SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Julius von Kugelgen $^{1,2}$ , Michel Besserve $^{1}$ , Liang Wendong $^{1}$ , Luigi Gresele $^{1}$ , Armin Kekic $^{1}$ , Elias Bareinboim $^{3}$ , David M. Blei $^{3}$ , and Bernhard Scholkopf $^{1}$
$^{1}$ Max Planck Institute for Intelligent Systems, Tübingen, Germany  
 $^{2}$ Department of Engineering, University of Cambridge, United Kingdom  
 $^{3}$ Columbia University, USA
We study causal representation learning, the task of inferring latent causal variables and their causal relations from high-dimensional functions ("mixtures") of the variables. Prior work relies on weak supervision, in the form of counterfactual pre- and post-intervention views or temporal structure; places restrictive assumptions, such as linearity, on the mixing function or latent causal model; or requires partial knowledge of the generative process, such as the causal graph or intervention targets. We instead consider the general setting in which both the causal model and the mixing function are nonparametric. The learning signal takes the form of multiple datasets, or environments, arising from unknown interventions in the underlying causal model. Our goal is to identify both the ground truth latents and their causal graph up to a set of ambiguities which we show to be irresolvable from interventional data. We study the fundamental setting of two causal variables and prove that the observational distribution and one perfect intervention per node suffice for identifiability, subject to a genericity condition. This condition rules out spurious solutions that involve fine-tuning of the intervened and observational distributions, mirroring similar conditions for nonlinear cause-effect inference. For an arbitrary number of variables, we show that at least one pair of distinct perfect interventional domains per node guarantees identifiability. Further, we demonstrate that the strengths of causal influences among the latent variables are preserved by all equivalent solutions, rendering the inferred representation appropriate for drawing causal conclusions from new data. Our study provides the first identifiability results for the general nonparametric setting with unknown interventions, and elucidates what is possible and impossible for causal representation learning without more direct supervision.
Causal representation learning (CRL) seeks to describe high-dimensional, low-level observations through a small number of interpretable, causally-related latent variables [108, 110]. In doing so, its goal is to combine the strengths of classical causal inference with those of modern machine learning. A causal model represents an entire family of distributions arising from interventions on a system of variables [10, 95, 100]. This provides a principled way for reasoning about distribution shifts, which facilitates out-of-distribution generalization and planning [9, 64, 96, 102]. However, causal models require that most (or at least some) relevant causal variables are directly observed. While reasonable in domains such as economics [7], social [91] or biomedical science [48, 57], this assumption has challenged the application of causal methodology to complex and high-dimensional data [85]. Machine learning, on the other hand, has proven successful at learning useful "representations"—latent vectors generating the observables via some nonlinear map—of high-dimensional data such as images, video, or text [16, 20, 77, 106]. However, most methods rely on independent and identically distributed (i.i.d.) data and only extract associational information. As a consequence, they often fail under distribution shifts and do not generalize beyond the training distribution [107], as exhibited by their reliance on spurious correlations [13, 87] or their vulnerability to adversarial examples [40, 119].
Figure 1: (Left) Data-Generating Process for Causal Representation Learning (CRL). Observations  $\mathbf{X}$  are generated by applying a nonlinear mixing function  $f$  to a set of causal latent variables  $\mathbf{V} = \{V_{1},\dots,V_{n}\}$ , which are related through a structural causal model (SCM) with independent exogenous variables  $\mathbf{U} = \{U_{1},\dots,U_{n}\}$ ; illustration by Ana Martín Larrañaga. (Right) Comparison to Structure Learning. The causal direction between two unconfounded observed variables (top) is uniquely identified from a single intervention [37]; for CRL (bottom), this is not the case, as the nonlinear mixing introduces additional ambiguity due to spurious representations. Shaded nodes are observed, white ones unobserved, and interventions highlighted as red diamonds.
Identifiability. It has been argued that to address these shortcomings, we ought to learn representations endowed with causal model semantics. However, a major challenge to this goal is that different representations can explain the same data equally well. Strictly simpler representation learning tasks, such as disentanglement [14] or independent component analysis (ICA) [53], are already non-identifiable in general [54, 83]. Identifiability studies are thus required to characterize additional assumptions under which the desired latent variables can be provably recovered [4, 17, 21, 31, 42, 43, 45, 46, 51, 52, 55, 56, 66, 69, 70, 75, 76, 84, 90, 101, 112, 114, 127, 141]. In CRL, we need not only identify the latents, but also the causal graph encoding their relations. Even in the fully observed case, this task of causal discovery, or structure learning, is very challenging: the graph can only be recovered up to Markov equivalence [115] based only on (observational) i.i.d. data [116], meaning that the direction of some edges cannot be determined. For CRL, the task gets strictly harder. For instance, if the causal latents  $V_{i}$  in Fig. 1 (Left) form a valid representation, then replacing them by the independent exogenous  $U_{i}$  might be considered an equally valid alternative.
What sets causal representations apart? A crucial feature of causal variables is that they are the ones on which interventions are defined and whose relations we are interested in [95]. Causal discovery and CRL thus often rely on non-i.i.d. data linked to interventions on the underlying causal variables [59, 72]. Unless all variables are subject to intervention, however, some fundamental differences between the fully observed and representation learning settings in the level of ambiguity in the graph remain [117], as illustrated in Fig. 1 (Right). In a sense, the non-identifiabilities of representation and structure learning combine, and both need to be addressed in conjunction.
Problem Setting. We study the general nonparametric CRL problem (§ 2.1) in which both the causal mechanisms and the mixing function are completely unconstrained. Our goal (§ 2.2) is to identify the latent causal variables up to element-wise nonlinear rescaling and their graph up to isomorphism (Defn. 2.6). As motivated above, doing so without further supervision requires access to interventional data. To this end, we consider learning from heterogeneous data from multiple related domains, or environments, that arise from interventions in a shared underlying causal model (§ 2.3).
Contributions. Our main contributions are theoretical in nature (§ 3). First, we establish the minimality of the targeted equivalence class (Prop. 3.1) in the sense that its ambiguities cannot be resolved from interventional data. We then present our main identifiability results. For the case of two latent causal variables, we show that an observational environment and one for each perfect intervention on either variable suffice (Thm. 3.2)—provided that the intervened and unintervened mechanisms are not "fine-tuned" to each other, which we formalize in the form of a genericity condition (3.2). For any number of latent causal variables, we prove that access to pairs of environments corresponding to two distinct perfect interventions on each node guarantees identifiability (Thm. 3.4). We then question how to use or interpret causal representations (§ 4), and show that certain quantities, such as the strengths of causal influences among variables, are preserved by all equivalent solutions (Thm. 4.2). We sketch possible learning objectives (§ 5), and empirically investigate training different generative models (§ 6), finding that only those based on the correct causal structure attain the best fit and identify the ground truth. We conclude by discussing limitations and extensions of our work (§ 7).
Related Work on Multi-Environment CRL. Most closely related are recent identifiability studies which also leverage multiple environments arising from single node interventions [5, 22, 117, 122], thus mirroring in different interventional setups the result of Brehmer et al. [18] based on counterfactual, multi-view data, which is harder to obtain. Squires et al. [117] provide results for linear causal models and linear mixing; Ahuja et al. [5] consider nonlinear causal models and polynomial mixings, subject to additional constraints on the latent support [125]; and Varici et al. [122] employ a score-based approach for nonlinear causal models and linear mixing. The concurrent work of Buchholz et al. [22] extends the results of Squires et al. [117] to general nonlinear mixings and linear Gaussian causal models. Liu et al. [82] leverage recent advances in nonlinear ICA [55, 65] to identify a linear Gaussian causal model with context-dependent weights and nonlinear mixing from sufficiently diverse environments. A more extensive discussion of other related works and a detailed comparison of existing results with the present study are provided in Appx. A and Tab. 1.
2 Nonparametric Causal Representation Learning
In this section, we describe the considered problem setting and state our main assumptions. First, we specify the assumed data generating process (§ 2.1) and learning task in the form of a target identifiability class (§ 2.2). We then demonstrate the hardness of our task from i.i.d. data or imperfect interventions and use this to motivate a multi-environment approach with perfect interventions (§ 2.3).
Notation. We use upper-case  $X$  for random variables and lower-case  $x$  for their realizations. Bold uppercase  $X$  denotes random vectors and lowercase  $x$  their realizations. We assume throughout that all distributions  $P_{X}$  possess densities  $p(x)$  with respect to (w.r.t.) the Lebesgue measure. We denote the pushforward of  $P_{X}$  through a measurable function  $f$  by  $f_{*}(P_{X})$ , and write  $[n] = \{1, \dots, n\}$ .
2.1 Data Generating Process
The assumed data generating process consists of a latent causal model and a mixing function, see Fig. 1 (left). For the former, we build on the structural causal model (SCM) framework [95, 100].
Definition 2.1 (Latent SCM). Let  $V = \{V_{1},\dots,V_{n}\}$  denote a set of causal "endogenous" variables, with each  $V_{i}$  taking values in  $\mathbb{R}$ , and let  $U = \{U_{1},\dots,U_{n}\}$  denote a set of mutually independent "exogenous" random variables. The latent SCM consists of a set of structural equations
$$
\left\{V _ {i} := f _ {i} \left(\mathbf {V} _ {\mathrm {p a} (i)}, U _ {i}\right) \right\} _ {i = 1} ^ {n}. \tag {2.1}
$$
where  $V_{\mathrm{pa}(i)} \subseteq V \setminus \{V_i\}$  are the direct causes, or causal parents, of  $V_i$ , and  $f_i$  are deterministic functions; and a fully factorized joint distribution  $P_U$  over the exogenous variables. The associated causal diagram  $G$ , a directed graph with vertices  $V$  and edges  $V_j \to V_i$  iff.  $V_j \in V_{\mathrm{pa}(i)}$ , is assumed acyclic.
By acyclicity, recursive substitution of the assignments in (2.1) yields the reduced form  $V = f_{\mathrm{RF}}(U)$ . The SCM thus induces a unique distribution  $P_V$  over the endogenous variables, given by the pushforward of  $P_U$  via (2.1), that is,  $P_V = f_{\mathrm{RF}^*}(P_U)$ . By construction,  $P_V$  is Markovian w.r.t. the causal graph  $G$  [95, 100], meaning that its density obeys the causal Markov factorization:
$$
p \left(v _ {1}, \dots , v _ {n}\right) = \prod_ {i = 1} ^ {n} p _ {i} \left(v _ {i} \mid \boldsymbol {v} _ {\mathrm {p a} (i)}\right). \tag {2.2}
$$
We place the following additional assumption on the distribution  $P_V$  induced by the SCM.
Assumption 2.2 (Faithfulness). The only conditional independence relations satisfied by  $P_V$  are those implied by  $\{V_i \perp V_{\mathrm{nd}(i)} \mid V_{\mathrm{pa}(i)}\}_{i \in [n]}$ , where  $V_{\mathrm{nd}(i)}$  denotes the non-descendants of  $V_i$  in  $G$ .
Asm. 2.2 ensures a one-to-one correspondence between (conditional) independence in  $P_V$  and graphical separation in  $G$  and is a standard assumption in causal discovery [115]. Faithfulness rules out cancellations of influences along different paths, which occurs with probability zero for random path-coefficients [121]. It can thus also be viewed as a minimality or genericity assumption.
In contrast to classical causal inference, we assume that both the exogenous variables  $\mathbf{U}$  and the endogenous causal variables  $\mathbf{V}$  are unobserved. Instead, we will only have access to  $d$ -dimensional nonlinear mixtures  $\mathbf{X}$  of  $\mathbf{V}$ . We therefore make the following additional assumption.
Assumption 2.3 (Known  $n$ ). The number  $n$  of latent causal variables is known.
Next, we specify the relationship between the unobserved causal variables  $V$  and the observed  $X$ .
Definition 2.4 (Mixing function). The observations  $\mathbf{X}$  are deterministically generated from  $V$  by applying a mixing function  $f: \mathbb{R}^n \to \mathbb{R}^d$  to  $V$ , that is,  $\mathbf{X} \coloneqq f(\mathbf{V})$ .
The terminology and setting of a deterministic mixing is rooted in the nonlinear ICA literature [56]. In deep generative models, it is commonly relaxed by considering additive noise in Defn. 2.4 [65, 90]. For representation learning scenarios, we are particularly interested in the case  $n \ll d$ . To allow for recovery of  $V$  from  $X$ , we assume that  $f$  is invertible, which is a standard assumption for identifiability.
Assumption 2.5 (Diffeomorphic mixing).  $f$  is a diffeomorphism<sup>1</sup> onto its image  $\operatorname{Im}(f) = \mathcal{X} \subseteq \mathbb{R}^d$ .
2.2 Learning Target: The CRL Identifiability Class  $\sim_{\mathrm{CRL}}$
Our goal is to infer the underlying latent causal variables  $\mathbf{V} = f^{-1}(\mathbf{X})$  and their causal relations. We therefore consider the true unmixing function  $f^{-1}$  and the causal graph  $G$  our joint learning target:  $f^{-1}$  informs us how to map observations  $\mathbf{X}$  to causal variables  $\mathbf{V}$ , and  $G$  tells us how to factorise the implied joint  $p(\mathbf{v})$  into the causal mechanisms  $p_i(v_i \mid v_{\mathrm{pa}(i)})$  from (2.2). Given only observations of  $\mathbf{X}$ , this is a challenging task since neither  $\mathbf{V}$  nor  $G$  are directly observed or known a priori.
When is a candidate solution  $(h, G')$  that satisfies a given learning objective (such as maximizing the likelihood, possibly subject to additional constraints) guaranteed to match the ground truth  $(f^{-1}, G)$ ? This is the subject of identifiability studies and the main focus of our work. A statistical model  $\mathcal{P} = \{p_{\theta} : \theta \in \Theta\}$  with parameter space  $\Theta$  is identifiable if the mapping  $\theta \mapsto p_{\theta}$  is injective [78]. For representation learning tasks, full identifiability is often not attainable, as there are some fundamental ambiguities that cannot be resolved. One therefore typically instead considers identifiability up to an appropriately chosen equivalence class in the model space [2, 65, 127].
For the assumed data generating process (§ 2.1), the order of the causal variables is arbitrary, since  $V$  is unobserved. We can therefore assume without loss of generality (w.l.o.g.) that the  $V_{i}$ 's are partially ordered w.r.t.  $G$ , that is,  $V_{i} \rightarrow V_{j} \Rightarrow i < j$ . Learning  $G$  thus reduces to inferring whether the edges  $\{V_{1}, \ldots, V_{i-1}\} \rightarrow V_{i}$  exist for  $i = 2, \ldots, n$ . The only remaining permutation ambiguity arises from permutations  $\pi$  that preserve the partial order: for example, if  $G$  is given by  $V_{1} \rightarrow V_{3} \leftarrow V_{2}$ , the order of  $V_{1}$  and  $V_{2}$  cannot be uniquely determined without further assumptions. Moreover, the scaling of the causal variables is also arbitrary: any invertible element-wise transformation can be undone as part of  $f$ . We therefore define the desired identifiability class through the following equivalence relation.
Definition 2.6 ( $\sim_{\mathrm{CRL}}$ -identifiability). Let  $\mathcal{H}$  be a space of unmixing functions  $h: \mathcal{X} \to \mathbb{R}^n$  and let  $\mathcal{G}$  be the space of DAGs over  $n$  vertices. Let  $\sim_{\mathrm{CRL}}$  be the equivalence relation on  $\mathcal{H} \times \mathcal{G}$  defined as
$$
\left(h _ {1}, G _ {1}\right) \sim_ {\mathrm {C R L}} \left(h _ {2}, G _ {2}\right) \quad \Longleftrightarrow \quad \left(h _ {2}, G _ {2}\right) = \left(\boldsymbol {P} _ {\pi^ {- 1}} \circ \phi \circ h _ {1}, \pi (G _ {1})\right) \tag {2.3}
$$
for some element-wise diffeomorphism  $\phi(\pmb{v}) = (\phi_1(v_1), \dots, \phi_n(v_n))$  of  $\mathbb{R}^n$  and a permutation  $\pi$  of  $[n]$  such that  $\pi : G_1 \mapsto G_2$  is a graph isomorphism and  $P_{\pi}$  the corresponding permutation matrix.
Remark 2.7. When  $G$  has no edges, any permutation is admissible and  $\sim_{\mathrm{CRL}}$  reduces to the standard notion of identifiability up to permutation and element-wise reparametrisation of nonlinear ICA [56].
The ground truth  $(f^{-1}, G)$  is identified up to  $\sim_{\mathrm{CRL}}$  by a given learning objective if any candidate solution  $(h, G')$  satisfies  $(h, G') \sim_{\mathrm{CRL}} (f^{-1}, G)$ . We seek to discover suitable conditions that ensure this.
2.3 Multi-Environment Data
Given only a single dataset of i.i.d. observations from  $P_{X}$ , there is no hope for  $\sim_{\mathrm{CRL}}$ -identifiability. Even for observed  $V$  (i.e., with  $n = d$  and known  $f = \mathrm{id}$ ),  $G$  can only be identified up to Markov equivalence [115]. With unknown mixing  $f$ , the degree of observational non-identifiability gets even worse: for example, by using the reduced form of the SCM (§ 2.1) we can express  $X$  in terms of the latent exogenous variables  $U$  via  $X = f(V) = f \circ f_{\mathrm{RF}}(U)$  [108]. This gives rise to a "spurious ICA solution"  $(f_{\mathrm{RF}}^{-1} \circ f^{-1}, G_{\mathrm{ICA}})$  where  $G_{\mathrm{ICA}}$  denotes the empty graph with independent components. Due to the non-identifiability of nonlinear ICA [54, 83], however, we cannot even learn the composition  $f \circ f_{\mathrm{RF}}$ , let alone separate it into its constituents  $f$  and  $f_{\mathrm{RF}}$  to isolate the intermediate causal variables  $V$ .
Motivated by these challenges to identifiability from i.i.d. data, we instead consider learning from multiple environments  $e \in \mathcal{E}$ . That is, we assume access to heterogenous data from multiple distinct distributions  $P_{X}^{e}$ . Environments can arise, for example, from different experimental settings or
A diffeomorphism is a bijective function  $f$  such that both  $f$  and  $f^{-1}$  are continuously differentiable.
<sup>2</sup>If they were not in such order to begin with, we could apply an appropriate permutation  $\tilde{\pi}$  to  $\mathbf{V}$  and incorporate the inverse permutation into the unknown mixing  $f$  without affecting  $\pmb {X} = f(\pmb {V}) = (f\circ \tilde{\pi}^{-1})(\tilde{\pi} (\pmb {V}))$  [117].
$\sim_{\mathrm{CRL}}$  satisfies symmetry and transitivity because permutations and element-wise functions commute.
Figure 2: Multi-Environment Setup with Single-Node Perfect Interventions and Shared Mixing Function. Illustration of the considered multi-environment setup for  $n = 2$  causal variables  $V = \{V_{1}, V_{2}\}$  with graph  $G$  given by  $V_{1} \rightarrow V_{2}$ , shared mixing function  $f$ , and environments  $\mathcal{E} = \{e_0, e_1, e_2\}$ , corresponding to the observational setting  $(e_0)$  and perfect stochastic interventions on  $V_{1}$  (in  $e_1$ ) and  $V_{2}$  (in  $e_2$ ). The learnt unmixing function, or encoder, is denoted by  $h$ , and the inferred latent representation by  $Z = h(\boldsymbol{X})$ . The corresponding inferred latent distributions  $Q_Z^{e_i} = h_*(P_X^{e_i})$  are Markovian w.r.t. the candidate graph  $G'$  (here, equal to  $G$ ). Since the intervention targets are not known, they may in principle differ in  $Q_Z^e$  as shown here. However, as we prove in Thm. 3.2, such misalignment is only possible if a certain genericity condition (3.2) is violated.
correspond to broader contexts such as climate or time. Previous work has shown that this setting can, in principle, provide useful causal learning signals [5, 8, 19, 32, 34, 47, 50, 59, 71-73, 82, 89, 98, 99, 102, 104, 109, 117, 120, 122]. However, multi-environment data is not necessarily useful if the corresponding distributions  $P_{\mathbf{X}}^{e}$  are allowed to differ in arbitrary ways. What makes this setting interesting is the assumption that certain parts of the causal generative process are shared across environments.
Here, we assume that all environments share the same invariant mixing function and underlying SCM, and that any distribution shifts arise from interventions on some of the causal mechanisms. General interventions can be modelled in SCMs by replacing some of the equations in (2.1) by new assignments  $V_{i} \coloneqq \tilde{f}_{i}(\mathbf{V}_{\mathrm{pa}(i)}, \tilde{U}_{i})$ , resulting in "intervened" mechanisms  $\tilde{p}_{i}(v_{i} \mid \mathbf{v}_{\mathrm{pa}(i)})$  replacing the corresponding conditionals  $p_{i}(v_{i} \mid \mathbf{v}_{\mathrm{pa}(i)})$  in (2.2) [29, 36, 95]. We summarise this as follows.
Assumption 2.8 (Shared mixing and mechanisms). Each environment  $e$  shares the same mixing  $f$ ,
$$
P _ {X} ^ {e} = f _ {*} \left(P _ {V} ^ {e}\right)
$$
and each  $P_V^e$  results from the same SCM through an intervention on a subset of mechanisms  $\mathcal{I}^e\subseteq [n]$
$$
p ^ {e} (\pmb {v}) = p ^ {e} (v _ {1}, \dots , v _ {n}) = \prod_ {i \in \mathcal {I} ^ {e}} p _ {i} ^ {e} \left(v _ {i} \mid \pmb {v} _ {\mathrm {p a} (i)}\right) \prod_ {j \in [ n ] \backslash \mathcal {I} ^ {e}} p _ {j} \left(v _ {j} \mid \pmb {v} _ {\mathrm {p a} (j)}\right).
$$
Importantly, the intervention targets  $\mathcal{I}^e$  are not assumed to be known.
An intervention on some  $V_{i}$  that fully removes the influence from its parents  $V_{\mathrm{pa}(i)}$  is referred to as perfect, otherwise as imperfect. It has been shown that imperfect interventions are generally insufficient for full identifiability [18], even in the linear case [117]. This is intuitive: if arbitrary imperfect interventions were allowed, including ones which preserve  $f_{i}\big(V_{\mathrm{pa}(i)},\cdot \big)$  and only replace  $U_{i}$  with some new  $\tilde{U}_i$ , then the spurious ICA solution  $(f_{\mathrm{RF}}^{-1}\circ f^{-1},G_{\mathrm{ICA}})$  should be indistinguishable from the ground truth. In line with prior work [5, 18, 117, 122], we therefore assume perfect interventions.
Assumption 2.9 (Perfect interventions). For all  $e \in \mathcal{E}$  and  $i \in \mathcal{I}^e$ , we have  $p^e(v_i \mid v_{\mathrm{pa}(i)}) = p^e(v_i)$ .
Based on the principle of independent causal mechanisms [100, 109], the sparse mechanism shift hypothesis [98, 110] posits that distribution changes tend to manifest themselves in a sparse or local way in the causal factorization. In this spirit, we will assume single-node ("atomic") interventions,  $|\mathcal{I}^e| = 1$ , for our main results, as also required for existing results [5, 18, 117, 122].