SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Figure 5: Representative cases for the CauCA settings described in Defn. E.2  $(i)$  -  $(iv)$ . Each row corresponds to a dataset where one perfect intervention is performed on one of the targets: Cloud (C), Sprinkler (S), Rain (R), and Wet grass (W). Each column corresponds to an admissible choice of intervention targets within each of the settings: the discrepancies between the intervention targets  $\tau^{\prime}(\mathrm{i}) - \tau^{\prime}(\mathrm{iv})$  and the ground truth targets  $\tau$  are meant to illustrate different degrees of ignorance about  $\tau$  across the settings in Defn. E.2  $(i)$  -  $(iv)$ . In the known intervention targets setting, which we focused on in the main paper,  $\tau^{\prime}(\mathrm{i})$  is the only possible choice: i.e., the targets must be perfectly aligned with the ground truth  $\tau$ . For the settings with known intervention targets up to graph automorphisms and with matched intervention targets,  $\tau^{\prime}(\mathrm{ii})$  and  $\tau^{\prime}(\mathrm{iii})$  represent admissible choices: identifiability results can be proved for both settings. We also show that in the setting of totally unknown targets, where  $\tau^{\prime}(\mathrm{iv})$  is an admissible choice, CauCA is not identifiable (see Remark E.9).
Remark E.3. In this general framework, the dataset  $\mathcal{D}_i\coloneqq \left(\{\mathbf{x}^{(j)}\}_{j = 1}^{N_i}\right)$ ,  $T_{i}$  denotes the nodes in  $V(G)$  ("cloudy", "sprinkler" etc) instead of nodes  $\tau_{i}\subset [d]$  in  $V(G)$  (2).
$^6$ Or  $(G,\mathcal{F},\mathcal{P}_G)$  for CauCA with known intervention targets.
If all  $d$  variables are included in the known targets, then there exists a unique bijection  $\sigma : V(G) \to V(\mathsf{G})$ . This implies that for any latent CBN  $(G, \mathbf{f}, (\mathbb{P}^i, \tau_i)_{i \in [0, K]})$  in  $(G, \mathcal{F}, \mathcal{P}_G)$ ,  $\tau_i = \sigma^{-1}(T_i)$  i.e. the targets  $\tau_i$  are uniquely defined by  $T_i$  in each interventional regime. In this case, without loss of generality, we can suppose that  $\forall i \in V(G)$ , if  $i \in \mathrm{pa}(j)$ , then  $i < j$ . This can be achieved by rearranging the nodes in the graph and, correspondingly, the coordinates of  $(\mathbb{P}^k, \tau_k) \forall k \in [0, K]$ . When the intervention targets are totally unknown,  $(\mathsf{G}, \mathcal{F}, \mathcal{P}_{\mathsf{G}})$  only gives us the information of an unordered graph  $\mathsf{G}$ , and in every interventional regime the candidate latent CBN models that achieve the same likelihood might intervene on totally different variables. In this case, we cannot rearrange the nodes without loss of generality.
Our main paper has shown identifiability results about Defn. E.2 (i). In the following, we generalize Thm. 4.2 to CauCA with known intervention targets up to graph automorphisms. We also generalize Prop. 4.6 to matched intervention targets in ICA setting. We also prove in Corollary E.8 that CauCA with unknown intervention targets is not identifiable. An open question is whether CauCA with a nontrivial graph is identifiable with matched intervention targets.
Assumption E.4 (Interventional discrepancy, general version). Given  $k \in [K]$ , we say that a stochastic intervention  $\tilde{p}_{\tau_k}$  satisfies general interventional discrepancy if for all  $i \in [d]$ ,
$$
\frac {\partial (\ln p _ {i})}{\partial z _ {i}} \left(z _ {i} \mid \mathbf {z} _ {p a (i)}\right) \neq \frac {\partial (\ln \widetilde {p} _ {\tau_ {k}})}{\partial z _ {\tau_ {k}}} \left(z _ {\tau_ {k}} \mid \mathbf {z} _ {p a ^ {k} \left(\tau_ {k}\right)}\right) \quad \text {a l m o s t e v e r y w h e r e (a . e .)}. \tag {40}
$$
Theorem E.5. For CauCA with known intervention targets up to graph automorphisms in  $(G, \mathcal{F}, \mathcal{P}_G)$ ,
(i) Suppose for each node in [d], there is one (perfect or structure-preserving) stochastic intervention such that Asm. E.4 is satisfied. Then CauCA in  $(G, \mathcal{F}, \mathcal{P}_G)$  is identifiable up to
$$
\mathcal {S} _ {A u t \bar {G}} = \left\{\mathbf {P} _ {\sigma} \circ \mathbf {h}: \mathbb {R} ^ {d} \rightarrow \mathbb {R} ^ {d} | \sigma \in A u t _ {G}, \mathbf {P} _ {\sigma} p e r m u t a t i o n m a t r i x o f \sigma , \right.
$$
$$
\left. \mathbf {h} (\mathbf {z}) = \left(h _ {i} \left(\mathbf {z} _ {\overline {{a n c}} (i)}\right)\right) _ {i \in [ d ]}, \mathbf {h} \text {i s} \mathcal {C} ^ {1} - d i f f e o m o r p h i s m \right\}.
$$
(ii) Suppose for each node  $i$  in  $[d]$ , there is one perfect stochastic intervention such that Asm. E.4 is satisfied, then  $\mathsf{CauCA}$  in  $(G,\mathcal{F},\mathcal{P}_G)$  is identifiable up to
$$
S _ {G - \text {s c a l i n g}} := \left\{\mathbf {P} _ {\sigma} \circ \mathbf {h} \mid \sigma \in \operatorname {A u t} _ {G}, \mathbf {P} _ {\sigma} \text {p e r m u t a t i o n m a t r i x o f} \sigma , \mathbf {h}: \mathbb {R} ^ {d} \rightarrow \mathbb {R} ^ {d}, \right.
$$
$$
\mathbf {h} (\mathbf {z}) = \left(h _ {1} \left(z _ {1}\right), \dots , h _ {d} \left(z _ {d}\right)\right) f o r s o m e h _ {i} \in \mathcal {C} ^ {1} (\mathbb {R}, \mathbb {R}) w i t h \left| h _ {i} ^ {\prime} (\mathbf {z}) \right| > 0 \quad \forall \mathbf {z} \in \mathbb {R} ^ {d} \}.
$$
Proof. Proof of (i): The proof is based on the proof of Thm. 4.2(i). Consider two latent models achieving the same likelihood across all interventional regimes:  $\left(G, \mathbf{f}, \left(\mathbb{P}^i, \tau_i\right)_{i \in [0,d]}\right)$  and  $\left(G, \mathbf{f}', \left(\mathbb{Q}^i, \tau_i'\right)_{i \in [0,d]}\right)$ . By the definition of latent CBN models with known targets up to graph automorphisms, there exists  $\sigma$  automorphism of  $G$  s.t. the targets  $(\tau_i')_{i \in [d]} = (\sigma(1), \dots, \sigma(d))$ . Since  $(\tau_i)_{i \in [d]}$  covers all  $d$  nodes in  $G$ , by rearranging  $(\tau_i)_{i \in [d]}$  we can suppose without loss of generality that  $\tau_i = i \forall i \in [d]$ . Namely, in the  $i$ -th interventional regime,
$$
\mathbb {P} ^ {i} = \widetilde {\mathbb {P}} _ {i} \left(Z _ {j} \mid \mathbf {Z} _ {\mathrm {p a} ^ {i} (j)}\right) \prod_ {j \neq i} \mathbb {P} _ {\left(Z _ {j} \mid \mathbf {Z} _ {\mathrm {p a} (j)}\right)}, \quad \mathbb {Q} ^ {i} = \widetilde {\mathbb {Q}} _ {\sigma (i)} \left(Z _ {\sigma (i)} \mid \mathbf {Z} _ {\mathrm {p a} ^ {\sigma (i)} (\sigma (i))}\right) \prod_ {j \neq i} \mathbb {Q} _ {\left(Z _ {\sigma (j)} \mid \mathbf {Z} _ {\mathrm {p a} (\sigma (j))}\right)}.
$$
Define  $\varphi \coloneqq \mathbf{f}^{\prime -1}\circ \mathbf{f}$ . Denote its  $i$ -th dimension output function as  $\varphi_{i}:\mathbb{R}^{d}\to \mathbb{R}$ . We will prove by induction that  $\forall i\in [d],\forall j\notin \overline{\mathrm{anc}} (i),\forall \mathbf{z}\in \mathbb{R}^{d},\frac{\partial\varphi_{\sigma(i)}}{\partial z_{j}} (\mathbf{z}) = 0.$
For any  $i\in [[0,d]]$ $\mathbf{f}_*\mathbb{P}^i = \mathbf{f}_*^\prime \mathbb{Q}^i$  . Since  $\varphi$  is a diffeomorphism, by the change of variables formula,
$$
p ^ {i} (\mathbf {z}) = q ^ {i} \left(\varphi (\mathbf {z})\right) | \det  \varphi (\mathbf {z}) |. \tag {41}
$$
For  $i = 0$ , factorize  $p_i$  and  $q^i$  according to  $G$ , then take the logarithm on both sides:
$$
\sum_ {j = 1} ^ {d} \ln p _ {j} \left(z _ {j} \mid \mathbf {z} _ {\mathrm {p a} (j)}\right) = \sum_ {j = 1} ^ {d} \ln q _ {j} \left(\varphi_ {j} (\mathbf {z}) \mid \varphi_ {(\mathrm {p a} (j))} (\mathbf {z})\right) + \ln | \det  D \varphi (\mathbf {z}) |. \tag {42}
$$
For  $i = 1$ ,  $\widetilde{\mathbb{Q}}_1$  has no conditionals, and so does  $Z_{\sigma(1)}$ . Thus  $q^i$  is factorized as
$$
q ^ {1} (\mathbf {z}) = \widetilde {q} _ {\sigma (1)} \left(z _ {\sigma (1)}\right) \prod_ {j \neq \sigma (1)} q _ {j} \left(z _ {j} \mid \mathbf {z} _ {\mathrm {p a} (j)}\right).
$$
So the equation (41) for  $i = 1$  after taking logarithm is
$$
\begin{array}{l} \ln \widetilde {p _ {1}} \left(z _ {1}\right) + \sum_ {j \neq 1} \ln p _ {j} \left(z _ {j} \mid \mathbf {z} _ {\mathrm {p a} (j)}\right) = \ln \widetilde {q} _ {\sigma (1)} \left(\varphi_ {\sigma (1)} (\mathbf {z}) \mid \varphi_ {\mathrm {p a} ^ {\sigma (1)} (\sigma (1))} (\mathbf {z})\right) \\ + \sum_ {j \neq \sigma (1)} q _ {j} \left(\varphi_ {j} (\mathbf {z}) \mid \varphi_ {\mathrm {p a} (j)} (\mathbf {z})\right) + \ln | \det  D \varphi (\mathbf {z}) |. \tag {43} \\ \end{array}
$$
Subtract (43) by (42),
$$
\ln \tilde {p} _ {1} \left(z _ {1}\right) - \ln p _ {1} \left(z _ {1}\right) = \ln \tilde {q} _ {\sigma (1)} \left(\varphi_ {\sigma (1)} (\mathbf {z})\right) - \ln q _ {\sigma (1)} \left(\varphi_ {\sigma (1)} (\mathbf {z})\right). \tag {44}
$$
$$
0 = \left[ \frac {\widetilde {q} _ {\sigma (1)} ^ {\prime} \left(\varphi_ {\sigma (1)} (\mathbf {z})\right)}{\widetilde {q} _ {\sigma (1)} \left(\varphi_ {\sigma (1)} (\mathbf {z})\right)} - \frac {q _ {\sigma (1)} ^ {\prime} \left(\varphi_ {\sigma (1)} (\mathbf {z})\right)}{q _ {\sigma (1)} \left(\varphi_ {\sigma (1)} (\mathbf {z})\right)} \right] \frac {\partial \varphi_ {\sigma (1)}}{\partial z _ {i}} (\mathbf {z}).
$$
By Asm. E.4, the term in the parenthesis is non-zero a.e. in  $\mathbb{R}^d$ . Thus  $\frac{\partial \varphi_{\sigma(1)}}{\partial z_i}(\mathbf{z}) = 0$  a.e. in  $\mathbb{R}^d$ . Since  $\varphi = \mathbf{f}'^{-1} \circ \mathbf{f}$  where  $\mathbf{f}, \mathbf{f}'$  are  $\mathcal{C}^1$ -diffeomorphisms, so is  $\varphi$ .  $\frac{\partial \varphi_{\sigma(1)}}{\partial z_i}$  is continuous and thus equals zero everywhere.
Now suppose  $\forall k\in [i - 1],\forall j\notin \overline{\mathrm{anc}} (k),\frac{\partial\varphi_{\sigma(k)}}{\partial z_j} (\mathbf{z}) = 0,\forall \mathbf{z}\in \mathbb{R}^d$  . Then for interventional regime  $i$
$$
\begin{array}{l} \ln \tilde {p} _ {i} \left(z _ {i} \mid \mathbf {z} _ {\mathrm {p a} ^ {i} (i)}\right) + \sum_ {j \neq i} \ln p _ {j} \left(z _ {j} \mid \mathbf {z} _ {\mathrm {p a} ^ {j} (j)}\right) = \ln \tilde {q} _ {\sigma (i)} \left(\varphi_ {\sigma (i)} (\mathbf {z}) \mid \varphi_ {\mathrm {p a} ^ {\sigma (i)} (\sigma (i))} (\mathbf {z})\right) \\ + \sum_ {j \neq i} q _ {\sigma (j)} \left(\varphi_ {\sigma (j)} (\mathbf {z}) \mid \varphi_ {\mathrm {p a} (\sigma (i))} (\mathbf {z})\right) + \ln | \det  D \varphi (\mathbf {z}) |, \\ \end{array}
$$
subtracted by (42),
$$
\begin{array}{l} \ln \tilde {p} _ {i} \left(z _ {i} \mid \mathbf {z} _ {\mathrm {p a} ^ {i (i)}}\right) - \ln p _ {i} \left(z _ {i} \mid \mathbf {z} _ {\mathrm {p a} (i)}\right) = \ln \tilde {q} _ {\sigma (i)} \left(\varphi_ {\sigma (i)} (\mathbf {z}) \mid \varphi_ {\mathrm {p a} ^ {\sigma (i)} (\sigma (i))} (\mathbf {z})\right) \\ - \ln q _ {\sigma (i)} \left(\varphi_ {\sigma (i)} (\mathbf {z}) \mid \varphi_ {\mathrm {p a} (\sigma (i))} (\mathbf {z})\right). \\ \end{array}
$$
For any  $j \notin \overline{\mathrm{anc}}(i), j \notin p a(i) \supset \mathrm{pa}^i(i)$  by assumption. Take partial derivative over  $z_j$ :
$$
\begin{array}{l} 0 = \frac {\sum_ {k \in \overline {{\mathrm {p a}}} ^ {\sigma (i)} (\sigma (i))} \frac {\partial \widetilde {q} _ {\sigma (i)}}{\partial \mathbf {x} _ {k}} \left(\varphi_ {\sigma (i)} (\mathbf {z}) \mid \varphi_ {\mathrm {p a} ^ {\sigma (i)} (\sigma (i))} (\mathbf {z})\right) \frac {\partial \varphi_ {k}}{\partial z _ {j}} (\mathbf {z})}{\widetilde {q} _ {\sigma (i)} \left(\varphi_ {\sigma (i)} (\mathbf {z}) \mid \varphi_ {\mathrm {p a} ^ {\sigma (i)} (\sigma (i))} (\mathbf {z})\right)} \\ - \frac {\sum_ {k \in \overline {{\mathrm {p a}}} (\sigma (i))} \frac {\partial q _ {\sigma (i)}}{\partial \mathbf {x} _ {k}} \left(\varphi_ {\sigma (i)} (\mathbf {z}) \mid \varphi_ {\mathrm {p a} (\sigma (i))} (\mathbf {z})\right) \frac {\partial \varphi_ {k}}{\partial z _ {j}} (\mathbf {z})}{q _ {\sigma (i)} \left(\varphi_ {\sigma (i)} (\mathbf {z}) \mid \varphi_ {\mathrm {p a} (\sigma (i))} (\mathbf {z})\right)}. \\ \end{array}
$$
Now prove that  $G_{\sigma (i)} = \sigma (G_i)$
$G_{\sigma(i)}$  is obtained by either a perfect stochastic or a structure-preserving stochastic intervention. For a structure-preserving stochastic intervention,  $G = G_{\sigma(i)} = \sigma(G_i)$ . For a perfect stochastic intervention, since  $\tau_i' = \sigma(\tau_i)$ , in  $G_{\sigma(i)}$  only the arrows towards  $\sigma(\tau_i)$  are deleted, which correspond to deleting arrows towards  $\tau_i$  in  $G_i$ .
Thus  $G_{\sigma(i)} = \sigma(G_i)$ . Thus  $\mathrm{pa}^{\sigma(i)}(\sigma(i)) = \mathrm{pa}^{\sigma(i)}(\sigma(i)) = \sigma(\mathrm{pa}^i(i))$ , the last equality by the definition of automorphism  $\sigma$ ,  $\sigma(\mathrm{pa}(i)) = \mathrm{pa}(\sigma(i))$ .
The assumption of induction says  $\forall k\in [i - 1]\quad \forall j\notin \overline{\mathrm{anc}} (k)$ $\frac{\partial\varphi_{\sigma(k)}}{\partial z_j} (\mathbf{z}) = 0\quad \forall \mathbf{z}\in \mathbb{R}^d$  . For all  $k\in \operatorname {pa}(i)\supset \operatorname{pa}^i (i)$  , since  $j\notin \overline{\mathrm{anc}} (i),j\notin \overline{\mathrm{anc}} (k)$  as well. By induction,  $\frac{\partial\varphi_{\sigma(k)}}{\partial z_j} (\mathbf{z}) = 0$  So the second sum of the right-hand side of (45) can be canceled except for  $k = \sigma (i)$  . Also,  $\mathrm{pa}(\sigma (i)) = \sigma (\mathrm{pa}(i)),\mathrm{pa}^{\sigma (i)}(\sigma (i)) = \sigma (\mathrm{pa}^i (i))$  . By the same assumption in the current induction, for all  $l\in \mathrm{pa}^{\sigma (i)}(\sigma (i)) = \sigma (\mathrm{pa}^i (i)),\frac{\partial\varphi_l}{\partial z_j} (\mathbf{z}) = 0$  . So the first sum of the right-hand side of (45) can be canceled except for  $k = \sigma (i)$
The equation rewrites after deleting the partial derivatives that are zero:
$$
0 = \left[ \frac {\frac {\partial \widetilde {q} _ {\sigma (i)}}{\partial \varphi_ {\sigma (i)}} \left(\varphi_ {\sigma (i)} (\mathbf {z}) \mid \varphi_ {\mathrm {p a} ^ {\sigma (i)} (\sigma (i))} (\mathbf {z})\right)}{\widetilde {q} _ {\sigma (i)} \left(\varphi_ {\sigma (i)} (\mathbf {z}) \mid \varphi_ {\mathrm {p a} ^ {\sigma (i)} (\sigma (i))} (\mathbf {z})\right)} - \frac {\frac {\partial q _ {\sigma (i)}}{\partial \varphi_ {\sigma (i)}} \left(\varphi_ {\sigma (i)} (\mathbf {z}) \mid \varphi_ {\mathrm {p a} (\sigma (i))} (\mathbf {z})\right)}{q _ {\sigma (i)} \left(\varphi_ {\sigma (i)} (\mathbf {z}) \mid \varphi_ {\mathrm {p a} (\sigma (i))} (\mathbf {z})\right)} \right] \frac {\partial \varphi_ {\sigma (i)}}{\partial z _ {j}} (\mathbf {z}).
$$
By Asm. E.4, the term in parenthesis is nonzero a.e., thus  $\frac{\partial\varphi_{\sigma(i)}}{\partial z_j} (\mathbf{z}) = 0$  a.e. Since  $\frac{\partial\varphi_{\sigma(i)}}{\partial z_j}$  is continuous, it equals zero everywhere.
The induction is finished when  $i = d$ . We have proven that  $\forall i \in [d], \forall j \notin \overline{\mathrm{anc}}(i), \forall \mathbf{z} \in \mathbb{R}^d, \frac{\partial \varphi_{\sigma(i)}}{\partial z_j}(\mathbf{z}) = 0$ , i.e.  $\left(\mathbf{P}_{\sigma}^{-1} D \varphi(\mathbf{z})\right)_{ij} = (D \varphi(\mathbf{z}))_{\sigma(i)j} = \frac{\partial \varphi_{\sigma(i)}}{\partial z_j}(\mathbf{z}) = 0$
Thus  $\mathbf{P}_{\sigma}^{-1}\varphi \in S_{\bar{G}}$ .  $\varphi \in S_{\mathrm{Aut}\bar{G}}$
By the result of (i),  $\psi \coloneqq \mathbf{f}^{\prime -1}\circ \mathbf{f}\in S_{\mathrm{Aut}\bar{G}}$  , thus there exists a permutation matrix  $\mathbf{P}_{\sigma}$  s.t.  $\mathbf{P}_{\sigma}^{-1}\psi \in S_{\bar{G}}$  Denote  $\varphi = \mathbf{P}_{\sigma}^{-1}\psi$  . Apply Thm. 4.2(ii), we can prove that  $\varphi \in S_{\mathrm{scaling}}$  . Thus  $\psi = \mathbf{P}_{\sigma}\varphi \in$ $S_{G\cdot \mathrm{scaling}}$
Proposition E.6. Suppose that  $G$  is the empty graph and that there are  $d - 1$  variables intervened on, with one single target per dataset, satisfying Asm. E.4. Then ICA with matched intervention targets in  $(G, \mathcal{F}, \mathcal{P}_G)$  with single-node interventions (Defn. E.2) is identifiable up to
$\mathcal{S}_{\text{reparam}} := \left\{\mathbf{g} : \mathbb{R}^d \to \mathbb{R}^d | \mathbf{g} = \mathbf{P} \circ \mathbf{h} \text{ where } P \text{ is a permutation matrix and } \mathbf{h} \in \mathcal{S}_{\text{scaling}}\right\}$ .
Proof. For an empty graph  $G$ ,  $\mathrm{Aut}_G = \mathfrak{S}_d$ . Thus ICA with matched intervention targets is the same as known intervention targets up to graph automorphisms. Consider two latent models achieving the same likelihood across all interventional regimes:  $\left(G, \mathbf{f}, \left(\mathbb{P}^i, \tau_i\right)_{i \in [0,d]}\right)$  and  $\left(G, \mathbf{f}', \left(\mathbb{Q}^i, \tau_i'\right)_{i \in [0,d]}\right)$ . By the definition of latent CBN models with known targets up to graph automorphisms, there exists  $\sigma$  automorphism of  $G$  s.t. the targets  $(\tau_i')_{i \in [d]} = (\sigma(\tau_1), \dots, \sigma(\tau_d))$ .
Apply Prop. 4.6 to  $\left(G, \mathbf{f}, \left(\mathbb{P}^i, \tau_i\right)_{i \in [0,d-1]}\right)$  and  $\left(G, \mathbf{f}' \circ \mathbf{P}_{\sigma}, \left(\mathbb{Q}^{\sigma^{-1}(i)}, \tau_i\right)_{i \in [0,d-1]}\right)$ , then  $\varphi := (\mathbf{f}' \circ \mathbf{P}_{\sigma})^{-1} \circ \mathbf{f} \in S_{\mathrm{scaling}}$ . Then for  $\left(G, \mathbf{f}, \left(\mathbb{P}^i, \tau_i\right)_{i \in [0,d-1]}\right)$  and  $\left(G, \mathbf{f}', \left(\mathbb{Q}^i, \tau_i'\right)_{i \in [0,d-1]}\right)$ ,  $\mathbf{f}^{\prime-1} \circ \mathbf{f} = \mathbf{P}_{\sigma} \circ \varphi \in S_{\mathrm{reparam}}$ .
Remark E.7. The setting of CauCA with matched intervention targets is similar to one scenario studied in causal representation learning. See, e.g., von Kugelgen et al. [55, Thm. 3.4]: the constraint on intervention targets expressed in their Asm. (A2') can be rephrased within our framework as the requirement that for any two latent CBN models, there exists a permutation  $\pi : V(G) \to V(G')$  such that the intervention targets  $\tau_{i,1}' = \tau_{i,2}' = \pi(\tau_{i,1}) = \pi(\tau_{i,2})$ , where  $\tau_{i,1}$  denotes the unknown target of the  $(i,1)$ -indexed interventional regime. Subsequent work by Varici et al. [53] also studied the setting with coupled environments. $^7$
<sup>7</sup>Varici et al. [53] additionally introduced a setting with uncoupled environments [53, Def. 2], relaxing the requirement of matched interventions. Note that the setting with uncoupled environments assumes strictly more information than Defn. E.2 (iv), since it requires that for each latent CBN model  $(G, \mathbf{f}, (\mathbb{P}^i, \tau_i)_{i \in [0,2d]})$ , for each node  $j \in V(G)$ , there exist  $k, l \in [1,2d]$  such that  $\tau_k = \tau_l = \{j\}$ .
Corollary E.8. [Corollary of Prop. 4.7] Given a DAG G, CauCA with unknown intervention targets is not identifiable in  $(\mathsf{G},\mathcal{F},\mathcal{P}_{\mathsf{G}})$  up to  $S_{\text{reparam}}$ .
Proof. Fix any  $G \models \mathsf{G}$  and  $\mathbf{f}$ . Without loss of generality, suppose  $K > d - 2$ . By Prop. 4.7, there exist  $\left(G, \mathbf{f}', \left(\mathbb{Q}^i, \tau_i\right)_{i \in [0,d-2]} \right)$  such that for all  $i \in [0,d-2]$ ,  $\mathbf{f}_* \mathbb{P}^i = \mathbf{f}_*' \mathbb{Q}^i$  and  $\varphi := \mathbf{f}'^{-1} \circ \mathbf{f} \notin S_{\text{reparam}}$ . Notice that for all  $i \in [0,d-2]$ ,  $\mathbb{P}^i$  and  $\mathbb{Q}^i$  are independent components and therefore Markov relative to  $G$ . Since the targets are unknown, we can construct a spurious solution as follows: for all  $i \in [d-1,K]$ , set  $\tau_i = \{1\}$ , choose  $\mathbb{P}^i$  to be any distribution that is composed of independent components, and set  $\mathbb{Q}^i := \varphi_* \mathbb{P}^i$ . Then for all  $i \in [0,K]$ ,  $\mathbf{f}_* \mathbb{P}^i = \mathbf{f}_*' \mathbb{Q}^i$  and  $\varphi := \mathbf{f}'^{-1} \circ \mathbf{f} \notin S_{\text{reparam}}$ .
Remark E.9. Although the proof above is almost trivial, it clarifies that a minimal requirement for identifiability up to permutation and scaling, both in ICA and in CRL, is that each latent variable must be intervened on at least once.
Relaxing the assumption of a known causal graph. Since we only require the distributions  $\mathbb{P}^k$  in Defn. 3.2 to be Markov to  $G$  (resp.,  $\mathbb{Q}^k$  to  $G'$ ), our theory suggests that it should be possible to identify latent variables up to the ambiguities characterised in our results even when the learned model assumes a denser DAG than the true one: in particular, a fully connected graph would always be a valid choice, thus partially relaxing the assumption of a known graph. A thorough characterisation of the performance of CauCA in this misspecified setting is left for future work.
F Relationship between CauCA and nonlinear ICA
ICA is a special case of CauCA. One way to show that ICA is a special case of CauCA is the following. In CauCA, we suppose the distributions are Markov to a given graph  $G$ , but not necessarily faithful to  $G$ . This implies that any CauCA model  $(G, \mathcal{F}, \mathcal{P}_G)$  may allow independent components: this holds even for models with a non-empty graph (in this case, the distribution would be unfaithful to  $G$ ). Since the distributions with independent components are a small subset within the set of distributions which are Markov to a non-trivial graph  $G$ , it follows that CauCA is a strictly more general, and harder, problem than ICA, and no trivial reduction of CauCA to ICA is possible.
Comparison to the results of Hyvärinen et al. [21]. For the special case of CauCA where there are no edges in the causal graph (i.e., ICA), Hyvärinen et al. [21, Thm.1] proved an identifiability result based on (in our terminology)  $2d$  interventional datasets with unknown intervention targets, plus one unintervened dataset. While our work takes inspiration from [21], our theoretical analysis presents several differences.
Our identifiability results can be compared with [21, Thm.1] in two cases:
(i) Trivial graph:
- Previous work: A core step in the proof of Thm. 1 of [21] is eq. (20,21), which is essentially the same as (33), (34) in our work. We will refer to the proof of our Prop. 4.9 in the following (since the proof of this proposition specifically is similar to the one of Thm.1 of [21]). The proof proceeds then as follows: we take twice the partial derivatives of (32)—i.e., we calculate the Hessian matrix of the two sides. We then obtain a system of  $2d$  equations, and identifiability corresponds to the uniqueness of the solution of the system  $\mathbf{0} = \mathbf{A}\mathbf{x}$ , where  $\mathbf{A}$  is in  $\mathbb{R}^{2d\times 2d}$ . So  $\mathbf{A}$  needs to be invertible—i.e., the "variability" assumption in [21, Thm.1].
- Our work: Our Prop. 4.6 and Corollary 4.8 show that, if the intervention targets are known, we can provide an identifiability proof which only requires the Jacobian of the above equation to be invertible (instead of Hessian, as in the previous results). The functions in  $\mathcal{F}$  can thus be  $\mathcal{C}^1$ , instead of  $\mathcal{C}^2$ , and the linear system will be of  $d$  equations instead of  $2d$ . In short, exploiting knowledge on the intervention targets allows us to provide a different proof, even for the special case of ICA.
In Tab. 2, we summarize our theoretical results for nonlinear ICA and compare them to [21, Thm.1].
(ii) Nontrivial graph:
As we mentioned in the previous paragraph, CauCA is a strictly more general problem than ICA. Our proof of Thm. 4.2 is therefore even farther from all ICA results. A peculiarity of ICA is that, after