SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} 0 = \sum_ {l = 1} ^ {n _ {k}} \left[ \left(\frac {q _ {k , l} ^ {s ^ {\prime}}}{q _ {k , l} ^ {s}}\right) ^ {\prime} (\varphi_ {l} (\mathbf {z} _ {\tau_ {k}})) - \left(\frac {q _ {k , l} ^ {0 \prime}}{q _ {k , l} ^ {0}}\right) ^ {\prime} (\varphi_ {l} (\mathbf {z} _ {\tau_ {k}})) \right] \frac {\partial \varphi_ {l} (\mathbf {z} _ {\tau_ {k}})}{\partial z _ {k}} \frac {\partial \varphi_ {l} (\mathbf {z} _ {\tau_ {k}})}{\partial z _ {j}} \\ + \left[ \frac {q _ {k , l} ^ {s \prime} \left(\varphi_ {l} \left(\mathbf {z} _ {\tau_ {k}}\right)\right)}{q _ {k , l} ^ {s} \left(\varphi_ {l} \left(\mathbf {z} _ {\tau_ {k}}\right)\right)} - \frac {q _ {k , l} ^ {0 \prime} \left(\varphi_ {l} \left(\mathbf {z} _ {\tau_ {k}}\right)\right)}{q _ {k , l} ^ {0} \left(\varphi_ {l} \left(\mathbf {z} _ {\tau_ {k}}\right)\right)} \right] \frac {\partial^ {2} \varphi_ {l} \left(\mathbf {z} _ {\tau_ {k}}\right)}{\partial z _ {k} \partial z _ {j}}. \tag {34} \\ \end{array}
$$
For  $1 \leqslant k < j \leqslant n_k$  there are  $\frac{n_k(n_k - 1)}{2}$  equations.
Define  $\mathbf{a}_l\left(\mathbf{z}_{\tau_k}\right) = \left(\frac{\partial\varphi_l}{\partial z_k}\left(\mathbf{z}_{\tau_k}\right)\frac{\partial\varphi_l}{\partial z_j}\left(\mathbf{z}_{\tau_k}\right)\right)_{1\leqslant k\leqslant j\leqslant n_k},$ $\mathbf{b}_l\left(\mathbf{z}_{\tau_k}\right) = \left(\frac{\partial^2\varphi_l\left(\mathbf{z}_{\tau_k}\right)}{\partial z_k\partial z_j}\right)_{1\leqslant k <   j\leqslant n_k}.$
Then the  $\frac{n_k(n_k - 1)}{2}$  equations can be written as a linear system
$$
\begin{array}{l} 0 = \sum_ {l = 1} ^ {n _ {k}} \mathbf {a} _ {l} \left(\mathbf {z} _ {\tau_ {k}}\right) \left[ \left(\frac {q _ {k , l} ^ {s \prime}}{q _ {k , l} ^ {s}}\right) ^ {\prime} \left(\varphi_ {l} \left(\mathbf {z} _ {\tau_ {k}}\right)\right) - \left(\frac {q _ {k , l} ^ {0 \prime}}{q _ {k , l} ^ {0}}\right) ^ {\prime} \left(\varphi_ {l} \left(\mathbf {z} _ {\tau_ {k}}\right)\right) \right] \\ + \mathbf {b} _ {l} (\mathbf {z} _ {\tau_ {k}}) \left[ \frac {q _ {k , l} ^ {s \prime} (\varphi_ {l} (\mathbf {z} _ {\tau_ {k}}))}{q _ {k , l} ^ {s} (\varphi_ {l} (\mathbf {z} _ {\tau_ {k}}))} - \frac {q _ {k , l} ^ {0 \prime} (\varphi_ {l} (\mathbf {z} _ {\tau_ {k}}))}{q _ {k , l} ^ {0} (\varphi_ {l} (\mathbf {z} _ {\tau_ {k}}))} \right]. \\ \end{array}
$$
Define  $\mathbf{M}_k(\mathbf{z}_{\tau_k}) = (\mathbf{a}_1(\mathbf{z}_{\tau_k}),\dots ,\mathbf{a}_{n_k}(\mathbf{z}_{\tau_k}),\mathbf{b}_1(\mathbf{z}_{\tau_k}),\dots ,\mathbf{b}_{n_k}(\mathbf{z}_{\tau_k})).$
Collect the equations for  $s = 1, \dots, 2n_k$ ,
$$
\begin{array}{l} \mathbf {0} = \mathbf {M} _ {k} \left(\mathbf {z} _ {\tau_ {k}}\right) \left(\mathbf {w} _ {k} \left(\varphi_ {\tau_ {k}} \left(\mathbf {z} _ {\tau_ {k}}\right), 1\right) - \mathbf {w} _ {k} \left(\varphi_ {\tau_ {k}} \left(\mathbf {z} _ {\tau_ {k}}\right), 0\right), \dots , \right. \\ \mathbf {w} _ {k} \left(\varphi_ {\tau_ {k}} \left(\mathbf {z} _ {\tau_ {k}}\right), 2 n _ {k}\right) - \mathbf {w} _ {k} \left(\varphi_ {\tau_ {k}} \left(\mathbf {z} _ {\tau_ {k}}\right), 0\right). \\ \end{array}
$$
By assumption, the matrix containing  $\mathbf{w}$  is invertible. Thus  $M_{k}(\mathbf{z}_{\tau_{k}}) = \mathbf{0}$ , which implies  $\mathbf{a}_{\ell}(\mathbf{z}_{\tau_{k}})$  are zero for all  $\mathbf{z}_{\tau_k}$ . By the same reasoning as [21], each row in  $D\varphi_{\tau_k}(\mathbf{z}_{\tau_k})$  has only one non-zero term, and this does not change for different  $z$ , since otherwise by continuity there exists  $\mathbf{z}$  such that  $D\varphi_{\tau_k}(\mathbf{z}_{\tau_k})$  is singular, contradiction with the invertibility of  $\varphi_{\tau_k}$  (Thm. 4.5). Thus  $\forall i \in \tau_k$ ,  $\varphi_i$  is a function of one coordinate of  $\mathbf{z}_{\tau_k}$ . Since  $\varphi_{\tau_k}$  is invertible,  $\det D\varphi_{\tau_k}(\mathbf{z}_{\tau_k}) \neq 0$ , so  $\exists \sigma$  permutation of  $\tau_k$  s.t.  $\forall i \in \tau_k$ ,  $\frac{\partial\varphi_{\sigma(i)}}{\partial z_i} (\mathbf{z}_{\tau_k}) \neq 0$ . Thus  $\varphi_{\tau_k} \in S_{\mathrm{reparam}}$ .
C A Counterexample for Thm. 4.2 (ii) when Asm. 4.1 is violated
One trivial case of violation of Asm. 4.1 is when  $\mathbb{P}_i$  and  $\widetilde{\mathbb{P}}_i$  are the same. In that case, the interventional regime  $i$  is useless, namely, it does not constrain at all the indeterminacy set. Our counterexample is in a non-trivial case of violation, where the interveneded mechanisms are not deterministically related, and do not share symmetries with the causal mechanisms.
We construct a counterexample for Thm. 4.2 (ii) when Asm. 4.1 is violated. The visualization of it is in Fig. 2. The counterexample is similar to the non-Gaussian case in the proof of Prop. 4.7.
Suppose that  $d = 2$ ,  $E(G) = \varnothing$  and that for all  $i \in \{1, 2\}$ ,  $\mathbb{P}_i$  has the same density  $p_{a,b}$ :
$$
p _ {a, b} (z) = \left\{ \begin{array}{l l} \exp (- a z ^ {2}) & z <   0 \\ 1 & 0 \leq z \leq 1 - \frac {1}{2} (\sqrt {\frac {\pi}{a}} + \sqrt {\frac {\pi}{b}}) \\ \exp (- b (z - (1 - \frac {1}{2} (\sqrt {\frac {\pi}{a}} + \sqrt {\frac {\pi}{b}}))) ^ {2}) & z > 1 - \frac {1}{2} (\sqrt {\frac {\pi}{a}} + \sqrt {\frac {\pi}{b}}) \end{array} \right. \tag {35}
$$
where  $\sqrt{\frac{\pi}{a}} < 1$ ,  $\sqrt{\frac{\pi}{b}} < 1$ . One can verify that  $p_a, p_b$  are smooth p.d.f.
Suppose that for all  $i \in \{1, 2\}$ , the intervened mechanism  $\widetilde{\mathbb{P}}_i$  has the same density  $p_{c,d}$ , defined in the same way as (35), such that  $\sqrt{\frac{\pi}{c}} < 1$ ,  $\sqrt{\frac{\pi}{d}} < 1$ , and  $c, d \notin \{a, b\}$ .
Set  $\lambda := \min_{(x,y) \in \{(a,b),(c,d)\}} \left(1 - \frac{1}{2}\left(\sqrt{\frac{\pi}{x}} + \sqrt{\frac{\pi}{y}}\right)\right)$ , then over  $(0,\lambda)^2$  all the densities are constant, violating Asm. 4.1.
$$
\varphi (\mathbf {z}) = \left\{ \begin{array}{l l} \mathbf {z} & | | \mathbf {z} | | \geq R \\ \left( \begin{array}{l} \cos (\alpha (| | \mathbf {z} - \mathbf {c} | | - R)) z _ {1} - \sin (\alpha (| | \mathbf {z} - \mathbf {c} | | - R)) z _ {2} \\ \cos (\alpha (| | \mathbf {z} - \mathbf {c} | | - R)) z _ {2} + \sin (\alpha (| | \mathbf {z} - \mathbf {c} | | - R)) z _ {1} \end{array} \right) & | | \mathbf {z} | | <   R \end{array} \right.
$$
where  $\mathbf{c} = (\frac{\lambda}{2},\frac{\lambda}{2})$  denotes the center of rotation,  $R\in (0,\frac{\lambda}{2} ]$  denotes the radius of the disk, and  $\alpha \neq 0$
Now let us prove that  $\varphi$  preserves  $\mathbb{P}^i$  for all  $i\in [[0,2]]$ . By shifting  $p_{a,b}$  by  $-\mathbf{c}$ , we only need to prove that the shifted  $\varphi$  preserves the uniform distribution over  $[-R,R]^2$ . One can verify that  $\varphi$  is a diffeomorphism over the 2-dimensional open disk  $D^{2}(\mathbf{0},R)\setminus \{\mathbf{0}\} \to D^{2}(\mathbf{0},R)\setminus \{\mathbf{0}\}$  and  $|\operatorname *{det}(\varphi (\mathbf{z}))| = 1$ . Thus  $p_a(\mathbf{z}) = p_a(\varphi (\mathbf{z}))|\operatorname *{det}(\varphi (\mathbf{z}))|\forall \mathbf{z}\in D^2 (\mathbf{0},R)\setminus \{\mathbf{0}\}$ . Since  $\varphi = Id$  outside of the disk, this change of variables formula holds almost everywhere in  $\mathbb{R}^2$ , thus  $\mathbb{P}_i = \varphi_*\mathbb{P}_i$ ,  $\widetilde{\mathbb{P}}_i = \varphi_*\widetilde{\mathbb{P}}_i$ , which implies that  $\varphi_{*}\mathbb{P}^{i} = \mathbb{P}^{i}\forall i\in [[0,2]]$ .
Thus for all  $\mathbf{f} \in \mathcal{F}$ ,  $\mathbf{f}_*\mathbb{P}^i = (\mathbf{f} \circ \varphi^{-1})_*\mathbb{P}^i$ . However,  $(\mathbf{f} \circ \varphi^{-1})^{-1} \circ \mathbf{f} = \varphi$ , which is not in  $S_{\text{reparam}}$  or  $S_{\text{scaling}}$ .
Remark C.1. The above example can be easily generalized to any  $p_{a,b}$  such that the constants on the plateau are different between  $\mathbb{P}_i$  and  $\widetilde{\mathbb{P}}_i$  and the domains of the plateau intersect on a nonzero measure set. For  $d > 2$ , the above example can be generalized by constructing the same  $\mathbb{P}_i$  and  $\widetilde{\mathbb{P}}^i$  for  $i = 1,2$ , and for  $i > 2$  we fix any  $\mathbb{P}_i$  and  $\widetilde{\mathbb{P}}_i$  verifying Asm. 4.1. Then one spurious solution  $\varphi$  is as follows: let  $\varphi_{[[1,2]]}$  be the same measure-preserving automorphism as in the previous counterexample, and  $\varphi_j = Id$  for  $j > 2$ .
Remark C.2. In CauCA, we suppose the distributions are Markov to a given graph  $G$ , but not necessarily faithful to  $G$ . This implies that independent components are in  $\mathcal{P}_G$  no matter which graph  $G$  is supposed given. Therefore, as long as Asm. 4.1 is not assumed, this counterexample applies to any CauCA model  $(G,\mathcal{F},\mathcal{P})$  with nonlinear  $\mathcal{F}$  and nonparametric  $\mathcal{P}$ .
D Identifiability by structure-preserving stochastic interventions
In this section, we extend the result of Thm. 4.2 (i) to the case when we have access to a higher number of imperfect interventions. Here we focus on one special case of imperfect interventions, structure-preserving interventions, i.e., the interventions that do not change the parent set.
Proposition D.1. For  $\text{CauCA}$  in  $(G, \mathcal{F}, \mathcal{P}_G)$  assume the assumptions in Thm. 4.2 (i) hold. Fix any  $i \in [d]$  such that  $pa(i) \neq \text{anc}(i)$ ,  $pa(i) \neq \varnothing$ , and define  $n_i := |\overline{pa}(i)|$ . If there are  $n_i (n_i + 1)$  structure-preserving interventions on node  $i$  such that the variability assumption  $V^i$  holds, then  $\text{CauCA}$  in  $(G, \mathcal{F}, \mathcal{P}_G)$  is identifiable up to
$$
\mathcal {S} _ {\overline {{G}} _ {i}} = \left\{\mathbf {h} \in \mathcal {C} ^ {1} (\mathbb {R} ^ {d}): \mathbf {h} (\mathbf {z}) = \big (h _ {j} (\mathbf {z} _ {\overline {{a n c}} (j)}) \big) _ {j \in [ d ]} \mid h _ {j} (\mathbf {z} _ {\overline {{a n c}} (j)}) = h _ {j} (\mathbf {z} _ {\overline {{p a}} (i)}) \forall j \in \overline {{p a}} (i) \right\}.
$$
Namely, for all  $\varphi \in S_{\overline{G}_i}$ , for all the nodes  $j \in \overline{pa}(i)$ , the reconstructed  $Z_j$  can at most be a mixture of variables corresponding to the nodes in the closure of parents of  $i$ , instead of the closure of ancestors of  $j$ .
The variability assumption  $V^i$  means
$$
\left( \begin{array}{c c} \mathbf {A} _ {i} ^ {1} (\mathbf {z}) & \mathbf {B} _ {i} ^ {1} (\mathbf {z}) \\ \vdots & \vdots \\ \mathbf {A} _ {i} ^ {n _ {i} (n _ {i} + 1)} (\mathbf {z}) & \mathbf {B} _ {i} ^ {n _ {i} (n _ {i} + 1)} (\mathbf {z}) \end{array} \right) \in \mathbb {R} ^ {n _ {i} (n _ {i} + 1) \times n _ {i} (n _ {i} + 1)}
$$
is invertible, where the symbols are defined as follows:
$$
\mathbf {A} _ {i} ^ {t} (\mathbf {z}) = \left( \begin{array}{c} \frac {\partial (g _ {i} ^ {s _ {1} , t} - h _ {i} ^ {s _ {1}})}{\partial x _ {r _ {1}}} (\varphi_ {i} (\mathbf {z}) | \varphi_ {p a (i)} (\mathbf {z})) \\ \vdots \\ \frac {\partial (g _ {i} ^ {s _ {k} , t} - h _ {i} ^ {s _ {k}})}{\partial x _ {r _ {m}}} (\varphi_ {i} (\mathbf {z}) | \varphi_ {p a (i)} (\mathbf {z})) \\ \vdots \\ \frac {\partial (g _ {i} ^ {s _ {n _ {i}} , t} - h _ {i} ^ {s _ {n _ {i}}})}{\partial x _ {r _ {n _ {i}}}} (\varphi_ {i} (\mathbf {z}) | \varphi_ {p a (i)} (\mathbf {z})) \end{array} \right) ^ {\top} \in \mathbb {R} ^ {1 \times n _ {i} ^ {2}},
$$
$$
\mathbf {B} _ {i} ^ {t} (\mathbf {z}) = \left( \begin{array}{c} \left(g _ {i} ^ {s _ {1}, t} - h _ {i} ^ {s _ {1}}\right) \left(\varphi_ {i} (\mathbf {z}) \mid \varphi_ {p a (i)} (\mathbf {z})\right) \\ \vdots \\ \left(g _ {i} ^ {s _ {n _ {i}}, t} - h _ {i} ^ {s _ {n _ {i}}}\right) \left(\varphi_ {i} (\mathbf {z}) \mid \varphi_ {p a (i)} (\mathbf {z})\right) \end{array} \right) ^ {\top} \in \mathbb {R} ^ {1 \times n _ {i}}
$$
where  $r_k, s_k$  are the  $k$ -th variable in  $\overline{pa}(i)$ , and
$$
g _ {i} ^ {k, t} \left(z _ {i} \mid \mathbf {z} _ {p a (i)}\right) := \frac {\frac {\partial \widetilde {q} _ {i} ^ {t}}{\partial z _ {k}} \left(z _ {i} \mid \mathbf {z} _ {p a (i)}\right)}{\widetilde {q} _ {i} ^ {t} \left(z _ {i} \mid \mathbf {z} _ {p a (i)}\right)}, \quad h _ {i} ^ {k} \left(z _ {i} \mid \mathbf {z} _ {p a (i)}\right) := \frac {\frac {\partial q _ {i}}{\partial z _ {k}} \left(z _ {i} \mid \mathbf {z} _ {p a (i)}\right)}{q _ {i} \left(z _ {i} \mid \mathbf {z} _ {p a (i)}\right)}
$$
where  $\widetilde{q}_i^t$  denotes the intervened mechanism in t-th interventional regime that has the interventional target  $i$ ,  $q_i$  denotes the causal mechanism on  $Z_{i}$ .
Proof. Based on the assumption of Thm. 4.2(i), reuse the proof of Thm. 4.2(i) from the equation (21):
$$
\ln \widetilde {p _ {i}} ^ {t} \left(z _ {i} \mid \mathbf {z} _ {\mathrm {p a} ^ {i} (i)}\right) - \ln p _ {i} \left(z _ {i} \mid \mathbf {z} _ {\mathrm {p a} (i)}\right) = \ln \widetilde {q _ {i}} ^ {t} \left(\varphi_ {i} (\mathbf {z}) \mid \varphi_ {\mathrm {p a} ^ {i} (i)} (\mathbf {z})\right) - \ln q _ {i} \left(\varphi_ {i} (\mathbf {z}) \mid \varphi_ {\mathrm {p a} (i)} (\mathbf {z})\right)
$$
where  $\widetilde{p}_i^t,\widetilde{q}_i^t$  denote the intervened mechanism in  $t$ -th interventional regime that has the interventional target  $i$ . Notice that by the assumption of structure-preserving interventions,  $\mathrm{pa}^i (i) = \mathrm{pa}(i)$ .
Thm. 4.2(i) has already concluded that  $\partial_j\varphi_i$  is constant 0 for all  $j\notin \operatorname {anc}(i)$ . Now we are interested in  $\partial_j\varphi_i\forall j\in \operatorname {anc}(i)$ . Take the partial derivative over  $z_{j}$  with  $j\in \operatorname {anc}(i)\setminus \operatorname {pa}(i)$  (non-empty by assumption):
$$
0 = \frac {\sum_ {k \in \overline {{\mathrm {p a}}} (i)} \frac {\partial \tilde {q} _ {i} ^ {t}}{\partial x _ {k}} \left(\varphi_ {i} (\mathbf {z}) \mid \varphi_ {\mathrm {p a} (i)} (\mathbf {z})\right) \frac {\partial \varphi_ {k}}{\partial z _ {j}} (\mathbf {z})}{\tilde {q} _ {i} ^ {t} \left(\varphi_ {i} (\mathbf {z}) \mid \varphi_ {\mathrm {p a} (i)} (\mathbf {z})\right)} - \frac {\sum_ {k \in \overline {{\mathrm {p a}}} (i)} \frac {\partial q _ {i}}{\partial x _ {k}} \left(\varphi_ {i} (\mathbf {z}) \mid \varphi_ {\mathrm {p a} (i)} (\mathbf {z})\right) \frac {\partial \varphi_ {k}}{\partial z _ {j}} (\mathbf {z})}{q _ {i} \left(\varphi_ {i} (\mathbf {z}) \mid \varphi_ {\mathrm {p a} (i)} (\mathbf {z})\right)} \tag {36}
$$
Recall that we define
$$
g _ {i} ^ {k, t} \left(z _ {i} \mid \mathbf {z} _ {\mathrm {p a} (i)}\right) := \frac {\frac {\partial \widetilde {q} _ {i} ^ {t}}{\partial z _ {k}} \left(z _ {i} \mid \mathbf {z} _ {\mathrm {p a} (i)}\right)}{\widetilde {q} _ {i} ^ {t} \left(z _ {i} \mid \mathbf {z} _ {\mathrm {p a} (i)}\right)}, \quad h _ {i} ^ {k} \left(z _ {i} \mid \mathbf {z} _ {\mathrm {p a} (i)}\right) := \frac {\frac {\partial q _ {i}}{\partial z _ {k}} \left(z _ {i} \mid \mathbf {z} _ {\mathrm {p a} (i)}\right)}{q _ {i} \left(z _ {i} \mid \mathbf {z} _ {\mathrm {p a} (i)}\right)}
$$
So (36) is rewritten as
$$
0 = \sum_ {k \in \overline {{\mathrm {p a}}} (i)} \left[ g _ {i} ^ {k, t} \left(\varphi_ {i} (\mathbf {z}) \mid \varphi_ {\mathrm {p a} (i)} (\mathbf {z})\right) - h _ {i} ^ {k} \left(\varphi_ {i} (\mathbf {z}) \mid \varphi_ {\mathrm {p a} (i)} (\mathbf {z})\right) \right] \partial_ {j} \varphi_ {k} (\mathbf {z})
$$
Choose any  $l \in \mathrm{pa}(i)$  (non-empty by assumption). Take the partial derivative of  $z_{l}$  on two sides:
$$
\begin{array}{l} 0 = \sum_ {k \in \overline {{\mathrm {p a}}} (i)} \left[ \sum_ {m \in \overline {{\mathrm {p a}}} (i)} \partial_ {m} \left(g _ {i} ^ {k, t} - h _ {i} ^ {k}\right) \left(\varphi_ {i} (\mathbf {z}) \mid \varphi_ {\mathrm {p a} (i)} (\mathbf {z})\right) \partial_ {j} \varphi_ {k} (\mathbf {z}) \partial_ {l} \varphi_ {m} (\mathbf {z}) \right] \\ + \left(g _ {i} ^ {k, t} - h _ {i} ^ {k}\right) \left(\varphi_ {i} (\mathbf {z}) \mid \varphi_ {\mathrm {p a} (i)} (\mathbf {z})\right) \partial_ {l} \partial_ {j} \varphi_ {k} (\mathbf {z}) \\ \end{array}
$$
$$
0 = \mathbf {A} _ {i} ^ {t} (\mathbf {z}) \mathbf {a} _ {j, l} ^ {i} (\mathbf {z}) + \mathbf {B} _ {i} ^ {t} (\mathbf {z}) \mathbf {b} _ {j, l} ^ {i} (\mathbf {z}) \tag {37}
$$
where  $\mathbf{a}_{j,l}^{i}(\mathbf{z}) = \left( \begin{array}{c}\partial_{j}\varphi_{p_1}(\mathbf{z})\partial_{l}\varphi_{p_1}(\mathbf{z})\\ \vdots \\ \partial_{j}\varphi_{p_k}(\mathbf{z})\partial_{l}\varphi_{p_m}(\mathbf{z})\\ \vdots \\ \partial_{j}\varphi_{p_{n_i}}(\mathbf{z})\partial_{l}\varphi_{p_{n_i}}(\mathbf{z}) \end{array} \right)\in \mathbb{R}^{n_i^2},$ $\mathbf{b}_{j,l}^{i}(\mathbf{z}) = \left( \begin{array}{c}\partial_{l}\partial_{j}\varphi_{p_1}(\mathbf{z})\\ \vdots \\ \partial_{l}\partial_{j}\varphi_{p_{n_i}}(\mathbf{z}) \end{array} \right)\in \mathbb{R}^{n_i},$
Collect  $n_i$  ( $n_i + 1$ ) equations, every one corresponding to one interventional regime, in the form of (37) for all  $t \in [n_i(n_i + 1)]$ :
$$
\mathbf {0} = \mathbf {M} _ {i} (\mathbf {z}) \left( \begin{array}{l} \mathbf {a} _ {j, l} ^ {i} (\mathbf {z}) \\ \mathbf {b} _ {j, l} ^ {i} (\mathbf {z}) \end{array} \right) \tag {38}
$$
$$
\mathbf {M} _ {i} (\mathbf {z}) := \left( \begin{array}{c c} \mathbf {A} _ {i} ^ {1} (\mathbf {z}) & \mathbf {B} _ {i} ^ {1} (\mathbf {z}) \\ \vdots & \\ \mathbf {A} _ {i} ^ {n _ {i} (n _ {i} + 1)} (\mathbf {z}) & \mathbf {B} _ {i} ^ {n _ {i} (n _ {i} + 1)} (\mathbf {z}) \end{array} \right) \in \mathbb {R} ^ {n _ {i} (n _ {i + 1}) \times n _ {i} (n _ {i + 1})} \tag {39}
$$
By assumption of variability  $V^i$ ,  $\mathbf{M}_i(\mathbf{z})$  is invertible for all  $\mathbf{z} \in \mathbb{R}^d$ . Thus (38) has a unique solution, which is  $\mathbf{a}_{j,l}^i = \mathbf{0}$ ,  $\mathbf{b}_{j,l}^i = \mathbf{0}$ .
$\mathbf{a}_{j,l}^{i}(\mathbf{z}) = \mathbf{0}$  implies  $\forall k, m \in \overline{\mathrm{pa}}(i), \partial_{j}\varphi_{k}(\mathbf{z})\partial_{l}\varphi_{m}(\mathbf{z}) = 0.$
Since  $l \in pa(i)$ ,  $\partial_l\varphi_l(\mathbf{z}) \neq 0$  is in  $\mathbf{a}_{j,l}^i (\mathbf{z})$ , so  $\forall k \in \overline{\mathrm{pa}} (i)$ ,  $\partial_j\varphi_k(\mathbf{z})\partial_l\varphi_l(\mathbf{z}) = 0$ , which implies  $\partial_j\varphi_k(\mathbf{z}) = 0$ . We have proven that for all  $j \in \operatorname {anc}(i)\setminus \operatorname {pa}(i)$ , for all  $k \in \overline{\mathrm{pa}} (i)$ , for all  $\mathbf{z} \in \mathbb{R}^d$ ,  $\partial_j\varphi_k(\mathbf{z}) = 0$ . Namely,  $\varphi_{k}$  only depends on  $\overline{\mathrm{pa}} (i)$ . Combining with the result in Thm. 4.2(i), we obtain the conclusion.
E Known vs. unknown intervention targets
In the main paper, for simplicity, we only provided the minimal set of notation required for describing the problem of CauCA with known intervention targets. However, we believe that a more general version of CauCA should also be considered for cases where the targets are unknown. In fact, in the following, we will distinguish many problem settings, ranging from totally known targets to totally unknown targets. Each setting may be more or less suited to model a collection of datasets, depending on the amount and kind of prior knowledge available.
In the following, we provide a general framework in which CauCA with unknown intervention targets can be rigorously formulated. Note that  $G$  denotes a DAG such that  $V(G) = [d]$ , indexed by natural numbers, which correspond to the indices of targets  $\tau_{i}$  of interventions.  $G$  denotes instead a DAG
equipped only with a partial order induced by the arrows, namely,  $V(\mathsf{G})$  is a set not necessarily indexed by natural numbers. For example, consider  $V(\mathsf{G}) = \{\text{"cloudy", "sprinkle", "raining", "wet grass"}\}$ . In this case, the probability distribution  $\mathbb{P}$  that is Markov relative to this graph is not defined because of the lack of indices:  $\mathbb{P}_1$  might denote the marginal of "cloudy", "sprinkle", "raining" or "wet grass". However,  $\mathbb{P}$  is Markov relative to an indexed DAG of  $\mathsf{G}$ , denoted  $G \models \mathsf{G}$ , which denotes that there exists a bijection  $\sigma$  s.t.  $(u,v) \in E(G)$  iff  $(\sigma(u),\sigma(v)) \in E(\mathsf{G})$ .
Definition E.1. Given two DAGs  $G \models \mathsf{G}$  and  $G' \models \mathsf{G}$ , an isomorphism from  $G$  to  $G'$  is a bijection of  $V(G) = [d]$  such that  $(i, j) \in E(G)$  if and only if  $(\sigma(i), \sigma(j)) \in E(G')$ . An automorphism of  $G$  is an isomorphism  $G \to G$ .
Definition E.2 (Identifiabilities of Causal component analysis, general setting). Given  $\mathsf{G}$  a partially ordered DAG,  $\mathcal{F}$  a class of diffeomorphisms,  $\mathcal{P}_{\mathsf{G}}$  a set of distributions such that for every  $\mathbb{P} \in \mathcal{P}_{\mathsf{G}}$  there exists  $G \models \mathsf{G}$  such that  $\mathbb{P} \in \mathcal{P}_G$ , we define  $(\mathsf{G}, \mathcal{F}, \mathcal{P}_{\mathsf{G}})$  as a class of latent CBN  $(G, \mathbf{f}, (\mathbb{P}^i, \tau_i)_{i \in [0, K]})$  such that  $G \models \mathsf{G}$ ,  $f \in \mathcal{F}$  and  $\mathbb{P} \in \mathcal{P}_G$ .
(i) We define  $\text{CauCA}$  with known intervention targets in  $(G, \mathcal{F}, \mathcal{P}_G)$  as a class of latent CBN models such that all latent CBN models have the same  $G$  and  $(\tau_i)_{i \in [0, K]}$ .
(ii) We define  $\text{CauCA}$  with known intervention targets up to graph automorphisms in  $(G, \mathcal{F}, \mathcal{P}_G)$  as a class of latent CBN models such that for any two latent CBN models  $(G, \mathbf{f}, (\mathbb{P}^i, \tau_i)_{i \in [0, K]})$  and  $(G, \mathbf{f}', (\mathbb{P}^i, \tau_i')_{i \in [0, K]})$  in the class, there exists  $\sigma$  an automorphism of  $G$  such that  $\tau_i' = \sigma(\tau_i)$  for all  $i \in [K]$ .
(iii) We define  $\text{CauCA}$  with matched intervention targets in  $(\mathsf{G},\mathcal{F},\mathcal{P}_{\mathsf{G}})$  as a class of latent CBN models such that for any two latent CBN models  $(G,\mathbf{f},(\mathbb{P}^i,\tau_i)_{i\in [[0,K]]})$  and  $(G',\mathbf{f}',(\mathbb{P}^i,\tau_i')_{i\in [[0,K]]})$  in the class with  $G,G' \models \mathsf{G}$ , there exists a permutation  $\pi \in \mathfrak{S}_d:V(G)\to V(G')$  such that  $\tau_i' = \pi (\tau_i)$  for all  $i\in [K]$ .
(iv) We define CauCA with unknown intervention targets in  $(\mathsf{G},\mathcal{F},\mathcal{P}_{\mathsf{G}})$  as a class of latent CBN models that contains all  $(G,\mathbf{f},(\mathbb{P}^i,\tau_i)_{i\in [[0,K]]})$  such that  $G\models \mathsf{G}$ ,  $f\in \mathcal{F}$  and  $\mathbb{P}\in \mathcal{P}_G$
We say that the CauCA in  $(\mathsf{G},\mathcal{F},\mathcal{P}_{\mathsf{G}})^6$  is identifiable up to  $\mathcal{S}$  if for any  $(G,\mathbf{f},(\mathbb{P}^i,\tau_i)_{i\in [0,K]})$  and  $(G^{\prime},\mathbf{f}^{\prime},(\mathbb{Q}^{i},\tau_{i}^{\prime})_{i\in [0,K]})$  in  $(\mathsf{G},\mathcal{F},\mathcal{P}_{\mathsf{G}})$ , the relation  $\mathbf{f}_*\mathbb{P}^i = \mathbf{f}_*\mathbb{Q}^i\forall i\in [0,K]]$  implies that there is  $\mathbf{h}\in S$  such that  $\mathbf{h} = \mathbf{f}^{\prime -1}\circ \mathbf{f}$  on the support of  $\mathbb{P}$ .
Dataset	τ	τ'(i)	τ'(ii)	τ'(iii)	τ'(iv)
D1	C	C	C	S	S
D2	C	C	C	S	R
D3	S	S	R	W	W
D4	R	R	S	C	W
D5	W	W	W	R	W