SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\mathcal {S} _ {\text {r e p a r a m}} := \left\{\mathbf {g}: \mathbb {R} ^ {d} \rightarrow \mathbb {R} ^ {d} \mid \mathbf {g} = \mathbf {P} \circ \mathbf {h}, \mathbf {P} \text {i s a p e r m u t a t i o n m a t r i x , h} \in \mathcal {S} _ {\text {s c a l i n g}} \right\}. \tag {6}
$$
A similar result in [49] shows that one intervention per node is necessary when the underlying graph is unknown. Prop. 4.3 shows that this is still the case, even when the graph is known.
Fat-hand interventions. A generalization of single-node interventions are fat-hand interventions—i.e., interventions where  $|\tau_k| > 1$ . In this section, we study this more general setting and focus on a weaker form of identification than for single-node intervention.
Assumption 4.4 (Block-interventional discrepancy). We denote  $\mathbb{Q}_{\tau_k}^0$  as the causal mechanism of  $\mathbf{Z}_{\tau_k}$  in the unintervened regime. For each  $k \in [K]$ , we denote  $\mathbb{Q}_{\tau_k}^s$  as the intervention mechanism in the  $s$ -th interventional regime that has  $\tau_k \subseteq [d]$  as targets of intervention, i.e.,  $\mathbb{Q}_{\tau_k}^s$  is a (conditional) joint distribution over  $\mathbf{Z}_{\tau_k}$ . Then the Block-interventional discrepancy for  $\tau_k$  is defined as follows:
- if there is no arrow into  $\tau_{k}$  (i.e.,  $\tau_{k}$  has no parents in  $[d] \setminus \tau_{k}$ ), suppose that there are  $n_{k}$  interventions with target  $\tau_{k}$  such that the following  $n_{k} \times n_{k}$  matrix
$$
\mathbf {M} _ {\tau_ {k}} := \left( \begin{array}{c c c} \frac {\partial}{\partial z _ {1}} (\ln q _ {\tau_ {k}} ^ {1} - \ln q _ {\tau_ {k}} ^ {0}) (\mathbf {z} _ {\tau_ {k}}) & \dots & \frac {\partial}{\partial z _ {n _ {k}}} (\ln q _ {\tau_ {k}} ^ {1} - \ln q _ {\tau_ {k}} ^ {0}) (\mathbf {z} _ {\tau_ {k}}) \\ \vdots & & \vdots \\ \frac {\partial}{\partial z _ {1}} (\ln q _ {\tau_ {k}} ^ {n _ {k}} - \ln q _ {\tau_ {k}} ^ {0}) (\mathbf {z} _ {\tau_ {k}}) & \dots & \frac {\partial}{\partial z _ {n _ {k}}} (\ln q _ {\tau_ {k}} ^ {n _ {k}} - \ln q _ {\tau_ {k}} ^ {0}) (\mathbf {z} _ {\tau_ {k}}) \end{array} \right) \tag {7}
$$
is invertible for  $\mathbf{z}_{\tau_k} \in \mathbb{R}^{n_k}$  almost everywhere, where  $q_{\tau_k,j}^s$  denotes the  $j$ -th marginal of  $q_{\tau_k}^s$ ,  $z_{\tau_k,j}$  denotes the  $j$ -th dimension of  $\mathbf{z}_{\tau_k}$ , and  $s = 0$  denotes the unintervened (observational) regime;
- otherwise, suppose that there are  $n_k + 1$  interventions with target  $\tau_k$  such that the matrix (7) is invertible for  $\mathbf{z}_{\tau_k} \in \mathbb{R}^{n_k}$  almost everywhere, where  $q_{\tau_k,j}^s$  and  $z_{\tau_k,j}$  are defined as before, but  $s = 0, \dots, n_k$  now indexes the  $n_k + 1$  interventions—i.e., without an unintervened regime.
Asm. 4.4 is tightly connected to Asm. 4.1: if  $G$  has no arrow, if  $\forall k : n_k = 1$ , then Asm. 4.4 is reduced to Asm. 4.1. However, for any  $G$  that has arrows, the number of interventional regimes required by Asm. 4.4 is strictly larger than Asm. 4.1.
Theorem 4.5. Given any DAG  $G$ . Suppose that our datasets encompass interventions over all variables in the latent graph, i.e.,  $\bigcup_{k\in [K]}\tau_k = [d]$ . For all  $k\in [K]$ , suppose the targets of interventions are strict subsets of all variables, i.e.,  $|\tau_k| = n_k$ ,  $n_k\in [d - 1]$ . Suppose the interventions over  $\tau_k$  are perfect, i.e. the intervention mechanisms  $\mathbb{Q}_{\tau_k}^s$  are joint distributions over  $\mathbf{Z}_{\tau_k}$  without conditioning on other variables. Suppose Asm. 4.4 is satisfied for  $\tau_k$ .
Then  $\text{CauCA in } (G, \mathcal{F}, \mathcal{P}_G)$  is block-identifiable (following [54]): namely, if for all  $k \in [K]$ ,  $\mathbf{f}_* \mathbb{P}^k = \mathbf{f}_*' \mathbb{Q}^k$ , then for  $\varphi := \mathbf{f}'^{-1} \circ \mathbf{f}$ , for all  $k \in [K]$ ,
$$
\left[ \boldsymbol {\varphi} (\mathbf {z}) \right] _ {\tau_ {k}} = \boldsymbol {\varphi} _ {\tau_ {k}} \left(\mathbf {z} _ {\tau_ {k}}\right). \tag {8}
$$
We illustrate the above identifiability results through an example in Tab. 1.
4.2 Special Case: ICA with stochastic interventions on the latent components
An important special case of CauCA occurs when the graph  $G$  is empty, corresponding to independent latent components. This defines a nonlinear ICA generative model where, in addition to the mixtures, we observe a variable  $\tau_{k}$  which indicates which latent distributions change in the interventional regime  $k$ , while every other distribution is unchanged. This nonlinear ICA generative model is closely related to similar models with observed auxiliary variables [21, 25]: it is natural to interpret
4The relationship between CauCA and nonlinear ICA is discussed in more detail in App. F.
Table 1: Overview of identifiability results. For the DAG  $Z_{1} \to Z_{2} \to Z_{3}$  from Fig. 1, we summarise the guarantees provided by our theoretical analysis in § 4.1 for representations learnt by maximizing the likelihoods  $\mathbb{P}_{\theta}^{k}(X)$  for different sets of interventional regimes.
Requirement of interventions	Learned representation \(\hat{\mathbf{z}} = \hat{\mathbf{f}}^{-1}(\mathbf{x})\)	Reference
1 intervention per node	\([h_1(z_1), h_2(z_1, z_2), h_3(z_1, z_2, z_3)]\)	Thm. 4.2 (i)
1 perfect intervention per node	\([h_1(z_1), h_2(z_2), h_3(z_3)]\)	Thm. 4.2 (ii)
1 intervention per node for \(z_1\) and \(z_2\), plus \(|\overline{pa}(3)|(|\overline{pa}(3)| + 1) = 2 \times 3\) imperfect interventions on \(z_3\) with “variability” assumption	\([h_1(z_1), h_2(z_2), h_3(z_2, z_3)]\)	Prop. D.1
1 perfect intervention on \(z_1\) and \(2+1=3\) perfect fat-hand interventions on \((z_2, z_3)\)	\([h_1(z_1), h_2(z_2, z_3), h_3(z_2, z_3)]\)	Thm. 4.5
$\tau_{k}$  itself as an auxiliary variable. As we will see, our interventional interpretation allows us to derive novel results and re-interpret existing ones. Below, we characterize identifiability for this setting.
Single-node interventions. We first focus on single-node stochastic interventions, where the following result proves that we can achieve the same level of identifiability as in Thm. 4.2 (ii), with one less intervention than in the case where the graph is non-trivial.
Proposition 4.6. Suppose that  $G$  is the empty graph, and that there are  $d - 1$  variables intervened on, with one single target per dataset, such that Asm. 4.1 is satisfied. Then CauCA (in this case, ICA) in  $(G, \mathcal{F}, \mathcal{P}_G)$  is identifiable up to  $S_{\text{scaling}}$  defined as in eq. (3).
The result above shows that identifiability can be achieved through single-node interventions on the latent variables using strictly fewer datasets (i.e., auxiliary variables) than previous results in the auxiliary variables setting ( $d$  in our case,  $2d + 1$  in [21, Thm. 1]). One potentially confusing aspect of Prop. 4.6 is that the ambiguity set does not contain permutations—which is usually an unavoidable ambiguity in ICA. This is due to our considered setting with known targets, where a total ordering of the variables is assumed to be known. The result above can also be extended to the case of unknown intervention targets, where we only know that, in each dataset, a distinct variable is intervened on, but we do not know which one (see App. E). For that case, we prove (Prop. E.6) that ICA in  $(G, \mathcal{F}, \mathcal{P}_G)$  is in fact identifiable up to scaling and permutation. Note that Prop. 4.6 is not a special case of Thm. 4.5 in which  $n_k = 1 \forall k$ , since it only requires  $d - 1$  interventions instead of  $d$ .
We can additionally show that for nonlinear ICA,  $d - 1$  interventions are necessary for identifiability.
Proposition 4.7. Given an empty graph  $G$ , with  $d - 2$  single-node interventions on distinct targets, with one single target per dataset, such that Asm. 4.1 is satisfied. Then CauCA (in this case, ICA) in  $(G, \mathcal{F}, \mathcal{P}_G)$  is not identifiable up to  $S_{\text{reparam}}$ .
Fat-hand interventions. For the special case with independent components, the following corollary characterises identifiability under fat-hand interventions.
Corollary 4.8. [Corollary of Thm. 4.5] Suppose  $G$  is the empty graph. Suppose that our datasets encompass interventions over all variables in the latent graph, i.e.,  $\bigcup_{k\in [K]}\tau_k = [d]$ . Suppose for every  $k$ , the targets of interventions are a strict subset of all variables, i.e.,  $|\tau_k| = n_k$ ,  $n_k\in [d - 1]$ .
Figure 3: We use the “ $\triangleright$ ” symbol together with a “times” symbol to represent how many interventions are required by the two assumptions. (Left) (Thm. 4.5) For Asm. 4.4, we need  $n_k$  interventions to get block-identification of  $\mathbf{z}_{\tau_k}$ . (Right) (Prop. 4.9) For the block-variability assumption, we need  $2n_k$  to get to elementwise identification up to scaling and permutation.
Suppose Asm. 4.4 is verified, which has a simpler form in this case: there are  $n_k$  interventions with target  $\tau_k$  such that  $\mathbf{v}_k(\mathbf{z}_{\tau_k}, 1) - \mathbf{v}_k(\mathbf{z}_{\tau_k}, 0), \dots, \mathbf{v}_k(\mathbf{z}_{\tau_k}, n_k) - \mathbf{v}_k(\mathbf{z}_{\tau_k}, 0)$  are linearly independent, where
$$
\mathbf {v} _ {k} \left(\mathbf {z} _ {\tau_ {k}}, s\right) := \left(\left(\ln q _ {\tau_ {k}, 1} ^ {s}\right) ^ {\prime} \left(z _ {\tau_ {k}, 1}\right), \dots , \left(\ln q _ {\tau_ {k}, n _ {k}} ^ {s}\right) ^ {\prime} \left(z _ {\tau_ {k}, n _ {k}}\right)\right), \tag {9}
$$
where  $q_{\tau_k}^s$  is the intervention of the  $s$ -th interventional regime that has the target  $\tau_k$ , and  $q_{\tau_k,j}^s$  is the  $j$ -th marginal of it.  $z_{\tau_k,j}$  is the  $j$ -th dimension of  $\mathbf{z}_{\tau_k}$ .  $s = 0$  denotes the unintervened regime.
Then  $\text{CauCA in } (G, \mathcal{F}, \mathcal{P}_G)$  is block-identifiable, in the same sense as Thm. 4.5.
Our interventional perspective also allows us to re-interpret and extend a key result in the theory of nonlinear ICA with auxiliary variables, [21, Thm.1]. In particular, the following Proposition holds.
Proposition 4.9. Under the assumptions of Thm. 4.5, suppose furthermore that all density functions in  $\mathcal{P}_G$  and all mixing functions in  $\mathcal{F}$  are  $\mathcal{C}^2$ , and suppose there exist  $k \in [K]$  and there are  $2n_k$  interventions with targets  $\tau_k$  such that for any  $\mathbf{z}_{\tau_k} \in \mathbb{R}^{n_k}$ ,  $\mathbf{w}_k(\mathbf{z}_{\tau_k}, 1) - \mathbf{w}_k(\mathbf{z}_{\tau_k}, 0), \ldots, \mathbf{w}_k(\mathbf{z}_{\tau_k}, 2n_k) - \mathbf{w}_k(\mathbf{z}_{\tau_k}, 0)$  are linearly independent, where
$$
\mathbf {w} _ {k} \left(\mathbf {z} _ {\tau_ {k}}, s\right) := \left(\left(\frac {q _ {\tau_ {k} , 1} ^ {s ^ {\prime}}}{q _ {\tau_ {k} , 1} ^ {s}}\right) ^ {\prime} \left(z _ {\tau_ {k}, 1}\right), \ldots , \left(\frac {q _ {\tau_ {k} , n _ {k}} ^ {s ^ {\prime}}}{q _ {\tau_ {k} , n _ {k}} ^ {s}}\right) ^ {\prime} \left(z _ {\tau_ {k}, n _ {k}}\right), \frac {q _ {\tau_ {k} , 1} ^ {s ^ {\prime}}}{q _ {\tau_ {k} , 1} ^ {s}} \left(z _ {\tau_ {k}, 1}\right), \ldots , \frac {q _ {\tau_ {k} , n _ {k}} ^ {s ^ {\prime}}}{q _ {\tau_ {k} , n _ {k}} ^ {s}} \left(z _ {\tau_ {k}, n _ {k}}\right)\right),
$$
where  $q_{\tau_k}^s$  is the intervention of the  $s$ -th interventional regime that has the target  $\tau_k$ , and  $q_{\tau_{k,j}}^s$  is the  $j$ -th marginal of it.  $z_{\tau_{k,j}}$  is the  $j$ -th dimension of  $\mathbf{z}_{\tau_k}$ .  $s = 0$  denotes the unintervened regime. Then  $\varphi_{\tau_k} \in S_{\text{reparam}} := \left\{\mathbf{g}: \mathbb{R}^{n_k} \to \mathbb{R}^{n_k} | \mathbf{g} = \mathbf{P} \circ \mathbf{h} \text{ where } \mathbf{P} \text{ is a permutation matrix and } \mathbf{h} \in S_{\text{scaling}}\right\}$ .
Remark 4.10. The assumption of linear independence  $\mathbf{w}_k(\mathbf{z}_{\tau_k}, s), s \in [2n_k]$  precisely corresponds to the assumption of variability in [21, Thm. 1]; however, we only assume it within a  $n_k$ -dimensional block (not over all  $d$  variables). We refer to it as block-variability.
Note that the block-variability assumption implies block-interventional discrepancy (Asm. 4.4): i.e., it is a strictly stronger assumption, which, correspondingly, leads to a stronger identification. In fact, block-interventional discrepancy only allows block-wise identifiability within the  $n_k$ -dimensional intervened blocks based on  $n_k$  interventions. In contrast, the variability assumption can be interpreted as a sufficient assumption to achieve identification up to permutation and scaling within a  $n_k$ -dimensional block, based on  $2n_k$  fat-hand interventions (in both cases one unintervened dataset is required), see Fig. 3 for a visualization. We summarise our results for nonlinear ICA in Tab. 2, App. F.
In [21], the variability assumption is assumed to hold over all variables, which in our setting can be interpreted as a requirement over  $2d$  fat-hand interventions over all latent variables simultaneously (plus one unintervened distribution). In this sense, Prop. 4.9 and block-variability extend the result of [21, Thm. 1], which only considers the case where all variables are intervened, by exploiting variability to achieve a strong identification only within a subset of the variables.
Our experiments aim to estimate a CauCA model based on a known graph and a collection of interventional datasets with known targets. We focus on the scenarios with single-node, perfect interventions described in § 4. For additional technical details, see App. H.
Synthetic data-generating process. We first sample DAGs  $G$  with an edge density of 0.5. To model the causal dependence among the latent variables, we use the family of CBNs induced by linear Gaussian structural causal model (SCM) consistent with  $G$ . For the ground-truth mixing function, we use  $M$ -layer multilayer perceptrons  $\mathbf{f} = \sigma \circ \mathbf{A}_M \circ \ldots \circ \sigma \circ \mathbf{A}_1$ , where  $\mathbf{A}_m \in \mathbb{R}^{d \times d}$  for  $m \in [1, M]$  denote invertible linear maps (sampled from a multivariate uniform distribution), and  $\sigma$  is an element-wise invertible nonlinear function. We then sample observed mixtures from these latent CBNs, as described by eq. (2).
Likelihood-based estimation procedure. Our objective is to learn an encoder  $\mathbf{g}_{\theta}:\mathbb{R}^d\to \mathbb{R}^d$  that approximates the inverse function  $\mathbf{f}^{-1}$  up to tolerable ambiguities, together with latent densities  $(p^{k})_{k\in \mathbb{I}[0,d]}$  reproducing the ground truth up to corresponding ambiguities (cf. Lemma 3.3). We
Additional experiments with nonlinear, non-Gaussian CBNs can be found in App. I.
Figure 4: Experimental results. Figures (a) and (e) present the mean correlation coefficients (MCC) between true and learned latents and log-probability differences between the model and ground truth  $(\Delta \log \mathrm{prob.})$  for CauCA experiments. Misspecified models assuming a trivial graph  $(E(G) = \emptyset)$  and a linear encoder function class are compared. All violin plots show the distribution of outcomes for 10 pairs of CBNs and mixing functions. Figures (c) and (d) display CauCA results with varying numbers of nonlinearities in the mixing function and latent dimension. For the ICA setting, MCC values and log probability differences are illustrated in (b) and (f). Baselines include a misspecified model (linear mixing) and a naive (single-environment) unidentifiable normalizing flow with an independent Gaussian base distribution (labelled i.i.d.). The naive baseline is trained on pooled data without using information about interventions and their targets. Figure (g) shows the median MCC for CauCA and the misspecified baseline  $(E(G) = \emptyset)$  as the strength of the linear parameters relative to the exogenous noise in the structural causal model generating the CBN increases. The shaded areas show the range between minimum and maximum values.
estimate the encoder parameters by maximizing the likelihood, which can be derived through a change of variables from eq. (1): for an observation in dataset  $k > 0$  taking a value  $\mathbf{x}$ , it is given by
$$
\log p _ {\boldsymbol {\theta}} ^ {k} (\mathbf {x}) = \log | \det  \mathbf {J} \mathbf {g} _ {\boldsymbol {\theta}} (\mathbf {x}) | + \log \widetilde {p} _ {\tau_ {k}} ((\mathbf {g} _ {\boldsymbol {\theta}}) _ {\tau_ {k}} (\mathbf {x})) + \sum_ {i \neq \tau_ {k}} \log p _ {i} ((\mathbf {g} _ {\boldsymbol {\theta}}) _ {i} (\mathbf {x}) | (\mathbf {g} _ {\boldsymbol {\theta}}) _ {\mathrm {p a} (i)} (\mathbf {x})), \tag {10}
$$
where  $\mathbf{J}\mathbf{g}_{\pmb{\theta}}(\mathbf{x})$  denotes the Jacobian of  $\mathbf{g}_{\pmb{\theta}}$  evaluated at  $\mathbf{x}$ . The learning objective can be expressed as  $\theta^{*} = \arg \max_{\pmb{\theta}}\sum_{k = 0}^{K}\left(\frac{1}{N_{k}}\sum_{n = 1}^{N_{k}}\log p_{\pmb{\theta}}^{k}(\mathbf{x}^{(n,k)})\right)$ , with  $N_{k}$  representing the size of dataset  $k$ .
Model architecture. We employ normalizing flows [40] to parameterize the encoder. Instead of the typically deployed base distribution with independent components, we use the collection of densities (one per interventional regime) induced by the CBN over the latents. Following the CauCA setting, the parameters of the causal mechanisms are learned while the causal graph is assumed known. For details on the model and training parameters, see App. H.
Settings. We investigate two learning problems: (i) CauCA, corresponding to § 4.1, and (ii) ICA, where the sampled graphs in the true latent CBN contain no arrows, as discussed in § 4.2.
Results. (i) For a data-generating process with non-empty graph, experimental outcomes are depicted in Fig. 4 (a, e). We compare a well-specified CauCA model (blue) to misspecified baselines, including a model with correctly specified latent model but employing a linear encoder (red), and a model with a nonlinear encoder but assuming a causal graph with no arrows (orange). See caption of Fig. 4 for details on the metrics. The results demonstrate that the CauCA model accurately identifies the latent variables, benefiting from both the nonlinear encoder and the explicit modelling of causal dependence. We additionally test the effect of increasing a parameter influencing the magnitude of the sampled linear parameters in the SCM (we refer to this as signal-to-noise ratio, see App. H.1 for details)—which increases the statistical dependence among the true latent components. The gap between the CauCA model and the baseline assuming a trivial graph widens (Fig. 4 (g)), indicating that correctly
modelling the causal relationships becomes increasingly important the more (statistically) dependent the true latent variables are. Finally, we verify that the model performs well for different number of layers  $M$  in the ground-truth nonlinear mixing (c) (performance degrades slightly for higher  $M$ ), and across various latent dimensionalities for the latent variable (d).
(ii) For data-generating processes where the graph contains no arrows (ICA), results are presented in Fig. 4 (b, f). Well-specified, nonlinear models (blue) are compared to misspecified linear baselines (red) and a naive normalizing flow baseline trained on pooled data (purple). The findings confirm that interventional information provides useful learning signal even in the context of nonlinear ICA.
Causal representation learning. In the present work, we focus on identifiability of latent CBNs with a known graph, based on interventional data, and investigate the nonlinear and nonparametric case. In CRL (unknown graph), many studies focus on identifiability of latent SCMs instead, which requires strong assumptions such as weak supervision (i.e., counterfactual data) [1, 3, 36, 54]. Alternatively, the setting where temporal information is available, i.e., dynamic Bayesian networks, has been studied extensively [29, 30, 33, 34, 62]. In a non-temporal setting, other works assume interventional data and linear mixing functions [49, 52]; or that the latent distributions are linear Gaussian [35]. Ahuja et al. [2] identify latent representations by deterministic hard interventions, together with parametric assumptions on the mixing, and an independent support assumption [56]. Concurrent work studies the cases with non-parametric mixing and linear Gaussian latent causal mode [5] or non-parametric latent causal model under faithfulness, genericity and Asm. 4.1 [55].
Prior knowledge on the latent SCM. Other prior works also leverage prior knowledge on the causal structure for representation learning. Yang et al. [61] introduce the CausalVAE model, which aims to disentangle the endogenous and exogenous variables of an SCM, and prove identifiability up to affine transformations based on known intervention targets. Shen et al. [46] also consider the setting in which the graph is (partially) known, but their approach requires additional supervision in the form of annotations of the ground truth latent. Leeb et al. [31] embed an SCM into the latent space of an autoencoder, provided with a topological ordering allowing it to learn latent DAGs.
Statistically dependent components. Models with causal dependences among the latent variables are a special case of models where the latent variables are statistically dependent [22]. Various extensions of the ICA setting allow for dependent variables: e.g., independent subspace analysis [16]; topographic ICA [20] (see also [24]); independently modulated component analysis [26]. Morioka and Hyvarinen [39] introduce a multi-modal model where within-modality dependence is described by a Bayesian network, with joint independence across the modalities, and a mixing function for same-index variables across these networks. Unlike our work, it encodes no explicit notions of interventions.
Limitations. (i) Known intervention targets: We proved that with fully unknown targets, there are fundamental and strong limits to identifiability (see Corollary E.8). We also studied some relaxations of this assumption (App. E), and generalized our results to known targets up to graph automorphisms and matched intervention targets (see Prop. E.6 and Prop. E.6). Other relaxations are left for future work; e.g., the case with a non-trivial graph and matched intervention targets is studied in [55], under faithfulness and genericity assumptions. (ii) Estimation: More scalable estimation procedures than our likelihood-based approach ( $\S 5$ ) may be developed, e.g., based on variational inference.
CauCA as a causal generalization of ICA. As pointed out in § 4.2, the special case of CauCA with a trivial graph corresponds to a novel ICA model. Beyond the fact that CauCA allows statistical dependence described by general DAGs among the components, we argue that it can be viewed as a causal generalization of ICA. Firstly, we exploit the assumption of localized and sparse changes in the latent mechanisms [42, 45], in contrast to previous ICA works which exploit non-stationarity at the level of the entire joint distribution of the latent components [17, 21, 38], leading to strong identifiability results (e.g., in Thm. 4.2 (ii)). Secondly, we exploit the modularity of causal mechanisms: i.e., it is possible to intervene on some of the mechanisms while leaving the others invariant [41, 43]. To the best of our knowledge, our work is the first ICA extension where latent dependence can actually be interpreted in a causal sense.
The authors thank Vincent Stimper, Weiyang Liu, Siyuan Guo, Junhyung Park, Jinglin Wang, Corentin Correia, Cian Eastwood, Adrián Javaloy and the anonymous reviewers for helpful comments and discussions.
Funding Transparency Statement
This work was supported by the Tübingen AI Center. L.G. was supported by the VideoPredict project, FKZ: 01IS21088.