Weakly supervised causal representation learning
Authors: Johann Brehmer
Pim De Haan
Phillip Lippe
Taco Cohen
Venue: NeurIPS 2022 Accept (2022)
Keywords: causal representation learning
causality
disentangled representation learning
causal discovery

Abstract: Learning high-level causal representations together with a causal model from unstructured low-level data such as pixels is impossible from observational data alone. We prove under mild assumptions that this representation is however identifiable in a weakly supervised setting. This involves a dataset with paired samples before and after random, unknown interventions, but no further labels. We then introduce implicit latent causal models, variational autoencoders that represent causal variables and causal structure without having to optimize an explicit discrete graph structure. On simple image data, including a novel dataset of simulated robotic manipulation, we demonstrate that such models can reliably identify the causal structure and disentangle causal variables.

Weakly supervised causal representation learning
Johann Brehmer" Qualcomm Al Research jbrehmer@qti. quw.tiiiuuuccmmmm.mmmm com
Pim de Haan"
Qualcomm AI Research
QUVA Lab, University of Amsterdam pimθqti. qualcomm....... com
Phillip Lippe QUVA Lab, University of Amsterdam p.lippe@uva.nk nnl
Taco Cohen
Qualcomm AI Research tacos@qti. qualcomm........
Abstract
Learning high-level causal representations together with a causal model from unstructured low-level data such pixels is impossible from observational data alone. We prove under mild assumptions that this representation is however identifiable in a weakly supervised setting. This involves a dataset with paired samples before and after random, unknown interventions, but no further labels. We then introduce implicit latent causal models, variational autoencoders that represent causal variables and causal structure without having to optimize an explicit discrete graph structure. On simple image data, including a novel dataset of simulated robotic manipulation, we demonstrate that such models can reliably identify the causal structure and disentangle causal variables.
Introduction
The dynamics of many systems can be described in terms of some high-level variables and causal relations between them. Often, these causal variables are not known but only observed in some unstructured, low-level representation, such as the pix- els camera feed. Learning the causal representations to- gether with the causal structure between them is a challenging problem and may be important for instance for applications in robotics and autonomous driving [1]. Without prior assump- tions on the data-ges gettraainnatingg process or supervision, is impos- sible to uniquely identify the causal variables and their causal structure [2. 3].
Intervention
X
In this work, we show that weak form of supervision is suf- ficient identify both the causal representations and the struc- tural causal model between them. We consider a setting in which we have access to data pairs, representing the system be- fore and after randomly chosen, unknown intervention while preserving the noise. This may approximate the generative pro- cess of data collected from video feed of an external agent demonstrator interacting with a system. Neither labels on the intervention targets active control of the interventions
Figure 1: We learn to represent pix- els z as causal variables z. The bot- tom shows the effect of intervening on one variable. We prove that vari- ables and causal model can be iden- tified from samples (x, i)
"Equal contribution Qualcomm Al Research initiative of Qualcomm Technologies, Inc.
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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necessary for our identifiability theorem, making this setting useful for offline learning, We prove that with this form of weak supervision, and under certain assumptions (including that the interventions are stochastic and perfect and that all interventions occur in the dataset), latent causal models (1 (LCMs)- structural causal models (SCMs) together with a decoder from the causal factors to the data space- are identifiable up to relabelling and elementwise reparameterizations of the causal variables.
We then discuss two practical methods for LCM inference. First, we define explicit latent causal models (ELCMs) as a variational autoencoder (VAE) [4] in which the causal variables are the latent variables and the prior is based on an SCM. While this approach works in simple problems, can be finicky and is difficult to scale. We trace this to a major challenge in causal representation learning, namely that it is daiikkn--nnd-eee problem: it can be difficult to learn the causal variables when the causal graph is not yet learned, and it is difficult to learn the graph without knowing the variables.
To overcome this optimization difficulty, we introduce second model class: implicit latent causal models (ILCCss). These models can represent causal structure and variables without requiring an explicit, discrete graph representation, which makes gradient-basee optimization easier. Nevertheless, these models still contain the causal structure implicitly, and we discuss two algorithms that can extract it after the model is trained. Finally, we demonstrate ILCMs on synthetic datasets, including the new CausalCircuit dataset of a robot arm interacting with a causally connected system of light switches. We show that these models can robustly learn the true causal variables and the causal structure from pixels.
2 Related work
Our work builds on the work of Locatello et al. [5] on disentangled representation learning. The authors introduce similar weakly supervised setting where observations are collected before and after unknown interventions. In contrast to our work, however, they focus on disentangled representations, e. (conditionally) independent factors of variation with a trivial causal graph, which our work subsumes as a special case. Other relevant works on disentangled representation learning and (nonlinear) independent component analysis include Refs. [6-12].
The problem of causal representation learning has been gaining attention lately, see the recent review by Schölkopf al. [1]. Lu et al. [13] learn causal representations by observing similar causal models in different environments. von Kügelgen et al. [14] use the weakly supervised setting to study self- supervised learning, using known but non-trivial causal graph between content and style factors. Lippe et al. [15] learn causal representations from time-series data from labelled interventions, assuming that causal effects are not instantancous but can be temporally resolved. Yang et al. [16] propose to train a VAE with an SCM prior, but require the true causal variables as labels. Other relevant works include Refs. [17-21]. To the best of our knowledge, our work is the first provide identifiability guarantees for arbitrary, unknown causal graphs in this weakly supervised setting.
Identifiability of latent causal models from weak supervision
In this section, we show theoretically that causal variables and causal mechanisms are identifiable from weak supervision. In Sec. 4 we will then demonstrate how we can learn causal models in practice by training a causally structured VAE.
3.1 Setup
We begin by defining latent causal models and the weakly supervised setting. Here, we only provide informal definitions and assume familiarity with common concepts from causality as introduced for instance in Ref. [22]. We provide a complete and precise treatment in Appendix A and discuss limitations of our setup and possible generalizations in Appendix B.
We describe the causal structure between latent variables as Structural Causal Model (SCM). An SCM C describes the relation between causal variables 21,..., Zn with domains 2, and noise variables 61,- -1.-... 6n with domains & along a directed acyelic graph (DAG) G(C). Causal mechanisms f x Iippa 2, + 2, describe how the value of a causal variable is determined from the associated noise variables, as well as the values of its parents in the graph. Finally, an SCM includes a probability measure for the noise variables.
2

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An SCM entails unique solution + Z defined by successively upplying the causal mechanisms. We require the causal mechanisms to be pointwise diffeo- morphic, that is, for any value of the par- ents ápa, we have that fi (: ?pa,) is invert- ible, differentiable, and its inverse is dif- ferentiable." Then a is also diffeomorphic and thus noise variables can be uniquely in- ferred from causal variables. This simpli- fies the weakly supervised distribution, the only stochasticity comes from the noise variables and the intervention. The SCM also entails an observational distribution Pc(2) (Markov with respect the graph of the SCM), which is the pushforward of pe through the solution.
Ixtevenne
Intervene
2
9
g'
X
X (xx,
Figure 2: In LCM M, 2, denotes whether the i-th stone from the front is standing. Intervening on the second variable, 22, leads to ž. The decoder g renders 2, 2 as images 2, Æ. LCM M' has an equivalent representation in which z denotes whether the i-th stone from the back has fallen. In Thm. 1, we prove that if and only if two causal models have the same pixel distribution p(x; .). there exists an LCM isomorphism yo: element reparameterization of the causal variables plus a permu- tation of the ordering that commutes with interventions and causal mechanisms.
perfect, stochastic intervention (1, (fr)eer) modifies an SCM by replacing for subset of the causal variables, called the intervention target set C {1,.... n.,, the causal mechanism with new mech- anism ft E 2,, which does not de- pend on the parents. The intervened SCM has new solution 8 & + 2. We call interventions atomic the number of tar- geted variables is one
We will reason about generative models in a data space X, in which the causal structure is latent. Also including a distribution of interventions (as in Ref. [23]), we define LCMs:
Definition (Latent causal model (LCM)). latent causal model M 9. consists of
an acyclic SCM C, which is faithful (all independencies are encoded in its graph [24]), observation space X,
a decoder -- X that is diffeomorphic onto its image,
set I of interventions on C, and a probability measure over I.
We define two LCMs equivalent if all of their components are equal up to permutation of the causal variables and elementwise diffeomorphic reparameteriizatoon of each variable, see Fig. 2.
Definition (LCM isomorphism (informal)). Let M = (C, C. I, PT) and M' 1C'X,,' X.G.I''' pT)) be two LCMs with identical observation space. LCM isomorphism between them is a graph isomorphism 1b : G(C) + G(C') together with elementwise diffeomorphisms for noise and causal variables that tell how to reparameterize them, such that the structure functions, noise distributions, decoder, intervention set, and intervention distribution of M' are compatible with the corresponding elements of M reparameterized through the graph isomorphism and elementwise difreomorphisms. M and M' are equivalent, M ~ M', if and only if there is an LCM isomorphism between them.
Following Locatello et al. [5], we define a generative process of pre- and post-interventionn data:*
Definition 3 (Weakly supervised generative process). Consider an LCM M where the underlying SCM has continuous noise spaces E,, independent probabilities pe., and admits solution s.
Unddr some mild smoothness assumptions, any SCM can be brought into this form by elementwise redefinitions of the variables, preserving the observational and interventional distributions, but not the weakly supervised/ counterfactual distribution.
'This construction is closely related to twinned SCMs [25, Def. 2.17]. typically used compute counterfac- tual queries p(,,,|z, (2, žT). We instead focus on the joint distribution of pre-intervention and post-interven post-intervention data.
3

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define the weakly supervised generative process of data pairs (a, ~ P²M 3) as follows:
<~ :
s(c),
x g(2),
I~p ViEI, PE. Vi & = c
2 = ()())
ī = g
(1)
3.2 Identifiability result
The main theoretical result of this paper is that an LCM M can be identified from p(z, 2)) up to a
relabeling and elementwise transformations of the causal variables:
Theorem (Identifiability R-valued LCMs from weak supervision). Let M = &,,
and M' (C', 1/ be LCMs with the following properties:
The LCMs have an identical observation space X.
The SCMs C and C' both consist of n real-valued endogeneous causal variables and corresponding exogenous noise variables, i. e. E, 2; 2 & R.
The intervention sets I and I' consist of all atomic, perfect interventions, I
{0, (zo)...., {2n}} and similar for I',
. The intervention distribution pr and p'z have full support.
Then the following two statements are equivalent:
1. The LCMs entail equal weakly supervised distributions, D (x, = (2,
2. The LCMs equivalent in the sense of Def. 2, M ~ M'.
Let us summarize the key steps of our proof, which we provide in its entirety in Sec. A in the supplementary material. The direction 2 = 1 follows from the definition of equivalence. The direction + 2 is proven constructively along the following steps:
1. We begin by defining a diffeomorphism şp =g' g 2 -
- Z' and note that if 2, 2 ~ p (z, the weakly supervised distribution of causal variables of model C, then *f=), pp() ~ E E²), The distribution over 2, å is mixture, where each intervention target 7 gives mixture component; each component is supported dimensional submanifold. Therefore, there exists bijection between the components vb [n] + [r]] that maps intervention targets / in M to intervention targets I' = (L) in M'. Furthermore, because the joint distribution 2, 3 is preserved by v, first mapping with
different (n + 1)-
then intervening, Z 2 Z' 4 Ż', equals Z 4 7 2 Z'.
2. Because I = { is perfect intervention, for the map 2 E, 2' - X,, is independent of z'. Thus, in both maps, 8 is independent of z. This means that for the path through 2, the intervention sample ž; transformed into E independently of z. For vR-ve variables, this statistical independence implies that the transformation is constant z, and thus ( is constant in for j * i. yo is therefore an elementwise reparametrization. 3. Using this, can show that s a causal graph isomorphism and that is compatible with the causal mechanisms. This proves LCM equivalence M ~ M'.
4 Practical latent causal models
Theorem I means that it is possible to learn causal structure from pixel-level data in the weakly supervised setting. Consider system that is described by unknown true LCM and assume that we have access to data pairs (z, .i) sampled from its probability density. Then we can train another LCM with learnable components by maximum likelihood. Assuming sufficient data and perfect optimization, this model's 's density will match that the ground-truth LCM. Our identifiability result guarantees that the trained LCM then has the same causal variables and causal structure the ground truth, up to relabelling.
In the following, describe two neural LCM implementations that can be trained data.

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Data
Noise encoding
Projection
Reconstruction
Encoder
Decoder
Solution function
Noise prior
Intervention encoder
Intervention target
Intervened causal variable
Intervention
prior
Encoder
Decoder
Data
Noise encoding
Projection
Reconstruction
Figure 3: ILCM architecture. Pre- and post-intervention data (left) are encoded noise encodings and intervention targets, which are then decoded back to the data space. To compute the prior probability density, the noise encodings transformed into causal variables with the neural solution function.
4.1 Explicit latent causal models (ELCMs)
To implement LCMs with neural networks,
the variational autoencoder (VAE) framework [4]. We first consider an approach where the causal variables (z, 8) are the latent variables. Data (ar, .) and latents (z, 2) are linked by stochastic encoder (<2(x)) and decoder p(æ[22; unlike the deterministic decoder from Sec. 3 this allows us to map high-dimensional data spaces (like images) to low-dimensional causal variables. The causal structure is encoded in the prior p(z, 2) and consists of a learnable causal graph and learnable causal mechanisms fi. The first contribution to the prior density the observational probability density p(z), which factorizes according the causal graph into components p (p,.) which are given by fixed base densities and the causal mechanisms. The second contribution is the interventional conditional density p()|z). which is also computable from the graph and causal mechanisms. The model can be trained on the ELBO loss, variational bound on - llo p(z, ž). For more details, see Appendix E.
We call this an explicit latent causal model (ELCM), as the model directly parameterizes all compo- nents of an LCM. In particular, ELCMs contain explicit representation of the causal graph and causal mechanisms. The graph can learned by an exhaustive search over all DAGs or through a differentiable DAG parameterization [26-29] and gradient descent.
In our experiments with ELCMs, which we describe Appendix E, we find that optimally trained ELCMs indeed correctly identify the causal structure and disentangle causal variables on simple datasets. However, jointly learning explicit graph and variable representations presents a challenging optimization problem. In particular, observe that the loss landscape has local minima correspond- ing to wrong graph configurations.
4.2 Implicit latent causal models (ILCMs)
To enable causal representation learning in a more robust, scalable way, we propose a second LCM implementation: Implicit Latent Causal Models (ILCMs). Like ELCMs. ILCMs are also variational autoencoders with causally structured prior. The key difference is that ILCMS represent the causal structure through neural solution functions s(e). Under our assumption of diffeomorphic causal mechanisms, the solution function whhihh-hhhhhhhh maps the vector of noise variables to the causal variables contains the same information as the causal graph and causal mechanisms that we parameterize with neural networks in ELCMs (see Appendix C). However, unlike ELCMs, this parameterization does not require an explicit graph parameterization. practice, II.CMs are thus easier train than EI.CMs.
The latent variables an ILCMs are noise encodings, defined through the inverse solution function as x (=)) and ह (E). The pre-intervention noise encoding is identical to the SCM noise variables. The post-intervention noise encoding ē corresponds to the value of the SCM noise variables that would have generated the post-intervention causal variables under the
Latents
5

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unintervened SCM mechanisms. ILCMs contain stochastic encoder q(e|2z) and decoder p(z|e) that map data (z, i) noise encodings (e, é).
Noise encodings have the convenient property that under an intervention with intervention targets 7, precisely the components 6, change value: & # in Appendix A. This means that from noise encodings é, we can infer interventions easily. We use simple heuristic intervention encoder that assigns higher intervention probability q(i E / [ ()) to component i the more this component of the noise encoding changes under interventions:
E 7 with probability 1. We prove this property
log qi [z h(ue (z); - (e()))
where He (a:) is the mean function of the noise encoder q(elæ:) and h is quadratic function with learnable parameters. Both the equality pattern of under interventions and this heuristic intervention encoder are similar to the ones used for disentangled representation learning in Ref. [5].
Prior Given encoders for noise encodings and intervention targets, let us now write down the prior p(e, é, I), which encodes the structure of the weakly supervised setting. The intervention-targ prior p(T) and the pre-intervention noise distribution p(e) are given by simple base densities, which choose as uniform categorical and standard Gaussian, respectively. The post-intervention noise encodings é follow the conditional probability distribution
az, 8e,
plele 1) II 5(c. e,) II I[ I
II 6(;; e.) VEI
I[
2, = (1
(3)
IEI
In the second equality we have parameterized the conditional density p(e:le) with a conditional normalizing flow consisting of a learnable diffeomorphic transformation é + 2, 5, (,; and base density p on 2, which we choose as standard Gaussian.
How does this prior encode causal structure? We rely on three key properties of SCMs, shown in Appendix A: 1) the noise variables e; are independent of each other; 2) upon intervening on variable i, the post-intervention causal variable 2; are independent of all e,; 3) while for the other variables j * i, the noise encodings are unchanged é, = ej. These three properties are ensured in the ILCM prior in Eq. (3). We show in Appendix C that therefore each ILCM is equivalent to unique ELCM. For each variable i, the ILCM function &; equal to the solution function 8; of the equivalent EL.CM, which maps from noise variables to causal variable
Thus, by learning transform é into ب s(ē;; e) in the ILCM, we learn the solution function of the corresponding ELCM. This implicitly describes both the causal graph and the causal mechanisms fi. We can thus learn causal model without ever explicitly modelling a graph.
The final question how to implement the first terms in Eq. (3), which encode that those noise encodings that are not part of the intervention targets I should not change value under the intervention. We enforce this in the encoder by setting the non- non-intervention components of e and è to the same value [similar 5]. In Appendix C this procedure described in more detail. We will refer this projective noise encoder as g(e, ē|a,, T, l).
Learning Putting everything together, an ILCM consists of an intervention encoder q(]|,r, noise encoder q(e, ē|.,, ã, I). noise decoder p(zle), and transformations solution functions s; (:)e), see Fig. 3. All of these components are implemented with neural networks and learnable, see Appendix C for details. The lower bound on the joint log likelihood pre-intervention and post- intervention data is given by
log logp(r,0) 2 E ste. logp(T) (T) + log p(e) + lole(,
-
log qle, ē|e, C. I) loge((le) + log logp(Ele)
(4)
'The solution function of an ELCM only depends on ancestors in the graph. The learned transformation s; (e<s; ex) of an ILCM should thus also depend only on ancestors of i. As each is constructed to be dificomorphism in its first argument, jointly they have triangular structure and thus diffeomorphism practice, however, the learned solution functions may still depend weakly on non ancestors. Therefore, to ensure that always forms a diffeomorphism, at some point in training. we test functional dependence to infer ancestral dependence, pick a topological ordering of variables conforming to the ancestry, and parameterize the solution functions s; only depend earlier variables in the ordering.
z.

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The model is trained by minimizing the corresponding VAE loss, learning to map low-level data to noise variables (with q) and to map noise variables to causal variables (with s). The expectation over T is computed via summation, but could alternatively be done with sampling.
In practice, we find it beneficial add regularization term to the loss that disincentivizes collapse to lower-dimensional submanifold of the latent space. For each batch of training data, we compute the batch-aggregate intervention posterior (I) bebhchh (g( 2 T)]. To the beta-VAE loss we then add the negative entropy of this distribution, weighted with a hyperparameter.
Downstream tasks us solve various tasks:
Despite the implicit representation of causal structure, argue that ILCMM let
Causal representation learning disentanglementt ILCMs allow us map low-level data to causal variables z by applying the encoder followed by the solution functions s. Intervention inference: It is also straightforward to infer intervention targets from an ob- served pair (z, 2t) of pre-intervention and gost-intervention data, as this just requires evaluat- encoder (x,
ing the intervention-
Causal discovery iiderr/iinnnniiiionnn We propose two methods to infer the causal graphs af- ter training an ILCMM. One is use an off-the-shelf method for causal discovery the learned representations. Since the ILCMM allows us to infer intervention targets, we can use intervention- based algorithms. In this paper, we use ENCO [28], recent differentiable causal discovery method that exploits interventions to obtain acyclic graphs without requir- ing constrained optimization. Alternatives to ENCO include DCDI [27] and GIES [30]. Alternatively, we can analyze the causal structure implicitly represented in the learned solution functions 8g. We propose heuristic algorithm that proceeds in three steps. First, it infers the topological order by sorting variables such that s; only depends on ej if 2; is after 2j in the topological order. It then iteratively rewrites the solution functions such that they only depend on ancestors in the topological order. Finally, it determines which causal ancestors are direct parents by testing the functional dependence of the causal mechanisms.
We describe this algorithm in more detail in Appendix C.
Generation of interventions and counterfactuals: The IICM entails a generative model for pairs of pre- and cost-intervention data. It is straightforward to sample from the joint distribution p(x, F, 1), from the conditional p(a, 7 [)), or from the conditional p(ī (x, I).
Experiments
Finally, we demonstrate latent causal models in practice. Here we focus implicit LCMs; explicit LCMs are demonstrated in similar experiments in Appendix E. We evaluate the causal graphs learned by the ILCM models either with ENCO (ILCM-E) with the heuristic algorithm described above (ILCM--).
Baselines Since are to the best of our knowledge the first to study causal representation learning in this weakly supervised setting, we are not aware of any baseline methods designed for this task. We nevertheless compare IL.CMs to three other methods. First, define disentanglement VAE that models the weakly supervised process, but assumes independent factors of variation rather than a non-trivial causal structure between the variables. This baseline is similar to the method proposed by Ref. [5]. but it differs in some implementation details to be more comparable to our ILCM setup. We infer the causal graph between the learned representations with ENCO (dVAE-E). We also compare to an unstructured 8- VAE that treats & and 7 as i. i. d. and uses a standard Gaussian prior. Finally, for the pixel-level data, we consider a slot attention model [31], which segments the image unsupervisedly into as many objects as there are causal variables. The latent representation associated to each object considered a learned causal variable.
5.1 2D toy experiment
We first demonstrate LCMs in a pedagogical toy experiment with X' 2 = R.. Training data is generated from a nonlinear SCM with the graph 2 - 22 and mapped the data space through randomly initialized normalizing flow.
An ILCM trained in the weakly supervised setting is able to reconstruct the causal factors accurately up
7

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True causal factors
Data
Learned causal factors
2
2
100
20
40
-2
-2-
-2
-2
2
X1
23
Figure 4: 2D toy data with graph zं + 22. The grey grids show the map between true causal factors, data, and latent causal factors learned by the LCM. The mint dots indicate the observational data distribution, the arrows from z to ž show interventions targeting 2 (red) or 2 (blue). The fact that axis-aligned lines in the true latent space are mapped to axis-aligned lines in the learned latent space implies that the disentanglement succeeded.
to elementwise reparameterizations,, shown in Fig. 4. In Tbl. we quantify the quality of the learned representations with the DCI disentanglement score [32]. We find that our LCM is able to disentangle the causal factors almost perfectly, while the baselines, which assume independent factors of variation, fail as expected. Both the ILCM and the dVAE baseline infer the intervention targets with high accuracy. Finally, we test the quality of the learned causal graphs. We infer the implicit graph with ENCO and the heuristic algorithm discussed above. In both cases, the learned causal graph is identical the correct one, whereas the representations found by the dVAEE baseline induce a wrong graph.
5.2
Cldsat3DIdent
We then turn to pixel-level data and more complex causal graphs. We test ILCMs
adaptation of the Cldsat dataset [14), which contains images of three-dimensional objects under variable positions and lighting conditions. We consider three causal variables representing object hue, the spotlight hue, and the position of the spotlight. We construct six versions of this dataset, each with a different causal graph, randomly initialized nonlinear structure functions, and heteroskedastic noise, These are mapped images with a resolution of 64 64, see Fig. 5 for examples.
Before
True intervention
IL.CMs again able to disentangle the causal variables reliably. The results in Tbl. show that the learned rep- resentations are more disentangled than those learned by methods that do not account for causal structure. The LCM as well the dVAE baseline can infer interven- tions with almost perfect accuracy. We demonstrate this in Fig. 5 by comparing true and inferred interventions, see Sec. D.3 of the supplementary material for details. The ILCMs also learn the causal graphs accurately, while the acausal dVAE-E baseline does in most cases not find the correct causal graphs.
ILCM intervention
Figure 5: Causal3Dldent before (top) and after (middle) interventions, and post-intervention samples generated from the ILCM under the intervention inferred from the data (bottom), indicat- ing we correctly learned intervene.
5.3 CausalCireuit
While Causal3Dident provides a good test of the ability to disentangle features that materialize in pixel space in different ways, like through the position of lights and the color of objects, the underlying causal structure we imposed may feel rather ad-hoc. To explore causal representation learning more intuitively causal setting, we introduce new dataset, which we call CausalCireuit.

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2
Interveging on Z1
Figure 6: Varying learned causal factors vs. intervening on them. With trained ILCM, we encode single test image (left column). In the top row, then vary the latent 21 independently, without computing causal effects, and show the corresponding reconstructed images. Only the robot arm position changes, highlighting that we learned disentangled representation. In the bottom instead intervene on 21 and observe the causal effects: the robot arm may activate lights, which in turn can affect other lights in the circuit.
The CausalCircuit system consists of robot arm that can interact with multiple touch-sensitive lights. The lights are connected with a stochastic circuit: light is more likely to be on if its button is pressed or if its parent lights are on. The robot arm itself can be seen as part of the causal system. Concretely, Fig. This system is observed from a fixed- fixed-position camera, and we generate samples in 512 512 resolution with MuloCo [33], Sec. D.4 of the supplementary material for more details.
Robot arm
Blue light
Green light
consider the causal graph shown in
Red light
ILCMs are again able to disentangle the causal variables reliably and better than the acausal baselines, see Tbl. As shown Appendix D, the slot attention model fails because the lights have no limited spatial extent and thus are not well represented by segments the image. Interventions are identified with high accuracy. ILCMs also correctly learn the causal graph shown in Fig. 6, both when extracted with ENCO and with our heuristic algorithm. In Fig. 6 we demonstrate how ILCMs let us infer and manipulate causal factors and reason about interventions.
Figure Causal graph of the CausalCireuit dataset.
By studying variations of this dataset, we tested the limitations of our method. We find that it works reliably only as long as the causal variables are continuous (that is, when we model the lights with continuous intensity).
soon as we consider discrete states, the assumptions of our identifiability theorem are violated and the model has difficulty disentangling these variables.
5.4 Scaling with graph size
Finally, we study how LCMs scale with the size of the causal system. We generate simple synthetic datasets with we generate three datasets, using linear SCMs with random DAGs, in which each edge in a fixed topological order is sampled from Bernoulli dis- tribution with probability 0.5. The causal variables are mapped the data space through a randomly sampled (O(n) rotation.
R For each dimension
DCI disentanglement score
0.8
0.4
We find that ILCMs are able to reliably disentan- gle the causal variables in systems with up to ap- proximately 10 causal variables, see Fig. 8. In this regime, the true causal graphs are also identified with good accuracy, see Sec. D.5 of the supplemen- tary material. larger causal systems, both dis- entanglement and graph accuracy become worse; more work is required to improve the scaling approach to causal representation learning.
ILCM (ours)
B-VAE
dVAE
0.0
10
15
20
Number of causal variables Figure 8: Scaling with graph size. LCMs dis- entangle causal variables robustly in simple sys- tems with up to ~ 10 causal variables.

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6 Discussion
What makes variable causal? One school of thought is that it that causal variables are those aspects of a system that can be intervened upon [34]. Following this logic, we find it in- teresting to ask: can we uniquely determine the causal variables underlying a system just by ob- serving the effect of interventions?
Table 1: Experiment results. We compare our ILCM-E (using ENCO for graph inference) and ILCM-H (with a heuristic for graph inference) to disentanglement VAE (dVAE-E), unstructured 3- VAE, and slot attention baselines. We show the DCI disentanglement score (D), the accuracy of intervention inference (Acc), and structural Ham- ming distance (SHD) between learned and true graph. Best results bold.
In this work we have found a partial answer to this question: we have shown in theory and prac- tice that under certain assumptions, causal vari- ables and their causal structure are identifiable from low-level representations like the pixels of a camera feed if the system is observed be- fore and after random, unlabeled interventions. Our identifiability theorem extends the results by Locatello et al. [5] from independent factors of variation (trivial causal graphs) to arbitrary causal graphs.
Dataset
Method
D
Acc
SHD
2D toy data
ILCM-E (ours)
0.99
0.96
0.00
ILCM-H (ours)
0.99
0.96
0.00
dVAE--
0.35
0.96
1.00
3-VAE
0.52
Causal3Dldent
ILCM-E (ours)
0.99
0.98
0.00
ILCM-H (ours)
0.99
0.98
0.17
dVAE-E
0.82
0.98
1.67
3-VAE
0.66
Latent causal structure can be described in vari- ational autoencoder setup. However, straight- forward, explicit parameterization of the causal structure requires simultaneously learning the variables and the causal graph. We found that leads to challenging optimization problems, es- pecially when scaling larger systems. As more robust alternative, we introduced implicit latent causal models (IICCs),, which parameterize
Slot attention
0.60
CausalCireuit
ILCM-E (ours)
0.97
1.00
0.00
ILCM-H (ours)
0.97
1.00
0.00
dVAE-E
0.34
1.00
5.00
3-VAE
0.39
Slot attention
0.38
causal structure without requiring an explicit graph representation. We also discussed two algorithms for extracting the learned causal mechanisms and graph after training.
In first experiments, we demonstrated that ILCMs let us reliably disentangle causal factors, identify causal graphs, and infer interventions from unstructured pixel data. For these experiments, we introduced the new CausalCireuit dataset, which consists of images of a robot arm interacting with connected switches and lights.
The setting consider is motivated by a potentially useful scenario: learning causal structure from passive observations of an agent (or demonstrator) interacting with a causal system. However, is currently far from practical. Our identifiability result relies on a number of assumptions, including that interventions are stochastic and perfect, that all atomic interventions may be observed, and that the causal variables are real-valued. In addition, realistic temporal sequence data are not likely exactly correspond to a causal system before and after an intervention (while preserving the noise variables); whether our causal abstraction provides useful approximation remains be tested. We discuss these requirements and their potential relaxation in Appendix B. Similarly, our practical implementation has so far been restricted simplified datasets with relatively few, continuous causal variables, and when trying to relax these limitations we saw the model performance decrease quickly. While more work will be required to make latent causal models applicable to real-world settings, we believe that our results demonstrate that causal representation learning is possible without explicit labels.
A Acknowledgeents We want to thank Joey Bose, Thomas Kipf, Dominik Neuenfeld, and Frank Rösler for useful discussions and Gabriele Cesa, Yang Yang, and Yunfan Zhang for helping with our experiments.
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References
[1] Bernhard Schölkopf, Francesco Locatello, Stefan Bauer, Nan Rosemary Ke, Nal Kalchbrenner, Anirudh Goyal, and Yoshua Bengio. Toward causal representation learning, Proceedings of the IEEE, 109(5)6612-66) 634, 2021.
[2] Frederick Eberhardt. Green and grue causal variables. Synthese, 193(4))1029-1045 2016.
[3] Francesco Locatello, Stefan Bauer, Mario Lucic, Gunnar Ractsch, Sylvain Gelly, Bernhard Schölkopf, and Olivier Bachem. Challenging common assumptions in the unsupervised learn- ing of disentangled representations. In Kamalika Chaudhuri and Ruslan Salakhutdinov, editors, Proceedings of the 36th International Conference Machine Learning, volume 97 of Proceed- ings of Machine Learning Research, pages 4114 4124. PMLR, 2019.
[4] Diederik P. Kingma and Max Welling. Auto-Encoding Variational Bayes. In 2nd International Conference on Learning Representations, ICLR 2014, Banff, AB, Canada, April 14-16, 2014, Conference Track Proceedings, 2014.
[5] Francesco Locatello, Ben Poole, Gunnar Rätsch, Bernhard Schölkopf, Olivier Bachem, and Michael Tschannen. Seakky-superrised disentanglement without compromises. In International Conference Machine Learning, pages 6348 6359.. PMLR, 2020.
[6] Aapo Hyvärinen and Erkki Oja. Independent component analysis: algorithms and applications. Neural Networks, 13:411-438 2000.
[7] Rui Shu, Yining Chen, Abhishek Kumar, Stefano Ermon, and Ben Poole. Weakly Supervised Disentanglement with Guarantees. In International Conference on Learning Representations, 2020.
[8] Ilyes Khemakhem, Diederik Kingma, Ricardo Monti, and Aapo Hyvarinen. Variational Autoen- coders and Nonlinear ICA: A Unifying Framework. In Silvia Chiappa and Roberto Calandra, editors, Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, volume 108 of Proceedings of Machine Learning Research, pages 2207-2217 PMLR, 2020.
[9] Hermanni Hälvä, Sylvain Le Corff, Luc Lehéricy, Jonathan So. Yongjie Zhu, Elisabeth Gassiat, and Aapo Hyvarinen. Disentangling Identifiable Features from Noisy Data with Structured Nonlinear ICA. In A. Beygelzimer, Y. Dauphin, P. Liang, and J. Wortman Vaughan, editors, Advances in Neural Information Processing Systems, 2021.
[10] Luigi Gresele, Julius von Kügelgen, Vincent Stimper, Bernhard Schölkopf, and Michel Besserve. Independent mechanism analysis, new concept? In A. Beygelzimer, Y. Dauphin, P. Liang, and
J. Wortman Vaughan, editors, Advances in Neural Information Processing Systems, 2021.
[11] Weiran Yao, Yuewen Sun, Alex Ho, Changyin Sun, and Kun Zhang. Learning Temporally Causal Latent Processes from General Temporal Data. In International Conference on Learning Representations, 2022.
[12] Sebastien Lachapelle, Pau Rodriguez, Rémi Le, Yash Sharma, Katie E Everett, Alexandre Lacoste, and Simon Lacoste-Julaes Disentanglement via Mechanism Sparsity Regularization: A New Principle for Nonlinear ICA. In First Conference on Causal Learning and Reasoning, 2022.
[13] Chaochao Lu, Yuhuai Wu, Jose Miguel Hermández-Lobato, and Bernhard Schölkopf. Nonlinear invariant risk minimization: A causal approach. arXiv preprint arXiv:2 2021.
[14] Julius von Kügelgen, Yash Sharma, Luigi Gresele, Wieland Brendel, Bernhard Schölkopf, Michel Besserve, and Francesco Locatello. Self Self-Supervised Learning with Data Augmentations Provably Isolates Content from Style. In Thirty-Fifth Conference on Neural Information Processing Systems, 2021.
[15] Phillip Lippe, Sara Magliacane, Sindy Löwe, Yuki M. Asano, Taco Cohen, and Efstratios Gavves. CITRIS: Causal Identifiability from Temporal Intervened Sequences. In Proceedings of the 39th International Conference on Machine Learning, ICML, 2022.
[16] Mengyue Yang, Furui Liu, Zhitang Chen, Xinwei Shen, Jianye Hao, and Jun Wang. Causal Disentangled representation learning via neural structural causal models. In Proceedings of the IEEF/CV Conference on Computer Vision and Pattern Recognition, pages 9593-9602, 2021.
[17] Jeffrey Adams, Niels Hansen, and Kun Zhang. Identification of partially observed linear causal models: Graphical conditions for the non-gaussian and heterogencous cases. In
11

---

M. Ranzato, A. Beygelzimer, Y. Dauphin, P.S. Liang, and J. Wortman Vaughan, editors, Advances in Neural Information Processing Systems, volume 34, pages 12822-22833 Cur- ran Associates, Inc., 2021. https: / metpeeeiddnnnnnneni...ps www.ineww....ppppeeeeeeee wtt.c//a/aae//2////eeeeeee c0f (atraaaaa58222222444444444446663
[18] Feng Xie, Ruichu Cai, Biwei Huang, Clark Glymour, Zhifeng Hao, and Kun Zhang. Generalized independent noise condition for estimating latent variable causal graphs. In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and Lin, editors, Advances in Neural Information Processing Systems, volume 33, pages 14891-14902. Curran As- sociates, Inc., 2020. URL pttpe://prooeeddiggs. www.gg....usss... catpsscccpaaaae cc/pppper//02/ffiiie 00w.4/66887000000000000077006600 Paper. pdf.
[19] Bohdan Kivva, Goutham Rajendran, Pradeep Ravikumar, and Bryon Aragam. Learning latent causal graphs via mixture oracles. In M. Ranzato, A. Beygelzimer, Y. Dauphin, P.S. Liang, and J. Wortman Vaughan, editors, Advances in Neurul Information Processing Systems, volume 34, pages 18087-18101. Curran Associates, Inc., 2021. /tLhttts::/prroeedaai http:////prooeddnnnneei..pppppp. neurips. cc/paper/2021/fiili //t/://6881111116555544444444000
Paper pdf.
[20] Krzysztof Chalupka, Pictro Perona, and Frederick Eberhardt. Visual causal feature learning. Uncertainty in Artificial Intelligence, December 2014.
[21] Sander Beckers and Joseph Y Halpern. Abstracting causal models. Proc. Conf. AAAI Artif. Intell., 33:2678-2685, July 2019.
[22] Jonas Peters, Dominik Janzing, and Bernhard Schölkopf. Flements of causal inference: founda- tions and learning algorithms. The MIT Press, 2017.
[23] Paul K Rubenstein, Sehastian Weichwald, Stephan Bongers, Joris M Mooij, Dominik Janzing, Moritz Grosse- and Bernhard Schölkopf. Causal consistency of structural equation models. Uncertainty in Artificial Intelligence, July 2017.
[24] Judea Pearl. Causality models, reasoning, and inference. Cambridge University Press, Cambridge, U.K. New York, 2000. ISBN 978-0521895606.
[25] Stephan Bongers, Patrick Forré, Jonas Peters, and Joris M. Mooij. Foundations of Structural Causal Models with Cycles and Latent Variables. Annals of Statistics, 49(5):2885-2915, 2021. doi: 10.121421-A0520044.
[26] Eric Jang. Shixiang Gu, and Ben Poole. Categorical Reparameterization with Gumbel-Softmax. In Sth International Conference on Leurning Representations, ICLR 2017, Toulon, France, April 24-26, 2017, Conference Track Proceedings, 2017.
[27] Philippe Brouillard, Sébastien Lachapelle, Alexandre Lacoste, Simon Lacoste-Julien, and Alexandre Drouin. Differentiable Causal Discovery from Interventional Data. In Hugo Larochelle, Marc' Aurelio Ranzato, Raia Hadsell, Maria-Florina Balcan, and Hsuan-Tien Lin, editors, Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NeurIPS 2020, December 6-12, 2020, virtual, 2020.
[28] Phillip Lippe. Taco Cohen, and Efstratios Gavves. Efficient Neural Causal Discovery without Acyclicity Constraints. In International Conference on Learning Representations, 2022.
[29] Bertrand Charpentier, Simon Kibler, and Stephan Günnemann. Differentiable DAG sampling.
March 2022.
[30] Alain Hauser and Peter Bühlmann. Characterization and Greedy Learning of Interventional Markov Equivalence Classes of Directed Acyclic Graphs. Journal of Machine Learning Research, 13(1):2409-2466 2012. ISSN 1532-4435.
[31] Francesco Locatello, Dirk Weissenborn, Thomas Unterthiner, Aravindh Mahendran, Georg Heigold, Jakob Uszkoreit, Alexey Dosovitskiy, and Thomas Kipf. Object-centric learning with slot attention. Advances in Neural Information Processing Systems, 53:11525-11523 2020.
[32] Cian Eastwood and Christopher K I Williams. A framework for the quantitative evaluation of disentangled representations. International Conference on Learning Representations, February 2018.
[33] Emanuel Todorov, Tom Erez, and Yuval Tassa. MuJoCo: A physics engine for model-based control. In 20/2 IEEE/RSJ International Conference on Intelligent Robots and Systems, pages 5026-5033, October 2012.
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[34] James Woodward. Making Things Happen: A Theory of Causal Explanation. Oxford University Press, USA, October 2005.
Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? [Yes]
(h) Did you describe the limitations of your work? [Yes] We strive to be transparent
about the limitations of our work. In the theory section we list the assumptions, in the experiment section we discuss observed failures, and in the conclusions and in Appendix B we discuss the requirements of our approach again.
(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Appendix F.
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
2. If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [Yes] We provide list of assumptions
Sec. and Appendix A and further discuss them Appendix B. (b) Did you include complete proofs of all theoretical results? [Yes] Our identifiability theorem is proven in Appendix A,
3. If you ran experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material as a URL)? [No] Not yet, but we aim to publish them as soon we obtain approval to do so.
(b) Did you specify all the training details (e.g.. data splits, hyperparameters, how they were chosen)? [Yes] We provide these details in Appendix D).
(c) Did you report error bars (e.g., with respect the random seed after running experi- ments multiple times)? [No]
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, cloud provider)? [No]
4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
(a) If your work uses existing assets, did you cite the creators? [Yes] We use the
Cldent dataset from von Kügelgen et al. [14].
(b) Did you mention the license of the assets? [No]
(c) Did you include any new assets either in the supplemental material or as a URL? [No] as possible.
We strive publish our CausalCireuit dataset
(d) Did you discuss whether and how consent was obtained from people whose data you're using/curating? [N/A] We only consider synthetie data showing simulated objects. (e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
5. If you used crowdsourcing or conducted research with human subjects...
(a) Did you include the full text of instructions given participants and screenshots, if applicable? [N/A]
(b) Did you describe any potential participant risks, with links Institutional Review Board (IRB) approvals, if applicable? [N/A]
(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]
13


Reviews:
[{"review_id": "xMhNmUv4C4v", "reviewer": "Reviewer_GuNY", "summary": "### Problem setting \nThe paper addresses the causal representation learning task, i.e., inferring high-level causal latent variables from low-level observations. Specifically, it considers a weakly-supervised setting in which pairs $(x,x’)$ of pre- and post-intervention observations are available. This can be seen as a generalization of Locatello et al. [5] to a setting in which the latents are not mutually independent but causally related. The paper formalizes this setting via latent causal models (LCMs) which consist of a structural causal model (SCM) over latent variables $z$ and a decoder $g$ mapping $z$ to observations $x$ (as also considered in previous works).\n\n### Theory\nThe main theoretical contribution (Thm. 1) is an identifiability result stating that, given an infinite number of pairs $(x,x’)$ resulting from all perfect stochastic single-node interventions (those that set a causal variable $z_i$ to a new noise variable, thus completely removing any influence from its causal parents), the true LCM is identified up to a permutation and element-wise invertible reparametrisation of the causal variables. \n\n### Algorithm/Method\nBased on this insight, the paper then proceeds to investigate algorithmic approaches to learning such LCMs from data. Specifically, two VAE-based approaches are investigated which either represent the causal variables and graph explicitly or only implicitly (ELCMs and ILCMs, respectively). ELCMs are only briefly discussed at a high level and are dismissed due optimization challenges. ILCMs, in which the latent space corresponds to the exogenous noise variables in the LCM, are portrayed as a more promising practical approach. ILCMs consist of an intervention encoder, a noise encoder and decoder, and a solution function (a map between exogenous noise and endogenous causal variables) and are trained by maximizing the corresponding ELBO. Two methods are proposed for post-hoc extracting causal structure from a trained ILCM. \n\n### Experiments\nIn experiments on three synthetic and image datasets, ILCMs are compared with other VAE architectures and SlotAttention w.r.t. DCI score, accuracy in inferring intervention targets, and structural Hamming distance and are shown to compare favourably. A new Mujoco-based dataset called CausalCircuit is also introduced in the process.  ", "questions": "### Main questions and comments\nMy main concerns and questions relate (i) to the concept of representing SCMs through the combination of exogenous variables and a solution function (which is also known as the reduced form SCM in the literature), as well as to (ii) the soundness of evaluation based on DCI scores.\n\n(i) To my understanding, this representation of SCMs is not equivalent to the original SCM, but strictly less expressive in that the reduced form SCM cannot model arbitrary interventions to the true causal mechanisms---see, e.g., Sec. 10 of (Schölkopf, B. & von Kügelgen, J. From Statistical to Causal Learning. 2022). Would the authors agree with this statement?\n\nI would like to better understand how the new solution functions $\\tilde{s}_I$ are defined:\n- Does this not require knowledge of the true mechanisms?  \n- How does this relate to the assumption of perfect, stochastic interventions? \n- What aspect of the theory and proposed method would break down, e.g., for hard interventions (which set $z_i$ to a constant) or for soft interventions that preserve some dependence on the parents?\n\nSome more specific questions regarding the solution function(s):\n- Before Defn. 1, the new mechanism $\\tilde{f}_i$ is defined over the same exogenous domain $\\mathcal{E}_i$, but in Defn. 3, $\\tilde{\\epsilon}_i$ seems to be a new noise variable with different domain. Can you clarify?\n- Regarding footnote 3, do you have a formal argument or reference to back up this claim? \n- In Sec. 4.2, paragraph “Latents”, the noise variables are defined as $e=s^{-1}(z)$ and $\\tilde{e}=s^{-1}(\\tilde{z})$ and it is correctly stated that the latter corresponds to noise value that would have generated  $\\tilde{z}$ under the *unintervened* SCM mechanisms. In the next paragraph, it is then stated that only those components of $\\epsilon$ corresponding to intervened nodes change. Where exactly in Appendix A is this proven?  While I found this result surprising at first, after looking at some toy examples, it seems to hold. However, the way in which $\\epsilon_I$ needs to change will depend on the value of the other noise terms $\\epsilon_{-I}$, thus rendering them dependent. Could you comment on the relevance of this? \n\n(ii) Based on my understanding, the DCI score of Eastwood and Williams is tailored to a setting with mutually independent ground truth factors since it is based on predicting different ground truth latents from the learnt representation. It would seem that this is no longer necessarily a sound evaluation when the $z_i$ are dependent.\n- Did you consider this aspect? \n- What was your choice for the feature importance matrix?\n- Do you report only the disentanglement (D) score or the whole DCI score (average of D, completeness C and informativeness I)? Since the abbreviation D was used, this was not completely clear. \n\n### Other more minor comments, questions, and suggestions\n- l.4: “this requires…” makes it sound like a necessary condition, whereas what is shown is that the considered setting is sufficient. Consider rewording, e.g., “this involves…” \n- the restriction to perfect stochastic interventions seems important enough to be mentioned in the abstract and introduction\n- Fig. 1: consider adding an explanation of what orange nodes represent (changed post-intervention values?)\n- Related work: I would consider adding the following causal representation learning references:\n- - Adams, J., Hansen, N., & Zhang, K. Identification of partially observed linear causal models: Graphical conditions for the non-gaussian and heterogeneous cases. Advances in Neural Information Processing Systems 34, 2021.\n- - Xie, F., Cai, R., Huang, B., Glymour, C., Hao, Z., & Zhang, K. Generalized independent noise condition for estimating latent variable causal graphs. Advances in Neural Information Processing Systems 33, 2020.\n- - Kivva, B., Rajendran, G., Ravikumar, P., & Aragam, B. Learning latent causal graphs via mixture oracles. Advances in Neural Information Processing Systems 34, 2021.\n- - Chalupka, K., Perona, P., & Eberhardt, F. Visual causal feature learning. Uncertainty in Artificial Intelligence, 2015.\n- - Beckers, S., & Halpern, J. Y. Abstracting causal models. AAAI Conference on Artificial Intelligence, 2019.\n- l.104: including a intervention distribution in defining SCMs relates to (Rubenstein, P., Weichwald, S., Bongers, S., Mooij, J., Janzing, D., Grosse-Wentrup, M., & Schölkopf, B. Causal Consistency of Structural Equation Models. Uncertainty in Artificial Intelligence, 2017.)\n- Defn. 2: What does “compatible” mean here? What implications does this have for the (obs./int./cf.) distributions implied by the two SCMs, will they match?\n- Thm. 1: consider adding “in the sense of Defn. 2” to statement 2.\n- l. 151: AFAIK, Pearl defines hard interventions as those which set a variable to a constant; in this sense, the considered stochastic interventions are not hard (in general)\n- Sec. 4.1: some more details on ELCMs (e.g., form of the prior $p(z,z’)$) would be helpful\n- Sec. 4.2: the idea of ILCMs where structure is embedded implicitly seems related to: Leeb, F., Lanzillotta, G., Annadani, Y., Besserve, M., Bauer, S., & Schölkopf, B. Structure by architecture: Disentangled representations without regularization. 2020.\n- l. 214: what if the true post-intervention distribution is non-Gaussian? Why is Gaussianity needed here, or could a more flexible density also be used? I understand this is only the base density used to encode a more complex distribution over the noise variable, but this seems somewhat counterintuitive: do we not mostly care about the distribution of causal variables in the end? In this sense, forcing it to be Gaussian seems restrictive. \n- L. 215 ff.: I was not able to really follow this and the next paragraph in satisfying detail; perhaps consider expanding/clarifying.\n- L. 238: since $q(I|x, x’)$ involves $\\mu_e$, does this not depend on the noise encoder? Is $\\mu_e$ shared between both?\n- L. 296: is the graph of dVAE-E not always the empty graph by construction? \n- L. 300: are objects/latents learned by SlotAttention assumed/constructed to be independent? \n- Evaluation: it could be interesting to also evaluate how well different approaches capture the true interventional distribution, e.g., by looking at an average KL between different true and inferred interventional distributions. \t\n- Fig.5: which ground truth variables are intervened upon here? Also, some more details on how the ILCM intervention is generated exactly would be helpful.\n- L.369: any intuition on what the failure mode is here? \n- (L.378) Generally, I think the pre- and post-intervention terminology is slightly misleading: Typically, interventions are associated with layer 2 queries in which the exogenous variables are not shared but randomly resampled. In the considered setting, the exogenous variables are shared which makes it a layer 3 / counterfactual query. This point has been made in a similar multi-view/weakly-supervised setting by von Kügelgen et al. [14, Sec. 3] and I think this could be articulated more clearly throughout. In particular, “pre-“ and “post-“ suggest a temporal succession, which is not the same as the considered *hypothetical* intervention under the same context/background condition/noise values; the analogy to agents observing the effect of actions may thus only hold approximately, as there some noise variables may change (naturally, i.e., not as the result of the intervention).", "limitations": "N/A", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Post-Rebuttal Update\nAfter discussion with the authors, some of my doubts and questions (particularly regarding the use of solution functions) have been resolved and addressed, and I have decided to increase my score as a result. I believe this is a solid paper that would benefit the NeurIPS community and recommend acceptance. \n\nI encourage the authors to take serious the suggestions brought up during the review phase to further improve the paper, particularly its accessibility and presentation, as they already indicated they would.\n____ \n\n*Disclaimer / review context: I have read the main paper carefully, but only skimmed the Appendix. Due to lack of familiarity with category theory, I did not check the proof of the main theorem and can thus not judge its soundness. I am quite familiar with the setting and related literature, but was not able to follow certain aspects in detail---specifically, the exact use of different solution functions---which I hope to be clarified during the discussion period. My score reflects my current impression of the paper, but I remain open to adjusting my score based on the authors' response.* \n\n### Strengths\n- The paper is generally well-written and well-motivated.\n- The problem setting of learning causal representation from weak supervision is novel and highly relevant.\n- The paper makes contributions to both theory (identifiability result) and algorithms (ELCMs and ILCMs) for causal representation learning.\n- The paper is well positioned in the related literature. \n- The paper is honest and transparent in discussing limitations, both regarding theoretical assumptions and implementation.\n- The paper introduces useful concepts such as isomorphisms between LCMs, which are of interest to the field of causal representation learning beyond the present work. \n\n### Weaknesses\n- Due to addressing both theory and algorithmic approaches, many technical details are deferred to the Appendix, which makes it hard to understand the proposed method in sufficient detail based on just the main text. \n- I have some doubts/reservations regarding the soundness of the proposed ILCMs (specifically, regarding the representation of SCMs through the noise variables and solution functions) and the evalution in terms of the DCI score (see below for more details).\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "wevYiyJVQ0L", "reviewer": "Reviewer_HAYV", "summary": "This paper provides identifiability results for causal discovery under a weakly supervised setting where we have access to data generated from perfect atomic interventions. Along with the theoretical results, it also proposes two algorithms for learning, one requiring an explicit DAG search and the other can learn the graph implicitly.\n", "questions": "As it stands, I think it is a very good paper and it should be accepted. A few questions and (minor) criticisms:\n\n- This is the question most directly related to the current paper - I didn't find it clear in the paper as to why ILCM should perform better than ELCM. I can see that ILCM doesn't require explicit graph search which is nice, but can it not also be stuck in local minima just like ELCMs do, if it has found an (implicit) incorrect graph? Perhaps you could clarify this.\n\n- The situation where we have interventional data from perfect and atomic interventions is not very applicable. Although the current work is a good starting point, further investigation into how to relax the interventional data assumption is needed.", "limitations": "The authors should be commended for calling out the limitation of their assumptions. I am quite interested in understanding what happens to the identifiability results if we still have data from heterogeneous regimes, but possibly from imperfect/non-atomic interventions? Can the results be extended? In the paper, the authors give a counter example for when the intervention is imperfect, but what about the case when we don't have atomic interventions?", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "This is a nice result and I enjoyed reading the paper! The strength of the paper are:\n\n- The theoretical result is novel and allows identifiability of causal graphs with interventional data. This is important and future algorithms can be made based on this.\n\n- The ILCM description is nice as it allows learning of the graph without explicit graph searching.\n\n- The experiments are non trivial and shows off the implication of the result.\n\n- The paper is well presented and written clearly. It was an enjoyable read.\n\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "Sk_Erlibqp", "reviewer": "Reviewer_UyEP", "summary": "This paper aims to identify the latent causal models from the observational data, where data can be intervened by some pre-defined distributions. To achieve this goal, the authors firstly demonstrate that if the observation data is generated according to equation 1, then the causal model can be identified. Based on this theory, the authors design two specific models to learn the causal variable and causal structure. The first model is based on VAE and the second one infers the intervention from the noisy variable. In addition, the authors also discuss the potential functions of their proposed models. In the experiments, the authors demonstrate the effectiveness of their model on the tasks of pedagogy, Causal3DIdent and CausalCircuit.", "questions": "See the Strengths And Weaknesses", "limitations": "The authors have discussed the limitations of their model. ", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 1, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "\nIn general, I believe the studied problem is interesting. However, I find the paper quite hard to read. There are so many contents referred to the appendix or other papers. For example, the formal definition of diffeomorphic function, at least, there should be some intuitive explanations. What is faithful in the causal domain? What is p_M^X? Ideally, the reader should clearly understand what the authors would like to deliver. However, I think there are a lot of concepts that should not be regarded as common knowledge (it is not necessary for the reviewers to read the appendix). Because of these unclear notations, I have trouble in reading this paper. \n\nThere should be an explicit algorithm on how to train the ELCM and ILCMs. I cannot capture how to relate the solution of ELCM and ILCMs with the theoretical results in section 3. Whether the results learned from ELCM and ILCMs can lead to a optimal result of theory 1?\n\nI'm wondering how accurate are the inferred interventions, and how the error of the inference influence the following causal prediction? Whether there are some strategies to overcome the inference error?\n\nConsidering that the paper seems to study a very interesting problem, I tend to be slightly positive. However, I can lower my rating if the other reviewers do not support this paper.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "G3svjvZ3WVj", "reviewer": "Reviewer_eczw", "summary": "This paper has contribution in two parts: Identifiability of causal models of high level (latent) variables from low-level data like that of pixels of images based on weak supervision, and two practical algorithms to learn causal models based on the weak supervision. The identifiability result is based on the assumption that pairs of datum where one of them is not intervened upon and the other is, after an atomic hard intervention to one of the high level variables, is available for training. The latent variables need to be continuous for the identifiability result to hold. \nIn addition, there are two algorithms, Explicit Latent Causal Models which are based on explicit modelling of causal graphs as well as Implicit Latent Causal Models which do not model the causal graph directly but rather parameterise the noise encodings. Experiments are performed on very simple datasets to demonstrate the practical algorithms.", "questions": "Here are some of the questions I have for the authors:\n\n- For Causal3Dindent dataset, only 6 graphs are considered, why not more? And are the considered six graphs random? Also please give the confidence intervals.\n\n- I do not get the intuition behind using a beta term for the ELBO. Is it not the case that the goal of the current setup is to get the correct causal graph? Encouraging disentanglement through the beta term might be encouraging more independent factors, however, that might not be necessarily the case with hierarchical latent spaces with causal graphs.\n\n- Does the ILCM also suffer from optimisation issues in higher dimensions, and if so, what advantages then practically speaking does it offer as against ELCM.\n\n ", "limitations": "One of the main limitations of the proposed approach is that the causal variables are assumed to be causally sufficient (i.e. they are apriori known or given in some way through slots etc and they alone define a causal model). However, this assumption is notoriously hard to be true in any realistic dataset. For any image, the cause of a particular variable could be outside the image itself, and hence hidden. This point should be discussed and highlighted. In my opinion, a true practical algorithm should be able to handle latent variables in causal models as well. Nevertheless, I feel that this aspect should be discussed as it is an important aspect of causal representation learning.", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**:\n\nThe identifiability result follows from the work of Locatello et al. 2020 where the identifiability of disentangled representations are presented. The authors in this paper do a good job of extending it to incorporate the notion of graphs in latent variable models and hence it enables the identifiability of arbitrary causal models from unknown interventions.\n\nIn addition, the idea of implicitly parameterising the causal model and then using the latent variable data to learn a causal model from an off-the-shelf algorithm is very interesting as this is something I believe has not been done before in causal representation learning.\n\n**Weaknesses**:\n\nThe identifiability result is based on some assumptions which are hard to verify. For any given dataset of samples, it is hard to verify, let alone know, if the data samples were produced from a single hard intervention on one of its latent variables or if there was a soft intervention or a multi-target intervention. So in this regard, I am a bit unsure in which real world image data does these assumptions hold. The presented results are on very simple and hand-designed datasets where the assumptions are made to hold. Also see below in limitations regarding other assumptions which are not addressed.\n\nThe main weakness of the paper is that the experiments are very weak, to say the least. The datasets are already simple in the sense that all the assumptions of the identifiability result are respected, and moreover, the number of high level variables on which the causal graph is defined is utmost 4, which is a very small number. For reference, for up to 4 variables, all the DAGs can even be enumerated and is a much simpler task. It is disappointing to see that even the implicit LCM algorithm which does not model a causal graph explicitly is not shown to work on more number of variables (10 or 20 for example). For the ELCM, the authors mention that it takes a performance hit when the number of variables increases due to optimisation issues. If it is the same case with ILCM as well, then I would argue that the presented algorithms are far from being practical. This could be a case of identifiability does not imply learnability, unless the authors convincingly demonstrate in higher dimensions that this works reasonably.\n\nAdding to the above point, I suggest the authors to consider a synthetic dataset (maybe with a random nonlinear SCM plus normalizing flow like in the pedagogical experiment) where they can create data samples such that number of high level variables are 10 (or more). Multiple such random datasets could be produced and trained on with an ILCM. This would at least highlight how the ILCM performs in higher dimensions. Given that ENCO can handle variables upto 1000 variables, I do not see any limitations in performing these experiments.\n\nThe proposed algorithm achieves both high disentanglement scores as well as SHD. But this seems a bit counterintuitive. Disentanglement measures how much each of the factors of variation are disentangled from each other while SHD measures how well the ground truth graph is recovered. If the ground truth graph is not extremely sparse, then both cannot be high. Maybe this is only a behaviour in low dimensions which the authors consider. In addition, it could be that DCI score maybe captures something similar to a structural metric. I personally do not think that it is important to measure disentanglement if the focus is to get the correct graph. If it is still important for some tasks, I encourage the authors to measure other disentanglement metrics as well.\n\nIn summary, I feel that the theoretical contribution is very interesting and can be a stepping stone for identifiability results with more relaxed assumptions. However, given that the experiments are weak and the practical algorithms need more realistic settings, I think that more work needs to be done in this aspect before the paper can be accepted.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept