SlowGuess/NeurIPS_2023_OCR huggingface dataset:
After all those preparations and presenting the proof sketch, we will finally prove our main theorem in full generality.
Full proof of Theorem 3. As the proof has to consider different cases and is a bit technical we structure it in a series of steps.
Labeling assumptions We can assume without loss of generality (by relabeling variables and interventions) that the variables  $\widetilde{Z}_i$  are ordered such that their natural order  $i = 1,2,\ldots$  is a valid topological order of the underlying DAG and that intervention  $1\leq i\leq n$  targets node  $Z_{i}$ , i.e.,  $\widetilde{t_i} = i$ . We make no assumption on the ordering of the  $Z_{i}$  or targets of  $t_i$  which are not necessarily assumed to be pairwise different. Recall that we defined (and showed existence of)  $h = \widetilde{f}^{-1}f$  in Lemma 2 (including the fact that their latent dimensions agree).
Induction claim We now prove by induction that  $h_k(z)$  is linear for  $k \geq 0$ . The base case  $k = 0$  is trivially true. For the induction step, assume that this is the case for  $j \leq k$ , i.e.,  $h_j(z) = m_j \cdot z$  for some  $m_j \in \mathbb{R}^d$ . We define  $M_k = (m_1, \ldots, m_k)^\top$  so that we can write concisely
$$
\left(h _ {1} (z), \dots , h _ {k} (z)\right) ^ {\top} = M _ {k} z. \tag {58}
$$
Note that since  $h$  is surjective  $M_{k}$  has full rank. To prove the induction step, we need to show that there exists  $m_{k + 1} \in \mathbb{R}^d$  such that  $h_{k + 1}(z) = m_{k + 1} \cdot z$ .
Precision matrix decomposition For the DAG  $\widetilde{G}$  underlying  $\widetilde{Z}_i$ , let  $\mathrm{PA}_i$  denote the indices of the set of parents of  $Z_i$  and let  $\overline{\mathrm{PA}}_i = \{i\} \cup \mathrm{PA}_i$ . Recall the notation  $\widetilde{s}^{(i)} = \widetilde{B}^\top e_i$  (using  $\widetilde{t}_i = i$ ) and  $\widetilde{r}^{(i)} = (\widetilde{B}^{(i)})^\top e_i$ . Now  $\widetilde{s}_j^{(i)} \neq 0$  only holds if  $j \in \overline{\mathrm{PA}}_i$  and we ordered the variables such that  $\overline{\mathrm{PA}}_i \subset [i]$ . In particular,  $\widetilde{s}_j^{(i)} = 0$  and  $\widetilde{r}_j^{(i)} = 0$  for  $j > i$  (interventions do not create new parents). By Lemma 3 we have
$$
\left(\widetilde {\Theta} ^ {(i)} - \widetilde {\Theta} ^ {(0)}\right) = \widetilde {r} ^ {(i)} \otimes \widetilde {r} ^ {(i)} - \widetilde {s} ^ {(i)} \otimes \widetilde {s} ^ {(i)} \tag {59}
$$
$$
\left(\widetilde {\Theta} ^ {(k + 1)} - \widetilde {\Theta} ^ {(0)}\right) _ {r s} = \widetilde {r} _ {r} ^ {(k + 1)} \widetilde {r} _ {s} ^ {(k + 1)} - \widetilde {s} _ {r} ^ {(k + 1)} \widetilde {s} _ {s} ^ {(k + 1)} = 0 \tag {60}
$$
if  $r > k + 1$  or  $s > k + 1$ . In particular, there exists a symmetric  $A \in \mathbb{R}^{(k + 1) \times (k + 1)}$  such that
$$
h (z) ^ {\top} \left(\widetilde {\Theta} ^ {(i)} - \widetilde {\Theta} ^ {(0)}\right) h (z) = \sum_ {r, s = 1} ^ {k + 1} h _ {r} (z) A _ {r s} h _ {s} (z). \tag {61}
$$
Now we decompose  $A$  as
$$
A = \left( \begin{array}{l l} A ^ {\prime} & b \\ b ^ {\top} & \delta \end{array} \right) \tag {62}
$$
for some  $A' \in \mathbb{R}^{k \times k}$ ,  $b \in \mathbb{R}^k$  and  $\delta \in \mathbb{R}$ . Using the induction hypothesis we find
$$
\begin{array}{l} h (z) ^ {\top} \left(\tilde {\Theta} ^ {(k + 1)} - \tilde {\Theta} ^ {(0)}\right) h (z) = \sum_ {r, s = 1} ^ {k + 1} h _ {r} (z) A _ {r s} h _ {s} (z) \tag {63} \\ = M _ {k} z \cdot A ^ {\prime} M _ {k} z + 2 h _ {k + 1} (z) b ^ {\top} M _ {k} z + \delta h _ {k + 1} (z) ^ {2}. \\ \end{array}
$$
We use the notation  $v_{:r} \in \mathbb{R}^r$  for the vector  $(v_1, \ldots, v_r)^\top$ . Then we can write
$$
\eta^ {(k + 1)} \widetilde {r} ^ {(k + 1)} \cdot h (z) = \eta^ {(k + 1)} \widetilde {r} _ {: k} ^ {(k + 1)} \cdot M _ {k} z + \eta^ {(k + 1)} \widetilde {r} _ {k + 1} ^ {(k + 1)} h _ {k + 1} (z) \tag {64}
$$
Now we have to consider different cases.
Case  $\delta \neq 0$  We assume first that  $\delta \neq 0$ . In this case we find using the last two displays
$$
\begin{array}{l} h (z) ^ {\top} \left(\tilde {\Theta} ^ {(k + 1)} - \tilde {\Theta} ^ {(0)}\right) h (z) - 2 \eta^ {(k + 1)} \tilde {r} ^ {(k + 1)} h (z) + c ^ {(k + 1)} \\ = z \cdot M _ {k} ^ {\top} A ^ {\prime} M _ {k} z + \delta \left(h _ {k + 1} (z) + \frac {b ^ {\top} M _ {k} z - \eta^ {(k + 1)} \widetilde {r} _ {k + 1} ^ {(k + 1)}}{\delta}\right) ^ {2} - \frac {1}{\delta} z \cdot M _ {k} ^ {\top} b b ^ {\top} M _ {k} z \tag {65} \\ + \frac {2}{\delta} \eta^ {(k + 1)} \widetilde {r} _ {k + 1} ^ {(k + 1)} b ^ {\top} M _ {k} z - \frac {1}{\delta} \left(\eta^ {(k + 1)} \widetilde {r} _ {k + 1} ^ {(k + 1)}\right) ^ {2} - 2 \eta^ {(k + 1)} \widetilde {r} _ {: k} ^ {(k + 1)} \cdot M _ {k} z + c ^ {(k + 1)}. \\ \end{array}
$$
Next we set
$$
g (z) = h _ {k + 1} (z) + \delta^ {- 1} b \cdot M _ {k} z - \delta^ {- 1} \eta^ {(k + 1)} \widetilde {r} _ {k + 1} ^ {(k + 1)}. \tag {66}
$$
Looking carefully at (65) we find that there is a symmetric matrix  $\widetilde{Q}$ , a vector  $\widetilde{w}$ , and a constant  $\widetilde{c}$  such that
$$
h (z) ^ {\top} \left(\tilde {\Theta} ^ {(k + 1)} - \tilde {\Theta} ^ {(0)}\right) h (z) - 2 \eta^ {(k + 1)} \widetilde {r} ^ {(k + 1)} h (z) = \delta g (z) ^ {2} + z \cdot \widetilde {Q} z + \widetilde {w} z + \widetilde {c}. \tag {67}
$$
We claim that  $g$  is continuous and surjective. Continuity follows directly from continuity of  $h$ . To show that  $g$  is surjective we can ignore the constant shift, and we note that  $\delta^{-1}v^{\top}M_{k}z = \delta^{-1}\sum_{r=1}^{k}v_{r}h_{r}(z)$  and thus surjectivity of  $h$  implies that  $g$  is surjective (for every  $c$  pick  $z$  such that  $h_{k+1}(z) = c$  and  $h_{r}(z) = 0$  for  $r \leq k$ ).
Using (19) from Lemma 2 we conclude that
$$
\begin{array}{l} \delta g (z) ^ {2} = z ^ {\top} \left(\Theta^ {(k + 1)} - \Theta^ {(0)}\right) z - 2 \eta^ {(k + 1)} r ^ {(k + 1)} z - z \cdot \widetilde {Q} z - \widetilde {w} z - \widetilde {c} (68) \\ = z \cdot Q z + w \cdot z + \alpha (69) \\ \end{array}
$$
for all  $z$  and some symmetric  $Q \in \mathbb{R}^{d \times d}$ ,  $w \in \mathbb{R}^d$ . Corollary 1 then implies that
$$
h _ {k + 1} (z) + \delta^ {- 1} v M _ {k} z + \delta^ {- 1} \eta^ {(k + 1)} \widetilde {r} _ {k + 1} ^ {(k + 1)} = g (z) = v \cdot z + c \tag {70}
$$
for some  $v \in \mathbb{R}^d$ ,  $c \in \mathbb{R}$ . Thus  $h_{k+1}$  is an affine function, i.e.,  $h_{k+1}(z) = m_{k+1}z + \alpha$ . But
$$
0 = \mathbb {E} \widetilde {Z} _ {k + 1} = \mathbb {E} Z \cdot m _ {k + 1} + \alpha = \alpha \tag {71}
$$
and therefore  $h_{k + 1}$  is linear.
Case  $\delta = 0, b \neq 0$  Now we consider the case that  $0 = \delta$ . Using (63), (64), and (19) we find similarly to above
$$
\begin{array}{l} M _ {k} z \cdot A M _ {k} z + 2 h _ {k + 1} (z) b ^ {\top} M _ {k} z - 2 \eta^ {(k + 1)} \widetilde {r} _ {: k} ^ {(k + 1)} \cdot M _ {k} z - 2 \eta^ {(k + 1)} \widetilde {r} _ {k + 1} ^ {(k + 1)} h _ {k + 1} (z) + \widetilde {c} ^ {(k + 1)} \\ = z ^ {\top} \left(\Theta^ {(k + 1)} - \Theta^ {(0)}\right) z - 2 \eta^ {(k + 1)} r ^ {(k + 1)} z \tag {72} \\ \end{array}
$$
Collecting all terms we find that  $h_{k + 1}$  satisfies a relation of the type
$$
h _ {k + 1} (z) (w \cdot z + \beta) = z \cdot Q z + w ^ {\prime} \cdot z + \alpha \tag {73}
$$
for suitable  $Q, \beta, w, w'$ , and  $\alpha$ . Note in particular that  $w^\top = 2b^\top M_k \neq 0$  because  $M_k$  has full rank. If  $w \neq 0$  then Corollary 2 implies that  $h_k(z)$  is affine and, as argued above, we conclude that  $h_k$  is linear.
Case  $\delta = 0, b = 0$  It remains to address the case  $b = 0$  and  $\delta = 0$ . Going back to the definition of  $b$  and  $\delta$  (see (61) and (62)) we find
$$
\left(b ^ {\top}, \delta\right) ^ {\top} = \widetilde {r} _ {: (k + 1)} ^ {(k + 1)} \widetilde {r} _ {k + 1} ^ {(k + 1)} - \widetilde {s} _ {: (k + 1)} ^ {(k + 1)} \widetilde {s} _ {k + 1} ^ {(k + 1)} \tag {74}
$$
Now  $\delta = 0$  implies  $\widetilde{s}_{k+1}^{(k+1)} = \widetilde{r}_{k+1}^{(k+1)}$  (because they are both positive). But then we conclude  $\widetilde{r}^{(k+1)} = \widetilde{s}^{(k+1)}$  from  $b = 0$  and  $\widetilde{r}_r^{(k+1)} = \widetilde{s}_r^{(k+1)} = 0$  for  $r > k + 1$ . This implies
$$
\widetilde {\Theta} ^ {(k + 1)} - \widetilde {\Theta} ^ {(0)} = \widetilde {r} ^ {(k + 1)} \otimes \widetilde {r} ^ {(k + 1)} - \widetilde {s} ^ {(k + 1)} \otimes \widetilde {s} ^ {(k + 1)} = 0 \tag {75}
$$
That is, this implies that intervention  $k + 1$  is a pure shift intervention, i.e.,  $B = B^{(k + 1)}$  and  $\eta^{(k + 1)} \neq 0$  (otherwise we do not actually intervene). In this case, we conclude from Lemma 2 and Lemma 6 (if  $\Theta^{(k + 1)} \neq \Theta$ )
$$
\eta^ {(k + 1)} \widetilde {r} ^ {(k + 1)} h (z) + c ^ {(k + 1)} = \left(z - \mu^ {\prime (k + 1)}\right) ^ {\top} \left(\Theta^ {(k + 1)} - \Theta\right) \left(z - \mu^ {\prime (k + 1)}\right) + c. \tag {76}
$$
But then  $\eta^{(k + 1)}\widetilde{r}^{(k + 1)}\nabla h(\mu^{\prime (k + 1)}) = 0$ , this implies that  $\nabla h$  is not invertible which is a contradiction to  $h$  being a diffeomorphism. Thus, we conclude that also  $\Theta^{(k + 1)} = \Theta$ , i.e., intervention  $k + 1$  is also a pure shift intervention for the  $Z$  variables. But then we find
$$
\widetilde {r} _ {k + 1} ^ {(k + 1)} h _ {k + 1} (z) + \widetilde {r} _ {: k} ^ {(k + 1)} M _ {k} z = \left(\widetilde {\eta} ^ {(k + 1)}\right) ^ {- 1} \left(\eta^ {(k + 1)} r ^ {(k + 1)} z + c ^ {(k + 1)}\right). \tag {77}
$$
So we finally conclude that also in this case  $h_{k + 1}$  is a linear function, this ends the proof.
B From Linear Identifiability to Full Identifiability
In this section, we provide the proof of Theorem 1 which is a combination of our linear identifiability result Theorem 3 proved in the previous section and the main result of [84] which shows identifiability for linear mixing  $f$ . However, we need to carefully address their normalization choices and in addition consider shifts.
We emphasize that the full identifiability results could also be derived from our proof of Theorem 3 by carefully considering ranks of quadratic forms and showing that we can inductively identify source nodes. We do not add these arguments as our reasoning is already quite complex and because for linear mixing functions this is not substantially different from the original proof in [84].
For the convenience of the reader, we will now restate the main results of [84]. For our work it is convenient to slightly deviate from their conventions, i.e., we define the causal order of a graph by  $i \prec j$  iff  $i \in \operatorname{an}(j)$ , in particular  $i \to j$  implies  $i \prec j$ .
We rephrase their results using our notation. Let  $S(G)$  denote the set of permutations of the vertices of  $G$  that preserve the causal ordering, i.e., all permutations  $\rho$  such that  $\rho(i) < \rho(j)$  if  $i \prec j$ . They consider linear mixing functions as specified next.
Assumption 5. Assume  $f(z) = Lz$  for some matrix  $L \in \mathbb{R}^{d' \times d}$  with trivial kernel and pseudoinverse  $H = L^+$  such that the largest absolute value of each row of  $H$  is one and the leftmost is positive.
Then the following result holds.
Theorem 5 (Linear setting, Theorems 1 and 2 in [84]). Suppose latent variables  $Z^{(i)}$  are generated following Assumptions 2 and 3 where  $\eta^{(i)} = 0$  (i.e., no shift) with DAG  $G$  and all  $A^{(i)}$  are lower triangular. Suppose in addition that Assumption 4 holds and the mixing function satisfies Assumption 5. Then we can identify intervention targets and the partial order  $\prec_{G}$  of the underlying DAG  $G$  up to permutation of the node labels. If we in addition assume that all interventions are perfect the problem is identifiable in the following sense. For every representation  $(\widetilde{B}^{(i)},\widetilde{H})$  such that all  $\widetilde{B}^{(i)}$  are lower triangular and  $\widetilde{L} (\widetilde{Z}^{(i)})\stackrel {\mathcal{D}}{=}X^{(i)}$  where  $\widetilde{L} = \widetilde{H}^{+}$  there is a permutation  $\sigma \in S(G)$  such that
$$
\tilde {H} = P _ {\sigma} ^ {\top} H, \quad \tilde {B} ^ {(i)} = P _ {\sigma} ^ {\top} B ^ {(i)} P _ {\sigma}. \tag {78}
$$
Remark 5. A few remarks are in order.
1. Compared to their statement in Theorem 2 we replaced  $P_{\sigma}$  by  $P_{\sigma}^{\top}$  because we used the transposed convention for the permutation matrices.
2. Note that their result per se doesn't mention identifiability up to scaling but that's because part of their assumptions involve normalizing  $f$ . Since we don't normalize the linear map  $f$ , we add scaling in the theorem statement.
3. For the case of imperfect interventions, their result talks about identifiability of the transitive closure  $\overline{G}$  of the graph  $G$ , but this is the same as identifiability of the partial order  $\prec_G$  that we state here.
4. We assume that interventions are not only acting on the noise but do change the corresponding row of  $A$ . Note that this is equivalent to their genericity assumption which states that
$$
r ^ {(i)} = \left(B ^ {(i)}\right) ^ {\top} e _ {t _ {i}} = \left(\operatorname {I d} - A ^ {(i)}\right) _ {t _ {i}} \cdot \left(D ^ {(i)}\right) _ {t _ {i} t _ {i}} ^ {- \frac {1}{2}} \tag {79}
$$
$$
s ^ {(i)} = \left(B ^ {(0)}\right) ^ {\top} e _ {t _ {i}} = \left(\operatorname {I d} - A ^ {(0)}\right) _ {t _ {i}}. \left(D ^ {(0)}\right) _ {t _ {i} t _ {i}} ^ {- \frac {1}{2}} \tag {80}
$$
are linearly independent. Indeed, this holds iff  $A_{t_i}^{(0)} \neq A_{t_i}^{(i)}$  (by acyclicity  $A_{t_i t_i}^{(i)} = 0$ ).
Let us provide a brief intuition why this result holds. The key ingredient is the same as in our proof of linear identifiability above, namely, Lemma 3 which reads
$$
\Theta^ {(i)} - \Theta^ {(0)} = r ^ {(i)} \otimes r ^ {(i)} - s ^ {(i)} \otimes s ^ {(i)}. \tag {81}
$$
This expression has rank 2 if  $t_i$  is not a source node in the causal graph and the intervention is not a pure noise intervention because  $s^{(i)}$  and  $r^{(i)}$  are linearly independent as explained in the remark above. On the other hand, this expression has rank one for source nodes because  $s^{(i)} \propto e_{t_i}$  in this case and the same is true for  $r^{(i)}$ . This allows us to identify interventions on source nodes. Moreover,  $\Theta^{(i)} - \Theta^{(k)}$  has rank 2 if intervention  $i$  and  $k$  target different source nodes and rank 1 if they target the same source node. Thus we can also identify the set of interventions targeting the same source node. One can complete the proof by an induction argument based on removing source nodes from the graph and studying the effect on the precision matrix of the latent variables.
We are now ready to prove our main theorems.
Proof of Theorem 1. Suppose there are two representations  $((B^{(i)},\eta^{(i)},t_i)_{i\in I},f)$  and  $((\widetilde{B}^{(i)},\widetilde{\eta}^{(i)},\widetilde{t}_i)_{i\in I},\widetilde{f})$  of the distributions  $X^{(i)},i\geq 0$ . As explained in the proof of Theorem 3 we can assume by relabeling the nodes that the matrices  $\widetilde{B}^{(i)}$  are lower triangular and the corresponding DAG  $\widetilde{G}$  has the property that for every edge  $i\rightarrow j$  we have  $i < j$ . We now apply Theorem 3 and find that there is an invertible linear map  $T$  such that  $\widetilde{f} = f\circ T$  and  $T^{-1}Z^{(i)}\stackrel {\mathcal{D}}{=}\widetilde{Z}^{(i)}$ . One minor difference to the setting in [84] is that we in addition consider shift interventions. They are simpler to analyze than a change of variance, but we can avoid additional arguments with the following trick. If  $\Sigma^{(i)} = \Sigma^{(0)}$  for some  $i$  we replace  $Z^{(i)} = N(\mu^{(i)},\Sigma^{(i)})$  by  $Z^{\prime (i)} = N(0,\Sigma^{(i)} + \mu^{(i)}(\mu^{(i)})^{\top})$ , otherwise we set  $Z^{\prime (i)} = N(0,\Sigma^{(i)})$  and similarly for  $\widetilde{Z}$ . Then  $T^{-1}(Z^{(i)})\stackrel {\mathcal{D}}{=}\widetilde{Z}^{(i)}$  implies  $T^{-1}(Z^{\prime (i)})\stackrel {\mathcal{D}}{=}\widetilde{Z}^{\prime (i)}$ . Using (13) and (17) it can be shown that this distribution corresponds to a change of variance of the noise, i.e., a change of  $D_{t_it_i}^{(i)}$ . This can also be seen directly from the relation  $Z^{\prime (i)}\stackrel {\mathcal{D}}{=}(B^{(i)})^{-1}(\epsilon +\eta^{(i)}\epsilon 'e_{t_{(i)}})$ , where  $\epsilon^{\prime}\sim N(0,1)$  and
independent of  $\epsilon$ . We can now apply Theorem 5 to the primed distributions. We first find a unique invertible  $\Lambda \in \mathrm{Diag}(d)$  such that  $\Lambda T$  has largest absolute value entry 1 in every row (and in case of ties the leftmost entry is 1). We can also find a permutation  $\rho \in S_d$  such that  $\bar{B}^{(i)} = P_{\rho}B^{(i)}\Lambda^{-1}P_{\rho}^{\top}$  is lower triangular for all  $i$  (note the distinction between  $\bar{B}^{(i)}$  and  $\widetilde{B}^{(i)}$ ). Indeed, this is possible since the underlying DAG  $G$  is acyclic by assumption, interventions do not add edges, and  $\Lambda$  is diagonal. Then we find
$$
\left(T ^ {- 1} \Lambda^ {- 1} P _ {\rho} ^ {\top}\right) \left(\bar {B} ^ {(i)}\right) ^ {- 1} \epsilon \stackrel {\mathcal {D}} {=} \left(T ^ {- 1} \Lambda^ {- 1} P _ {\rho} ^ {\top}\right) \left(\bar {B} ^ {(i)}\right) ^ {- 1} P _ {\rho} \epsilon \tag {82}
$$
$$
= T ^ {- 1} \left(B ^ {(i)}\right) ^ {- 1} \epsilon \tag {83}
$$
$$
\stackrel {\mathcal {D}} {=} \left(\widetilde {B} ^ {(i)}\right) ^ {- 1} \epsilon . \tag {84}
$$
Here the last step follows from the fact that  $T^{-1}Z^{(i)} \stackrel{\mathcal{D}}{=} \widetilde{Z}^{(i)}$  which implies that the centered distributions agree.
Now we can apply Theorem 5 (with  $B$  replaced by  $\bar{B}$  and identity mixing) and find that for the alternative representation in terms of  $\bar{B}$
$$
P _ {\rho} \Lambda T = P _ {\sigma} ^ {\top} \mathrm {I d}, \quad \bar {B} ^ {(i)} = P _ {\sigma} ^ {\top} \widetilde {B} ^ {(i)} P _ {\sigma} \tag {85}
$$
for some  $\rho \in S(\widetilde{G})$  because by assumption  $\bar{B}^{(i)}$  and  $\widetilde{B}$  are lower triangular. This implies that
$$
T = \Lambda^ {- 1} P _ {\rho} ^ {\top} P _ {\sigma} ^ {\top}, \tag {86}
$$
$$
B ^ {(i)} = P _ {\rho} ^ {\top} \bar {B} ^ {(i)} P _ {\rho} \Lambda = P _ {\rho} ^ {\top} P _ {\sigma} ^ {\top} \widetilde {B} ^ {(i)} P _ {\sigma} P _ {\rho} \Lambda . \tag {87}
$$
To summarize, there is a permutation  $\omega = (\rho \circ \sigma)^{-1}$  (note that  $P_{\sigma}P_{\rho} = P_{\rho \circ \sigma}$ ) and a diagonal matrix  $\Lambda$  such that
$$
B ^ {(i)} = P _ {\omega} \widetilde {B} ^ {(i)} P _ {\omega} ^ {\top} \Lambda , \quad T = \Lambda^ {- 1} P _ {\omega}, \quad Z ^ {(i)} = T \widetilde {Z} ^ {(i)} = \Lambda^ {- 1} P _ {\omega} \widetilde {Z} ^ {(i)}, \tag {88}
$$
We also find the relation  $\omega(t_i) = \widetilde{t}_i$  from here because  $P_{\omega}e_i = e_{\omega^{-1}(i)}$ . Equating the means of  $T^{-1}Z^{(i)}$  and  $\widetilde{Z}^{(i)}$  we find
$$
\widetilde {\eta} ^ {(i)} \left(\widetilde {B} ^ {(i)}\right) ^ {- 1} e _ {\tilde {t} _ {i}} = \eta^ {(i)} T ^ {- 1} \left(B ^ {(i)}\right) ^ {- 1} e _ {t _ {i}}. \tag {89}
$$
Multiplying by  $\widetilde{B}^{(i)}$  we find
$$
\widetilde {\eta} ^ {(i)} e _ {\tilde {t} _ {i}} = \eta^ {(i)} \widetilde {B} ^ {(i)} T ^ {- 1} \left(B ^ {(i)}\right) ^ {- 1} e _ {t _ {i}} = \eta^ {(i)} P _ {\omega} ^ {\top} e _ {t _ {i}} \tag {90}
$$
and we conclude that  $\widetilde{\eta}^{(i)} = \eta^{(i)}$ .
Theorem 2. Suppose we are given the distributions  $X^{(i)}$  generated using a model  $((B^{(i)},\eta^{(i)},t_i)_{i\in \bar{I}},f)$  with causal graph  $G$  such that the Assumptions 1-4 hold and none of the interventions is a pure noise intervention. Then for any other model  $((\widetilde{B}^{(i)},\widetilde{\eta}^{(i)},\widetilde{t}_i)_{i\in \bar{I}},\widetilde{f})$  with causal graph  $\widetilde{G}$  generating the same observations the latent dimension  $d$  agrees and there is a permutation  $\omega \in S_d$  such that  $\widetilde{t_i} = \omega (t_i)$  and  $i\prec_{\widetilde{G}}j$  iff  $\omega (i)\prec_G\omega (j)$ , i.e.,  $\prec_G$  can be identified up to a permutation of the labels.
Proof of Theorem 2. Suppose there are two representations  $((B^{(i)},\eta^{(i)},t_i)_{i\in I},f)$  and  $((\widetilde{B}^{(i)},\widetilde{\eta}^{(i)},\widetilde{t}_i)_{i\in I},\widetilde{f})$  of the observational distribution. As shown in Theorem 3 there is a linear map such that  $T^{-1}Z^{(i)} = \widetilde{Z}^{(i)}$ . Then the same is true for the transformed variables as explained above. Viewing  $\widetilde{Z}^{(i)}$  as new observations obtained through a linear  $T$  mixing we conclude from Theorem 5 that  $\prec_G$  and the intervention targets are identifiable up to relabeling.
C A comparison to related works
In this section, we will discuss the relation of our results to the most relevant prior works and put them into context.
[84] This work considers linear mixing functions  $f$ , and we directly generalize their work to nonlinear  $f$ . In particular, their techniques are linear-algebraic and do not generalize to non-linear  $f$  (see more technical details in proof intuition in Section 4); we handle this using a mix of statistical and geometric techniques. Notably, their finding that the linearly mixed latent variables are recoverable after observing one intervention per latent node is closely related to an earlier result in domain generalization, which showed that a number of environments equal to the number of "non-causal" latent dimensions can ensure recovery of the remaining latents [71], and later work showing that this requirement can be reduced further under linear mixing with additional modeling structure [11, 72]. This connection is not coincidental—that analysis was for invariant prediction with latent variables, which was originally presented as a method of causal inference using distributions resulting from unique interventions [62]. Later this approach was applied for the purpose of robust prediction in nonlinear settings [25], but the identifiability of the latent causal structure under general nonlinear mixing via interventional distributions remained an open question until this work.
[88] This work also considers linear mixing  $f$ , but allows for non-linearity in the latent variables. They use score functions to learn the model and they raise as an open question the setting of non-linear mixing, which we study in this work. While the models are not directly comparable, our model is more akin to real-world settings since it's not likely that high-level latent variables are linearly related to the observational distribution, e.g. pixels in an image are not linearly related to high-level concepts. Moreover, Gaussian priors and deep neural networks are extensively used in practice, and our theory as well as experimental methodology apply to them.
[48] While they do not discuss interventions, this work considers the setting of multi-environment CRL with Gaussian priors and their results can be applied to interventional learning. When applied to interventional data as in our setting (as also shown in [84, Appendix D]), they require additional restrictive assumptions such as  $d = d'$ , a bijective mixing  $f$  and  $e \leq d$  where  $e$  is the number of edges in the underlying causal graph whereas we make no such restrictions on  $d$ ,  $f$  and  $e$  (therefore, the maximum value of  $e$  can be as large as  $\approx d^2 / 2$  in our setting). And when these assumptions are satisfied, they require  $2d$  interventions whereas we show that  $d$  interventions are sufficient as well as necessary.
[2] This work also considers the setup of interventional causal representation learning. However, their setting is different in various ways and we now outline the key differences.