation implies the same for the set
[0x)e ((az) Jee[d]. i.e., for each 1,....d d:
Au # Aij.
v,=1,-.,, d.j #
(22)
Now consider the eigenvalue À of A: we show that the associated eigenspace is equal to the span of a single vector the standard basis. Being A diagonal, there is 1,... d such that À As Consider the eigenvector (U1, .., vd) such that:
Av Xv A
(23)
Being A diagonal, for each j = I,... d, component-wise we have:
(Av), = Agguy.
(24)
Equations
and 24 together imply:
Anny Aijul = (Ax Aij)es = 00, Vj= = =1,..,d d.
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By Equation (2), for j, Ais Ajj. meaning that Vy 0. Then, eigenvector of A must be aligned with the basis vector e;:
(x(A) span {-.).
(25)
With analogous computations, we find:
Ex = span
(26)
with e, potentially different from e, Given that by Equation 20 have ME)(A) (B), the last two equations imply
M (pan = span(es), M Jn (s).
We conclude that (s) maps one vector in the standard basis to another (up to rescaling), proving that Jn DP with D invertible diagonal and P permutation. We recall that by Equation (7) have Jr- J s.t.
Jf (x) pd --
By Lemma Reizinger et al. (2023), the permutation indeterminacy can uniquely determined and thus removed. Given that by Assumption 3 the Jacobian of Jr-1 (x) is faithful to the causal graph, the claim is verified.
D INDEPENDENT COMPONENT ANALYSIS
In this section, present primer the problem of Independent Component Analysis (ICA), based on the content of Section 2 in [Bcchhllz et al. (2022). ICA seeks to recover latent sources from their observed mixtures. We assume hidden random vector S Fd with independent coordinates and observations generated by
d
X = f(S):
p(s) = II pi(s:),
(27)
where f: Rd >> Rd is diffeomorphism. The goal of ICA is find unmixing map f-1; Rd B such that the components ff are independent-ideally achieving blind source separation (BSS), meaning f xs up to standard symmetries. Informally, for å = (x)) we call an ICA solution when
f(S) f(ss)
(equality is in distribution). general, we would like an ICA solution to be as close as possible to the real function f. To formalize this concept, known as identifiability, let F(A, B) be a class of invertible maps A + B (assumed diffcomorphisms) and let P C MM (R)&d be a family of product measures. Let S denote the group of admissible symmetries (e.g., permutations and coordinate-wic rescalings) up to which we agree to identify sources.
Definition 5 (Identifiability) ICA in (F,P) is identifiable up to S if. for any f, f E F and P, e P,
f(S) 2 f($) with S ~~ P. S ~ P.
(28)
implies the existence of h E S such that = o f on the support of P.
In general (i.e., for (F, arbitrarily large), the ICA problem is not identifiable for reasonable S. Notable example comes from the Darmois construction or constructions based on measure-presedd measure-preserving transformations. Several results in the literature have studied which conditions on (F, P) can help identifiability. Most notably, Buchholz et al. (2022) shows that when JF represents the class of conformal maps, identifiability is guaranteed up to trivial indeterminacies. If heterogeneous data are considered (c.g., in the multi-environment setting of this paper), identifiability was shown in the general case (Hyvärinen & Moriokal 2016).
E EXPERIMENTS APPENDIX
E.
COMPUTATIONAL RESOURCES
All experiments have been run on personal laptop, Lenovo ThinkPad T14 Gen 5, for a run time of approximately 6 hours.
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E.2 STRUCTURAL CAUSAL MODEL IDENTIFIABILITY FROM OBSERVATIONAL DATA
Without sufficiently restrictive modeling assumptions, causal discovery is ill-posed: the distribution of the data is compatible with many distinct graphs that define an equivalence class, the most one can hope to identify in the general case with i.i.d. observations. Unique graph recovery requires restrictions on the class of functional mechanisms and noise distributions of the underlying causal model: in what follows, we briefly introduce the four classes of causal models that are known to be identifiable. We always assume that the underlying graph is DAG.
Linear Non-Gaussian Model (LiNGAM).
A linear SCM over E R defined by
X BX
(29)
where B Radd collects the coefficients expressing each X; as a linear function of its parents plus disturbance S, With mutually independent, non- non-Gaussian noise terms, the model is identifiable; this known as the Linear Von-Gaussian Acyelic Model (LiNGAM) (Shimizu et al.) 2006).
Peters et al. Additive Noise Model (ANM). An Additive Noise Model (ANM) (Hoyer et al. 2008 2014) defines each causal variable as a function of (potentially) nonlinear mechanisms and an additive noise contribution:
X; Ja(PA,) + S; i= I.....,,d
(30)
The noise terms are required be mutually independent.
Post-Nonlinear Model (PNL). The most general class with known sufficient conditions for identi- fiability of the graph is the Post-Nonlinear (PNL) model (Zhang & Hyvärinen 2009), in which
X; = (.(.( + S:), i l..,....d
(31)
with f; and 9i both potentially nonlinear, 9i invertible, and mutually independent noises.
Location Scale Noise Model (LSNM) The LSNM (Immer et al. 2022) extends ANMs by allowing heteroscedastic noise follows:
X; (PPA,) (PA,) S,, i 1= where and gi 0 may be nonlinear and noise terms are jointly independent with zero mean and unit variance.
(32)
E.3 DETAILED PSEUDOCODE OF ALGORITHM [
Algorithm 2 provides a detailed pseudocode of the algorithm adopted in our experiments of Section 5 and sketched in Algorithm [
E.4 EXPERIMENTS BEYOND GAUSSIANITY
In this section, we present additional experimental results on bivariate graphs underlying synthetically generated structural causal models. The causal mechanisms are the same already described in Section 5.T The difference, here, is that we generate the independent sources from a Gamma distribution, which violates the assumptions of our theory. We sample the scale parameter 0 ~ U1.75, 2.25), and consider two different parameterizations of the shape a of the base environments: in the first case, ~ U0.55 1); in the second case (2, 2.5). What makes the Gamma density interesting it that can be flexibly modified by changing the values of its parameters, as shown in Figure
Gamma distribution with vanishing gradient. Figure 2a illustrate how the Gamma density function varies at a = and different values of 8. It is interesting to note that the gradient of the density function never vanishes, making this setup adversarial the assumptions of Theorem []] In line with this, in Figure 3 we see that generally our algorithm struggles to infer the causal direction for this class of structural causal models.
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Algorithm 2: Estimating supp Jr-1 from the data
Data: D E Rexuxd
V d-dimansional observations
1,2c [k]
Set of indices splitting the environments groups
Result: Estimate of supp Jp-1
$ score_ (aore__stimaa(()) E Rxxxx
H hess Mess_estimaae(2 Rexxxxxd
mean Bacphpparsssi___ indxs Rex2
1/ Pair of indices correspoading to observations at the mean
/7 For each environment e, find i s.t. (X](e,)) = ws
for e= 1,,........... k do
Ax E Rax
11 nor the difference of chservations from distinct envs
pairs E N
1/ Pair of indices 4, such that X(O, a X(e,5] 11 Container for norm of the differences in the score
score_ ditf__dffi_eeee + +00 E IR
for I,....I do
for j=/....,, do
Ax + - end
j + arg min [ pairs[i] jjj score score_diffs{(1 + || Ste,
1/ x[0, z Xle,al
end
m < arg min score_diffs
// Paired observatiors betweer enve (0, ext. diff. 0.
mean paias p + m, pairs[m] // The score dLEr, vanishes when source mean
end
Difference Bessians mean (i.e. Equations Ham + 0 22xxx
LG
for l 1,2 do
for e Le do
m1 m. < mcan Bactes_aiis_____eeee _ddssssds]]] [0, m1] ] A [e, me]
A н
Hans [8] +- Hanss [6] Ah.
end
end
M A Han. [2] R JASt 1
11 Ham Jf-1 Dedg91 by Equations
Jr-1 + diagonalizo (M =e DP
return supp (Jr-
11 can be found using the acyolicsty of the causal graph,
Gamma distribution with vanishing gradient. Figure 2b illustrates how the Gamma density function varies a and different values of 0. We can see that, in this case, the density achieves a maximum: we point to our analysis in Section 4. (the paragraph Theorem / beyond Gaussianity), where we discuss when and why is reasonable expect that Theorem extends to any source distribution that achieves a maximum or minimum in the interior of its domain. A word of caution is needed: despite the fact that the Gamma density with a E [2, 2.5) does have a vanishing gradient, the points of the domain at which the critical values occur are not preserved by our rescaling interventions (as is clear by inspection of Figure 2bb. Hence, the requirements of the Theorem not fully met (where it's implicit that the rescaling interventions do not change the location of the modes): this makes the experiments of Figure 4 an interesting challenge for our algorithm. The outcomes exciting: assumption violations, imposes enough constraints to infer the causal direction in the majority the experimental setups with 80% accuracy. This is of double interest: first, we have some empirical
that increasing the number of available environments, despite the
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density
density
0.6-
de
Dw
decxity
0.3-
dersity
0.3
0.2
61-
0.0
6.0
5.0
7.5
12.0
15.0
17.5
0.0
2.5
5.0
10.0
12.5
15.0
17.5
20.0
(a) Gamma density with 1.
(b) Gamma density with a = 2.
Figure 2: We plot the Gamma density for different values of shape and scale. The left plot fixes the shape I; the right plot fixes 2. We let 0 vary to illustrate how the distribution changes between the rescaling environments of our experiments. We note that for a the density doesn't have a finite critical point.
Gamma noise, [0.55 1]
environments
km3
0.8
k-6
Mean SHD (# (std)
0.6
0.4
0.2
0.0
noniinear anm
End
Ismm
fieea anm
arbirrary nonlinear (1)
arbitrary nontinear (i)
arbitrary nonfinear (ii)
Figure 3: Average SHD (0 is best, 1 is worst) achieved by Algorithm over 50 seeds on binary graphs. The sources are sampled from a gamma distribution with a E [0.5, 1]. In line with our theory, when the sources are generated according a density that doesn't 't have critical points, our algorithm generally fails to infer the causal direction.
evidence supporting the hypothesis that our theory can be extended beyond Gaussianity, Second, we see that this seems to be achieved thanks to the constraints from many environments, in contrast with what we observe when experiments are run on SCMs with Gaussian noise (Figure [D) where increasing environments do not translate into better accuracy. These empirical findings, despite being preliminary, should provide an incentive to pursue identifiabiliyy theory beyond Gaussianity.
E.5 EXPERIMENTS ON HIGHER DIMENSIONAL GRAPHS
In this section, we present and analyse experimental results graphs in dimensions higher than Our finding shows that, according to our theory, 2 sufficiently different auxiliary environments enough to infer about the causal order, even in cases known be non-identifiable with pure observations.
Metric. We monitor the error in the inferred causal order via the topological order divergence, first adopted in Rolland et al. (2022). Given a directed acyelic graph with d nodes, a causal order (or topological order) is permutation of the set [d] such that node in the ordering only of the nodes appearing after it in the same ordering. For example, the only graphs compatible with the topological order (2, 1) X₂ X, the empty graph. Consider causal order f. and
be parent
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Gamma noise, [2, 2.5]
1.0
k-3
0.8
Mean SHD (tt sstdd)
0.6
0.4
0.2
0.0
moniineer anm
pnd
Innn
linerr anm
arbirary nonlinear (1)
artitrary nonlinear (9)
arbitrary nonfinetr (14)
Figure 4: Average SHD (0 is best, 1 is worst) achieved by Algorithm over 50 seeds on binary graphs. The sources are sampled from gamma distribution with a E 2.5). which guarantees at least one point where the gradient of the oggliikelhood vanishes (see Figure 2b). Interestingly, this appears to enable accurate inference the causal graph when the number of environments increases.
binary adjacency matrix A representing directed acyclic graph (Ai = i PA,). The topological order divergence is defined as:
Dep ( A)
Σ 1=1 f.
Σ Aij.
where n > R, means that node successive to j in the order. If A is the right topological order for A, then Diop (ft, A) = 0. Else, Drop ( A) counts the number of edges that cannot be recovered due to the choice of topological order. For example, given a graph X + X2 + X3 with adjacency A, the causal order म {1, 3, 2 does not allow an edge X2 X3, and Drop (ft A) 1. Given that Theorem T concerns the identifiability the causal order, and our goal is to empirically support our theoretical findings, the topological order divergence is the right metric monitor. In Figure J Figure we report the average Drop over 20 seeds, and the error bars are 95% confidence intervals.
Random baseline. The performance of our algorithm is compared with that of random baseline: in particular, in the graph we report the mean accuracy of an algorithm that randomly sample a causal order among all possible permutations of the set {1, (1.. d), being the number of nodes. If the upper boundary of the 95% confidence intervals around the mean accuracy of our method are lower than the mean of the random baseline, that's statistically significant empirical evidence in support of our theory.
Next, we proceed to analyse the experiments. We separately consider the case of inference on linear and nonlinear structural causal models.
E.5.1 EXPERIMENTS ON LINEAR SCMS
When synthetic data are generated according to a linear model X = AS (A being the mixing matrix), the Hessian of the log-likelln likelihood is equal the inverse of the covariance matrix Σx (the Hessian, in this case, takes the name of precision mairix). For this reason, in the linear setting, we replace the Stein gradient estimator of the Hessian with simple approximation of the covariance Σx via averaging. The motivation is two-fold: (i) Hessian estimation via the Stein gradient is unstable the dimension of the graph grows (see, e.g., (Montagna et al. 2023al); (ii) the average estimator much faster, which allows us scale experiments to higher dimensions, In the linear case, our method similar to the BACKSHIFT algorithm (Rothenhäusler et al. 2013).
Synthetic data generation. We analyse the performance of Algorithm I on graphs with {10, 20, 50} nodes, respectively with number of edges (10, 40, 100). Graphs are generated via the Erdös-Rénj model (Erdos& & Renyi 1960). For each graph, experiments with {3, 6, 9} environments.
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Disp for linear anm (Gaussian noise)
nodes=10
nodies=20
nodes=50
Mean 15
15-
10-
Figure 5: Mean Deop (the lower, the better) of Algorithm on data generated with a synthetic linear SCM and graphs with different number of nodes (10, 20, 50). Error bars are 95% confidence intervals. refers the number of environments. We note that, line with our theory, 3 environments are sufficient to infer causality much better than random.
Rescaling coefficients for the source variance are uniformly sampled between 2 and min 10), [G] being the number of nodes the considered graph. dataset from a single environment consists of 2000 i.i.d. samples. The linear regression coefficients are uniformly sampled from [2, 5]. and the sign of the coefficient is randomly flipped.
Analysis of the experiments. Figure we see that even in high dimensions, our method can infer causality linear Gaussian models with as few three environments. In particular, 10 nodes, the mean error reduced by 75% compared to the random baseline; on 20 nodes, we improvements of = 45%; on 50 nodes, the error decreases by 30%. It's remarkable how the method's accuracy does not improve with more than 3 environments. This line with our theory, which demonstrates that sufficiently different environments guarantee identifiability of the causal graph.
E.5.2 EXPERIMENTS ON NONLINEAR SCMs
We now consider the empirical performance of Algorithm I on nonlinear structural causal models with nodes. With already 10 nodes, we observe that our method infers a causal order that is, on average, no better than random, suggesting that further research for good algorithmic implementation of our theoretical findings is necessary. To put this in perspective, we remark the goal of our experiments, and more generally, of the paper: the contribution of our work is devoted to establishing novel identifiability results for causal discovery with multiple environments, leveraging the duality between ICA and structural causal models; on the contrary. the goal is not present novel algorithmic solutions based on these results. With this in mind, we design Algorithm [ as a simple implementation of the steps in the proof of Theorem T we do not claim that this is a good strategy beyond our purpose of validating the theory with toy examples. In fact, according to the literature and our experience, multi-environment causal discovery with ICA is challenging problem on its own (see the discussion in Appendix B.2):: as such, we leave it to future research. Our experiments only serve the purpose of demonstrating that our theoretical results and our proof techniques are correct. In line with this goal, find that method only requires 3 environments to infer causal directions significantly better than random on nodes, even in challenging nonlinear scenarios.
Synthetic data generation. We consider synthetic data generated with nonlinear structural causal models that are not identifiable from pure observations, and satisfy the assumptions of Theorem I In particular, given variable zÿ and its parents ZPA,, mechanisms are defined follows: first we define cause random variable == ІTA, ΣxEPA, 26 as the mean of the parents; then, given the noise 8j. we consider the following causal mechanisms: (i) H; == cos(c)s; + aretan(s)): (ii) tanh(c) (xctan(s;) ) + s²; (iii) sin(c) + arctan(c)s, + cos(c)s3. Note that, differently from the experiments in Section 5 on bivariate graphs, we wrap the cause in trigonometric functions and avoid
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Drop for nonlinear scm (Gaussian noise)
km3
Random order baseline
Mean Due
arbitrary nonlinear
arbitrary nonlinear (i0) Mechanism lype
arbitrary nonilnear (ii)
Figure Mean Diop (the lower, better) of Algorithm 1 on data generated with synthetic nonlinear SCMs with 5 variables. Error bars 95% confidence intervals. refers to the number of environments. We note that, in line with our theory, 3 environments are sufficient to infer causality better than random, and adding environments does not decrease the error.
polynomials. This is prevent the variance from growing polynomially in the causal direction (a well-known phenomenon in simulated SCMs (Reisach et al. (02I)), which we observed to cause all values in the Hessian of the log-likelihooe to collapse zero. Graphs generated via the Erdös-Rényi model (Erdos & Renyi, 1960). For each graph, we run experiments with {3. 6, 9} environments. The rescaling coefficients per-environment of the source covariance uniformly sampled between 2 and 10. A dataset from a single environment consists of 2000 i.dd. samples.
Analysis of the experiments. Figure 6 shows that, for structural causal models with nodes, edges and nonlinear mechanisms, information about the causal order be inferred by method: in particular, compared to random baseline, whose expected Drop 2.5, our method with 3 environments yields improvements between ss 30% (on nonlinear mechanisms of type (i)) and a 25% (for mechanisms of type (iii).. Notably, in line with our theory, adding environments does not decrease the average error across seeds, showing that only sufficiently different environments are needed for inference.
F ASSUMPTIONS DEEPDIVE
We present further discussion on the assumptions of our theory and potential extensions beyond them.
F.1 BEYOND GAUSSIANITY
One of the key restrictions of our theory is that it requires the independent noise terms to be Gaussian. In the main paper, we discuss how this can be relaxed to noise distributions whose gradient of the log-likelihood has a critical point. Here, we expand on the discussion of Section 4.T to illustrate the fundamental limit of our proof technique address the we provide step-by-step mathematical intuition of why Gaussianity is crucial for our proof. The key ingredient of our theory is the analysis of the Hessian of the log-likelihood. By the chain rule of differentiation, can be verified that the score function at a data point x, under environment i, satisfies:
of general noise distributions. To begin,
V Vlogp' (x) V (s).
(33)
Applying once again the chain rule, one can easily verify the following expression of the Hessian of the log-likeling log-likelihood:
Jr-1 (x)" D3 log (s))p'sssss-io (x) DR log Jr-1 (x) + Σd asy log (sz) (x).
(34)
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The information about the causal graph is contained in the product of Jacobians Jx-1 (x-x() (the diagonal Hessian in between doesn't 't play a significant role). To access this information from the Hessian of the log- ligelee,,,odd, need to get rid of:
1. The log-det term D log 1 (x)];
2. The summation Σラ 0s; log (o, (s))
Being the mechanisms f invariant across the environments, it is immediate to see that log JJ - (x) vanishes in the difference D log p(x) D log (x). The assumption of Gaussianity, instead, is crucial to get vanishing summation: in fact, we know that the mean of the sources s = μs is a critical point of logps.
ps. This clarifies why the assumption of Gaussianity is crucial for our theory.
A natural question is whether our theory can extend to structural causal models with more general classes of noise distributions. Beyond density functions with a critical point, the answer is generally negative. To show why this is the case, we consider the exponential family, which encompasses large class of common distributions. Let S distributed according to the exponential family with the vector of parameters (in the Gaussian case, (ug, Σs)). Then:
log p(s) log h(s) + (0) T(s) A(n),
(35)
where h(s)) is the so called base measure, (7()) is the vector of the natural parameters, T(s) is the vector of sufficient statistics, and A(1) is the partition function. funcction Now, assume that, akin to the Gaussian case, we define auxiliary environments (Definition) 3) by changing 6 parameters for each environment i. The difference of the score of the observed vanables x, in this case, becomes:
V logp(s) V lo (s) = T(s) (n)() - m())))
Assuming that 0 + 0¹ in each component, we get that the score of the sources vanishes if and only if T(s) or orthogonal to r7) (0)). Clearly, orthogonality can not be enforced unless we carefully craft the intervention on 0. It remains to consider whether the T(s) vanishes at any point. A simple inspection of the sufficient statistics of the density functions in the exponential family reveals that this is often not the case.
The takeaway of our discussion are: (i) that, as far concerns our methodology, vanishing gradient of the log-likelihood at one point at least is necessary; when this is not the case, we can not extract the product of Jacobian matrices (hence, the DAG information) from the Hessian of the log-likelihood. This is in line with previous work (Montagna et al. 2023a 2025). showing that the Hessian matrix can only inform about the equivalence class of the ground truth graph. (ii) For wider classes of noise distributions, in general, extension of our results requires substantial additional research in terms of proof techniques.
not hope that the vanishing gradient condition is satisfied. Thus,
F.2 BEYOND CAUSAL SUFFICIENCY
In this section, we address the question of whether methodology be adapted demonstrate the identifiability of parts of the causal graph in potentially confounded scenarios. The duality between ICA and causal discovery that is key this paper remains relevant even in this scenario. This was explicitly highlighted in Ding et al. (2019), where, in the context of linear SCMs with latent confounders, causal discovery is phrased and analysed as an overcomplete ICA problem. For general nonlinear structural causal models, the presence of latent confounders induces an ICA model X f(S with f: RRd. + Rd= and d, > dz. First, we discuss why our proof technique can not be generalized to this scenario when f is nonlinear. Then, we show that in the case of linear structural causal models, our findings can be used to derive known theory of identifiability of SCMs without causal sufficiency.
We remind that the key theoretical result that enables identifiability our setting (Theorem is Lemma I which we report below.
Lemma 1. Let f(s) f(8), where = us.. Let Assumptions 2 and satisfied. Then:
DE log p(x) D2 log p¹ (x) (x) 22 Jxx J Jq
(9)
ieh
Σ D& ((x) D& dogp'(xx JF 1xx) S2 (xx (x) (10)
ieh
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Clearly, the result above relies the invertibility of the causal mechanism f. Moreover, it is easy to show that Q4 S2 are diagonal, which is key the proof of Theorem Unfortunately, in overcomplete ICA:
It is trivial that is not invertible.
2. Less trivially, computations based on the coarea formula (Negro 2021 show that 92 22 are non-diagonal.
From this, we conclude that generalizing our method for arbitrary nonlinear and confounded SCMs a feasible route, and more elaborate tools and ideas required. We note that, exceptionally, the Hessian of the log-likelihood is still informative about the causal graph in case of linear and overcomplete SCMs: fact, its inverse the covariance of the data, namely, (D2 log p(x)) Σx = AEsAA, for a structural model of the form X = AS, with A rectangular, wide, matrix. Notably, this setting, rank constraints and trek separations (Sullivant al. 2010) are informative about the causal graph.
G ADDITIONAL CONTENT
In this section, we collect some useful results and notes relevant to the main paper.
G.1 GRAPH THEORY
Directed graphs and DAGs. Let Xi...., Xa be a vector of random variables. A graph G = ( E) consists of a vertex set {X,}; ). and an edge set E. We recall few basic notions for directed graphs.
directed edge X, + X, indicates that X, is a parent of X; (and X, a child of X2). PA, [d] denotes the index of the parent nodes of X; in the graph G. CH; C [d] denotes the children. A path in G is a sequence of least two distinet vertices æ X X,.. such that each consecutive pair Xi, and Xi is joined by an edge for k- = .,.., 1. If every edge along the path is
oriented forward, Xix + Xix+19 we call it a directed path; then X,, is an ancestor of Xi, and X;m. descendant of Xi.
G.2 FROM SCM TO ICA MODELS
Equation claims that structural causal models can be expressed in the form of ICA models. Here, we show how this can be achieved, Consider set of causal variables X (X,) and without loss of generality, assume that the causal order is 1, 1,.., d. According to Equation for each i 1, .... d, we have:
X, EXPPA S.).
with S - (S.) the vector of mutually independent noise terms. An inductive argument shows the existence function f; SAN, + X,, where AN, denotes the indices of the ancestor nodes X; in the causal graph. Given the causal order 1, .... d, the base case is given for X₁ (i such that fi := Fj. The inductive step is as follows: assume that there is n < d such that X, = f.(S AN. S.) for all = 1, .... Then, there is map (|m ((n) The causal order L,... d implies AN n+1 C [m]. so that there is a map S [mj r XAN.+1 given that PA C AN there is map Sin) H
XPA.. from the structural equation Xn+1 == Fa+1 1, = ( ), Sn+1),
we conclude that there is fa+1 SAN. Sa+1 + X,+1. Then, we define f == (ft) and find
X == f(S).
important note that the DAG structure of the causal graph is reflected in the Jacobian of the mixing function f, which can be shown to be lower triangular.
G.3 HESSIAN OF THE LOG-DENSITY OF INDEPENDENT RANDOM VARIABLES
In the main paper we mention that the 2,, 22 matrices defined in Equation 8 are diagonal; here, we discuss why this is true. More generally, it is well known that for vector of independent random
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variables Z E F with density p. the following holds:
log p(Z) 0 - 2 2, 23 {2, 2,},
(36)
where Z; 11 2,
1 2n) indicates that Z; Zj independent conditional on all the remaining random variables in the vector Z. This result was shown in Lin (1997) and SSpantnni et al. (2018) (Lemma 4.1) and extensively adopted in the context of causal discovery (e.g., |Montagna al. (2023a 2025 ) By Equation (36) it is immediate to see that independence of Z entries implies that Di log pl Z) diagonal.
G.4 MEASURE THEORETIC ARGUMENTS IN SUPPORT OF THE ASSUMPTIONS
First, we show that Assumption 5
generically holds.
Proposition (Assumption 5 holds almost surely). Let L; = ...., X and X3 + 0, i 1.... k. Assume the joint law of the array A (25)je(4,,eEt) <E[M] is absolutely continuous with respect to Then, with probability one over the draw of A: for every j [d].
Lebesgue measure
(R) {0))
Σ
* k.
sE[R
Proof. Fix j [d]. Write & [7| and (2))EE()) E (IR) (0))*. Consider the smooth map F: (R \ {0))* + R.
k F(A) Σ xx²² - k. r=1
Its gradient is VF(A) = (-2X ..... 10 2x + 0 on the domain, so 0 is a regular value. By the regular level-set theorem, F (0) is (k -)ddimensional embedded submanifold of g and hence has Lebesgue measure zero. Because the k-tuple À (A))ee)) has a distribution absolutely continuous with respect to Lebesgue measure, get
Σ JE
= 0.
Taking the finite union over j I,.... d preserves measure zero, with probability one none of these equalities occurs.
Next, we show that the assumption of pairwise distinct {(2, )e])) elements (definition at Equation (8) generically holds.
Proposition (Pairwise distinct diagonal ratios hold almost surely). Let 11, 12 C. [k] 3. For each environment let L diag(>)...... X²) with X3 # O. Assume the joint law of the array
(24) (ed)) absolutely continuous with respect to Lebesgue measure on (R {0)) Suppose moreover that Qe is diagonal with entries
dk
=
(Dt) O
(xx)22-11) #0.
(1,),
j
VEL
Then, with probability one over the draw of Q2 invertible and the diagonal entries of 22 '22 are pairwise distinct.
Proof. Write
(971 = Eiel, (2)) 17
B
4,
E B -00)22
Σier, (X))
A
SET.
SEL2
By Proposition
Ag # 0 and Bj * for all with probability one, such that Ω is invertible.
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Fix j # f. The collision event (92 (2
e is equivalent to
Bj Be
== = A.B; O.
A₁ Ae
Let t =- -1a) and view Fje smooth function of the 2 variables (t)keEm U (fhhe For any fixed io IT,
aF 0A
A aB, - B A 0 Br.
B,
0t 0t
0t
Since B₁ * O, we have Vfje on the set under consideration, so is regular value of Fje. By the regular level-set theorem, the set {Fe 0} is a (2) - 1)-dimensional embedded submanifold of absolutely continuous w.r.. the
I2* hence has Lebesgue measure zero. Because the law Lebesgue measure,
P ((2 (2T (2)e O.
Taking the finite union over all pairs + yields that, with probability one, no two diagonal entries coincide; that is, {(2) 12) j= are pairwise distinct.
G.5 FIXED MECHANISMS ENVIRONMENTS REAL-WORLD WORLD DATA
In this section we briefly discuss the assumption of fixed mechanisms across environments implied by Definition 3 given two environments X f(S*). X' f(S)). they share the same causal mechanism f. In particular, we present examples from the domain of single-cell and gene perturbation causality studies where multiple environments with fixed mechanisms are commonly hypothesized. This suggests that our modeling assumptions, hence our theory, have practical relevance.
Liu et al. (2025) and (Lopez et al., 2023) assume an SCM and explicitly model gene and single- cell (respectively) perturbations as changes in the distribution of causal variables, while leaving all SCM mechanisms fixed. Similarly, but without an explicit assumption structural causal model, Zhhan et al. (2023) consider interventions on latent factors that leave causal mechanisms unchanged. Meinshausen et al. (20T6) studies the problem of gene perturbation through the Invariance Causal Prediction framework (Peters et al. 2015): in this context, they discuss the example of environments defined with fixed causal mechanisms and noise variance affected by a multiplier that environment dependent. This is precisely in line with the modelling assumptions of our theory.
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Reviews:
[{"review_id": "Fonr1TBVLG", "reviewer": "Reviewer_WHVU", "summary": "This paper studies the identifiability of causal graphs when data come from multiple environments. The authors prove that for structural causal models (SCMs) with arbitrary nonlinear mechanisms and Gaussian noise, the full causal graph can be uniquely identified using data from only two distinct environments. The work builds on the theoretical duality between Independent Component Analysis (ICA) and causal discovery, offering a new identifiability result that links second-order statistics (Hessians of log-likelihoods) across environments to the Jacobian structure of the causal model. Synthetic experiments on bivariate models validate the theoretical claims.", "questions": "1. For proposition~1, I believe there is a typo that should be $J_f^{-1}(x)_{ij} \\not=0$.\n\n2. Can the authors discuss the potential reasons why including more domains causes performance to decrease? I agree that in theory, two domains are enough. However, for nearly all CRL papers, we benefited from additional domains. Is that an optimization issue?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "I appreciate the authors’ clear and well-organized presentation of their work. The paper carefully states its assumptions, proofs, and detailed algorithms, demonstrating both theoretical rigor and practical insight. Most notably, it shows that full causal graph identifiability can be achieved without the number of required environments scaling with dimensionality—a highly appealing property for real-world applications, which I consider the paper’s most significant strength.", "weaknesses": "So far, the largest weakness of the paper is the narrowness of the experiment. They only test on simple two-variable synthetic examples. Although the authors acknowledge this as a limitation, I believe that experiments on multivariate settings are necessary. Without trying it on higher-dimensional problems or actual datasets, it's hard to gauge whether this really works in practice.\n\nAnother potential weakness lies in the assumption of Gaussian noise variables. At first glance, this may not seem overly restrictive for two reasons.\n\n1. Noise is often regarded as the sum of many small unobserved factors, which should approximate normality according to the central limit theorem.\n\n2. Even if the true noise is non-Gaussian, it can theoretically be expressed as an invertible transformation of a Gaussian variable, and such a mapping could be absorbed into the causal mechanisms without affecting the identifiability of the causal graph.\n\nHowever, the non-Gaussian experiments reported in the appendix raise some concerns—the results are not entirely convincing. This suggests that, in practice, the Gaussianity assumption may indeed be a substantive limitation of the current work.", "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": "CHHT67F06k", "reviewer": "Reviewer_FXre", "summary": "The authors establish that \"the full causal graph of an invertible SCM with arbitrary nonlinear mechanisms is identifiable from the model’s distribution and data gathered from only two sufficiently variant environments\", providing the first theoretical guarantee for complete causal graph recovery under a constant number of environments, in contrast to prior nonlinear identifiability results which require #environmental which scales linearly with data dimension. The analysis assumes Gaussian noise, yet the authors discuss how to relax this condition to a more general class of noise distribution (having at least one point with zero gradient of the likelihood, not necessarily at the mean as for Gaussian) . Their proof introduces a novel application of the recently highlighted duality between (non-linear) SCM and ICA identifiability, showing that causal discovery requires far less auxiliary information than nonlinear ICA. Empirical validation on synthetic data corroborates the theory, showing successful causal direction recovery in previously non-identifiable bivariate cases.\n\nOverall assessment: The paper is well written and generally interesting. It builds on the recent duality between identifiability in non-linear SCM and ICA and suggests promising directions for advancing identifiability theory and scalable algorithms for high-dimensional, multi-environment causal discovery. While I enjoyed reading the paper, I would have appreciated more comments/remarks/analysis on limitations, for example for unobserved confounders and a more general class of distributions. Nevertheless, I do believe the paper brings useful insight to ICLR, and more specifically causality, research communities.", "questions": "- I think there is a typo in the equation of Proposition 1: The RHS should be $j\\notin {\\text{PA}}_{I}$, no?\n- I think using an index for environment (say t or r)  different than that of variables (i) could make your presentation less confusing; see for example definition 4 and how it references equation (1). Also, a superscript for environment versus a subscript for variable could be defined in the notation section at the beginning of Section 3.1. \n- I think that a closer mathematical explanation in the last part of Section 4.1 (Theorem 1 beyond Gaussianity) could be beneficial to inexperienced reader.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "This work provides a first theoretical result toward identifiability of the full causal structure with as few as two environments with sufficient variability with respect to a base environment as specified in Assumption 5. The novel proof technique of exploiting the established duality of SCMs and ICA brings interesting methodology for subsequent analysis. The paper is well written and manifest a good balance between main body details and further explanations in the appendix. Empirical validation on synthetic data complements the theory, showing successful causal direction recovery in previously non-identifiable bivariate cases.", "weaknesses": "- Recent work shows that overcomplete ICA is beneficial in scenarios with unobserved confounders and a single environment. I think a discussion about the limitation of \"extending this work with multiple environments\" to causal diagrams with unobserved confounders could be of great interest to the community. For instance, what are the fundamental limitations to extend proof techniques to over complete non-linear ICA (more sources than variables)?\n\n- The authors acknowledged the limitation of \"Gaussian noise assumption\" and commented on what other noise distributions to which their method extends (at least one point with zero gradient). I would have appreciated also a discussion about the fundamental limitation of the method to wider classes of distributions (i.e., from the perspective of necessary as opposed to sufficient conditions).", "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": "3waeKDT0Qk", "reviewer": "Reviewer_ZrxN", "summary": "This paper considers the problem of recovering the causal graph in nonlinear structural causal model based on nonlinear independent analysis. Specifically, unlike previous results on causal discovery using nonlinear ICA, where the number of environments is required to be proportional to the number of sources for identifiability of the full model, the authors show that, if the task is to only recover the causal relations, under certain assumptions on the data generating mechanism, only three environments are required. Further, the authors provide an algorithm for graph identifiability based on estimation of the Jacobin matrix of the mixing function, and demonstrated the effectiveness of the proposed method through simulations based on synthetic data.", "questions": "1. How are $L_i$ selected in the numerical simulation? Specifically, in the settings with 6 and 9 environments, are $L_i$ different across all environments?\n2. Since the task is to recover the graph structure instead of the causal model, can the theoretical results provided in this paper be extended to the case where the causal graphs across environments are the same but the functional mechanisms are different? Note that some of the the existing works allow for different mechanisms across environments, such as Jaber et al. (2020).", "limitations": "", "rating": 2, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. The problem formulation is clear. Specifically, the authors clearly state that the task is to identify the causal graph, which is different from causal discovery where the task is to identify the full causal model.\n2. The theoretical results are clearly explained with detailed proofs.", "weaknesses": "1. Assumption 2 is very restrictive, making the theoretical results not novel. Specifically, Assumption 2 requires that the noises across the environments are \"rescaled\", i.e., $s_i^{k}=\\lambda_i^{(k)} s_i^{0}$ for all $i,k$, where $s_i^k$ represents the noise term $s_i$ in environment $k$. This is a very strong assumption and asserts dependencies among the noises across environments. On the contrary, most of the existing results on nonlinear ICA only assumes that the noises are jointly gaussian with environemnt-specific covariance matrix.\n2. The setting in the simulation results is over-simplified, which only considers the graph with two variables. In this case, the graph structure only has three possible choices, making the recovery task much easier. It would be better if the author could test the performance with more observed variables (say around 5). Also, it lacks comparison with baseline methods that use nonlinear ICA for causal discovery.", "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": "TKzuv7xsGD", "reviewer": "Reviewer_eLzQ", "summary": "This paper proposes novel results for causal discovery from multi-environmental data, by relying on the relationship between representational identifiability and the identifiability of the causal graph. Their resulst provides novel insights into this relationship by characterizing the complexity of the causal discovery task in terms of the sufficient variability conditions prevalent in the ICA and CRL literatures.\n\nAlthough the paper uses similar proof techniques as prior work (relying on the score and the support of the Jacobian), I believe their insight is extremely important (not to mention that it works for Gaussian sources, which is often a barraier to identifiability). The authors provide synethetic experiments to corroborate their findings (that causal discovery is independent of the number of performance, when their assumptions are met).\n\nI have a few minor remarks for improvement (see below); nonetheless, **I am strongly in favor of the acceptance of the paper, and would recommend it for an oral if my concerns are addressed.**", "questions": "- What does \"distribution of a structural causal model\" mean in the abstract?\n- L046: How do you prove that your results holds for arbitrary SCMs?\n- L363: Could you please explain why the SCMs you used cannot be reparametrized as PNL or location-scale models?", "limitations": "", "rating": 10, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "- The paper is generally well-written\n- The assumptions are clearly stated\n- The theoretical results are well explained and contextualized (the \"relation with ICA identifiability paragraph is very instructive\")\n- The experiments back up the theoretical findings (and even provide insights into how robust the findings are, i.e., when some of the assumptions are violated - I especially liked the ones with gamma distribution and vanishing/not vanishing gradients in E.4)\n- The topic is important, especially that it shows a fundamental (and quantifiable) connection between representation identifiability (ICA) and causal discovery - in terms of the number of environments", "weaknesses": "My concerns are mainly about phrasing and writing; thus, minor (also see my clarifying questions below).\n\n- abstract: please make it clear *two environments* is your best-case scenario. As of now, the current phrasing in the abstract might be misleading.\n- L226: please put a reference where you discuss why Assm.4 is needed\n\t- In general, I'd consider putting all assumptions into a single listing\n- Lem. 1.: for insights about the score function, there was previous work by Burak Varici and collaborators, I strongly recommend citing their papers.", "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": "rY0wub4tTQ", "reviewer": "Reviewer_5jiF", "summary": "This paper studies the condition for the unique acyclic causal graph identification from multiple environments.  Under assumptions around nonlinear ICA, the authors show that only two sufficiently different environments is enough for the identifiability.  The technique is by probing the Jacobian mixing functions' support.", "questions": "Could the authors elaborate more on the role of acyclicity in their framework?\n\nGiven the close connection between causal discovery and ICA, I am curious whether acyclicity is truly essential for the results in this work.  For instance, In the linear non-Gaussian case, as shown in https://arxiv.org/abs/1206.3273, the presence of cycles does not create technical difficulties.  One can simply run ICA and interpret the resulting demixing matrix differently (not necessarily to be lower triangular;  as long as it's nonzero entries on diagonal). \n\nIs the same reasoning applicable in the nonlinear ICA setting here, or does acyclicity play some more essential roles here?\n\n\nAnd minor, at L133 notations, \"Also, we use [d] := {1,...,d}.\" should be stated before being used at \"supp(M) := {(i,j)|i∈[m],j ∈[n] \".", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The **main theorem is well motivated and looks correct to me;** I didn't check for details though.\n\n2. The authors **do a good job in discussing the connection between causal discovery and independent component analysis (ICA),** and explain why causal discovery can be easier than full ICA recovery -- one cares the support of the mixing matrix at one point, while another cares the independent sources recovery at each point.\n\n3. **The overall presentation is good.**  The assumptions are clearly listed, and the writing is still clear with heavy notations.", "weaknesses": "1. **My major concern is that the assumptions are way too strong, untestable, and goes a bit against the core goal interventional causal discovery:**\n- Though the conclusion \"only two environments needed; this number does not depend to the number of vertices in graph\" may sound appealing at first, it comes with the price that the causal mechanisms (the entire mixing function) must remain fixed across all environments, the noise components are required to be Gaussian, and only variance rescaling is allowed between interventions.\n- These conditions are too strong merely for the goal of \"two environments needed\".  **This goal itself somewhat goes against the goals for both interventional causal discovery and the characterization of environments needed:**\n   - **For interventional causal discovery,** the main idea is to fully use the available information from multiple environments' data (while allow the changes across environments to be arbitrarily flexible).  It is good as long as we can identify more than from only one environment (e.g.,  the CPDAG).  The goal is not the exact DAG recovery at any cost, but rather to make full use of available information in practice. Otherwise, if the goal is just exact DAG recovery even with model misspecification, why bother to use the difficult estimations involved in this work, instead of just running LiNGAM on one domain?\n   - **For characterization of environments needed** for exact DAG recovery, or other characterizations to the sizes of the equivalence class under interventions: the main purpose is to characterize the internal randomness from data induced by the causal graph, and to help design experiments -- e.g., about how many interventions and which targets are needed for the exact DAG recovery, so as to do the exact effect estimation.  The dependence of this number to the graph size is not a bad thing.  And again, experiments in real world need to be flexible and the strict assumptions in this work are not desired.\n\n2. **Assumptions can be stated more clearly:**\n\nFor instance, in Assumption 2 about noise rescaling:\n - Is mean shift also allowed, as long as the variance changes?\n - Does L have to be diagonal, or is it enough to be orthogonal against Si's diagonal variances?\n\n3. **Experiments are limited to bivariate case with synthetic data.**  This is minor and acceptable though, given the theoretical focus of this paper.  The authors also acknowledge this in the appendix, which is fair.", "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)