Learning Structural Causal Models from Ordering: Identifiable Flow Models

Kien Do; Minh Khoa Le; Truyen Tran

REVIEW 4 major objections 8 minor 18 references

Learning Structural Causal Models from Ordering: Identifiable Flow Models

T0 review · 4 major / 8 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A family of flow models trained only on observational data and a valid causal ordering recovers the structural causal model, not just the observational distribution, and answers observational, interventional, and counterfactual queries in…

desk verdict Solid flow-based SCM learning with a clean parallel inference trick, but the identifiability claim is narrower than stated because it silently assumes TMI maps, which some of the paper's own benchmarks violate. read the letter →

arxiv 2412.09843 v1 pith:RQUWOQCL submitted 2024-12-13 cs.LG cs.AIstat.ML

Minh Khoa Le , Kien Do , Truyen Tran This is my paper · ORCID

classification cs.LGcs.AIstat.ML

keywords structuralcausalmodelsidentifiabilitynormalizingflowsflowmatchingorderingcounterfactualinferenceinterventionaltriangularmonotonemaps

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that flow models can learn a structural causal model (SCM) from observational data plus a valid causal ordering of the variables, without seeing the full causal graph or using interventional data. The construction routes each variable's exogenous noise through an ordinary differential equation whose solution is a triangular, monotone map from noise to observed value; identifiability theory then says such maps are recoverable up to per-variable invertible transformations. The authors add a masked autoregressive velocity network and an endogenous predictor so that abduction, prediction, interventional sampling, and counterfactual answers all run in $O(n)$ time, where $n$ is the number of integration steps and the cost is independent of the number of causal variables. If correct, this makes deep SCM learning practical for graphs with many nodes while preserving the structure needed for intervention and counterfactual questions.

What carries the argument

The load-bearing object is the triangular monotonically increasing (TMI) map: a function $f(u,t)$ whose $i$-th output depends only on $u_1,\dots,u_i$ and which is monotone increasing in $u_i$. The paper obtains it as the time-$1$ solution of an initial-value problem for each node, with velocity $v_i(z_i^t, u_{<\pi_i}, t)$; continuity in $t$ and Lipschitz continuity in $z_i^t$ (enforced by finite weights and 1-Lipschitz activations, with weight regularization) make the solution a TMI map via the Picard–Lindelöf uniqueness theorem. This one geometric property is what plugs the flow into existing identifiability results for nonlinear independent component analysis: a TMI generator with an independent, fully supported latent distribution is identified up to invertible component-wise transformations. The architecture then makes inference fast: MAVEN (a masked autoregressive velocity network using MADE-type masking plus a shared MLP) encodes lower-ordered variables in parallel for abduction, and an endogenous predictor $EP_\theta$ approximates $x$ from $(z_t,u,t)$ so the reverse ODE can be solved in parallel for prediction. The basic construction is the Causal Flow Model (CFM), with S-CFM and P-CFM as the efficient variants.

What would settle it

Train the proposed flow on a known acyclic SCM with a smooth but non-monotone mechanism (for example, $x_1 = u_1$ and $x_2 = (u_2)^2 + x_1$ with independent standard Gaussian noises), using the true causal ordering, and compare counterfactual values with the ground truth: if the counterfactual errors are no smaller than those of a transport baseline that ignores the true mechanism, the identifiability claim does not extend to non-TMI SCMs.

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Extended reading notes

Core claim

The central claim is that flow-based generative models whose per-variable velocity depends only on lower-ordered variables are identifiable structural causal models when trained on observational data and a valid topological ordering. The paper proves that the map $f(u,t)$ defined by the initial-value problems $dz_i^t = v_i(z_i^t, u_{<\pi_i}, t) dt$, $z_i^0 = u_i$, is a triangular monotonically increasing (TMI) map of the exogenous vector $u$ whenever each velocity is continuous in $t$ and Lipschitz in $z_i^t$; monotonicity follows by contradiction from the Picard–Lindelöf uniqueness theorem, triangularity by construction. Because the exogenous distribution is a fully supported product of Gaussians and the map is TMI, the nonlinear-ICA identifiability theorem yields recovery of the exogenous variables up to invertible, component-wise transformations, and the triangular dependence structure matches the true causal dependency pattern. The authors further show that a masked autoregressive velocity network gives parallel abduction and a learned endogenous predictor makes prediction parallel, so observational, interventional, and counterfactual queries all run in $O(n)$ time, where $n$ is the number of discrete steps and the cost is independent of the graph size $d$.

Load-bearing premise

The result assumes the true structural causal model's map from exogenous to endogenous variables is triangular and monotone increasing in each noise term, possibly after component-wise reparameterization; non-monotone mechanisms are outside the guarantee.

Editorial extensions

If this is right

Only a valid causal ordering, not the full causal graph, is needed to learn an identifiable SCM from observational data; interventions and counterfactuals are then answerable without additional data collection.
The $O(n)$ query complexity (where $n$ is the number of integration steps, independent of the number of variables $d$) means the method scales to graphs with many nodes: the reported experiments show nearly constant inference time from 5 to 50 nodes, versus linear growth for a diffusion baseline.
Because the learned map is triangular and monotone, the method handles both additive and non-additive noise structural equations, and it outperforms the compared normalizing-flow and variational baselines on most of the reported observational, interventional, and counterfactual metrics.
When the full causal graph is known, conditioning the velocity on the actual parents instead of all lower-ordered variables keeps the method consistent with the SCM, so the approach also covers the fully observed graph setting.
On a real two-node fMRI study, the method gives interventional predictions with median absolute error slightly better than the compared causal-flow, CAREFL, additive-noise, and linear baselines.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper, the identifiability guarantee is only as broad as the TMI assumption: for genuinely non-monotone mechanisms (including some non-additive equations used in the paper's own appendix, such as Diamond Nonadditive), the learned monotone transport may realize the observational distribution through a different SCM, so counterfactual estimates could be biased even when observational and
Beyond the paper, since the velocity for node $i$ depends on all lower-ordered variables through the masked encoder, the learned model could be pruned by inspecting which encoded components $c_i$ are actually used; that would turn the identifiable flow into a causal-discovery tool that recovers parent sets from the ordering assumption alone.
Beyond the paper, the $O(n)$ query complexity shifts the practical bottleneck to the ODE solver: for very large graphs the cost of the shared MLP evaluation per step will dominate, so tabulating the trade-off between integration steps and counterfactual accuracy would give practitioners a direct tuning rule.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a solid flow-matching approach to learning SCMs from observational data plus an ordering, with a genuinely useful parallel inference trick. The main thing to know is that the advertised identifiability result is narrower than the paper's language suggests: it holds only when the true SCM's noise-to-data map is triangular and monotone in each noise (TMI), and the paper never states that as an assumption. Several NADD benchmarks in the appendix violate it, e.g., Diamond Nonadditive f4 = (x2+x3+u4/4-7)^2, which is not invertible in u4. So the counterfactual numbers in those rows are not covered by the theory.

What's actually new: applying continuous flow matching to triangular causal maps, with a shared MADE velocity network, plus an endogenous predictor that makes abduction and prediction O(n) in discretization steps and independent of the number of variables. That is a clean architectural contribution. The Theorem 1 proof is correct: the Picard-Lindelöf uniqueness argument gives monotonicity in each noise coordinate. The empirical work is broad (several synthetic graphs, fMRI, Morpho-MNIST) and the speedup plot is convincing.

The soft spots, in order: (1) The TMI assumption is hidden inside 'diffeomorphic structural equations,' which is ambiguous and, under any reasonable reading, not satisfied by some of the paper's own datasets. The KR transport perspective means the learned model is a valid SCM, but it need not be the true one; counterfactuals will be biased whenever the true mechanism is non-monotone. (2) Identifiability is only 'up to component-wise invertible transformations,' which is fine for observational and interventional queries under the graph, but not for exact counterfactual recovery, since the abduction step depends on the specific noise coordinates. The contribution bullet 'We prove the identifiability' should be qualified. (3) P-CFM's endogenous predictor is a learned approximation and has no causal consistency guarantee; the theory covers the exact ODE, not the approximate parallel version. (4) Algorithm 2 looks like it has reversed arguments in the EPθ call; likely a typo, but it should be fixed.

None of these are fatal. The paper does what it claims in the experiments, and the empirical gains over CausalNF are plausible. I'd say this deserves a serious referee. The right outcome is likely 'revise': state the TMI assumption, temper the counterfactual claims for non-TMI regimes, and clarify the parallel variant's guarantees.

Referee Report

4 major / 8 minor

Summary. The paper proposes flow-based generative models (CFM, S-CFM, P-CFM) for learning structural causal models from observational data plus a known causal ordering. The authors prove (Theorem 1) that an ODE-based flow with node-wise velocity fields produces a triangular monotonically increasing (TMI) map from exogenous to endogenous variables, and they invoke the identifiability result of Xi and Bloem-Reddy (2023) to claim that such flow models recover the true SCM up to component-wise invertible transformations of the exogenous noise. They further introduce a masked autoregressive velocity network (MAVEN) and an endogenous predictor to enable parallel abduction and prediction, claiming O(n) inference complexity independent of the number of variables. Experiments on synthetic SCMs (additive and non-additive noise) and a real fMRI dataset compare against CausalNF and VACA, reporting favorable observational, interventional, and counterfactual metrics, and a scalability plot.

Significance. If the identifiability claim is restricted to the TMI class and the complexity and algorithm issues are resolved, the paper makes a useful contribution: it connects ODE-based flow matching to the known identifiability theory for triangular monotone maps, and it proposes a concrete parallel architecture that avoids per-variable sequential inference. The theoretical argument is transparent and checkable, relying on a standard uniqueness theorem and an external identifiability proposition quoted in the appendix. The empirical study is broad, with multiple SCM families and a real fMRI benchmark. However, as written, the paper overstates both the coverage of the identifiability result and the computational complexity, and one of the algorithms has an inconsistent input signature; these issues prevent me from recommending acceptance without revision.

major comments (4)

[Method, 'Identifiable Causal Flow Models'; Appendix 'Causal inference experiments, Datasets'] The identifiability claim (Contribution 1 and the 'Identifiable Causal Flow Models' subsection) applies only when the true SCM's noise-to-data map is triangular and monotone increasing in each exogenous variable. Theorem 1 shows that the proposed flow construction is always TMI, but it does not make the true SCM TMI. The paper's stated assumption, 'Markovian, acyclic with diffeomorphic structural equations,' is not sufficient as written: 'diffeomorphic' is undefined (diffeomorphic in which variables?), and monotonicity of each structural equation in its noise variable is not stated. If the intended assumption is that each f_i(xpa_i,.) is a C^1 diffeomorphism of R, then strict monotonicity follows and the identifiability argument is sound after component-wise sign flips; please state this precisely and prove that the noise-to-data map is TMI under this assumption. This is not a side issue: the appendix's non-additive benchmarks (e.g., Diamond Nonadditive f2 and f4, Y Nonadditive f3 and f4, LargeBD NADD) violate monotonicity and invertibility in u_i, so the reported counterfactual numbers on these datasets are not covered by the theorem. For a TMI method, the learned solution is the Knothe–Rosenblatt transport, which generally corresponds to a different SCM when the true generator is non-TMI; the paper needs either to restrict the empirical claims to the TMI regime or to provide a misspecification argument for the non-TMI cases.
[Method, 'Efficient Causal Flow Models' and 'Do operator'] The claim that abduction and prediction are O(n) 'independent of the number of causal variables' is not supported by the described implementation. Each forward pass of the MAVEN velocity network and of the endogenous predictor processes a d-dimensional input and produces d outputs; with a fixed hidden width, a single forward pass costs O(d) operations, so the total cost of n ODE steps is O(nd) in the standard sequential computation model. The O(n) statement is at best the number of sequential steps, valid only under an unstated assumption of parallel processing with d processors at constant per-node cost. Figure 3 measures wall-clock time for d up to 50, which does not demonstrate an asymptotic independence of d; the figure actually shows some growth for S-CFM. Please revise the complexity claim to 'O(n) sequential function evaluations, each of cost O(d)', or specify and justify a parallel model in which per-step time is O(1).
[Method, 'Efficient Causal Flow Models', Algorithm 2 and Eq. (12)] Algorithm 2 calls EPθ(z1, ztn, tn), with z1 = u and ztn the current latent state, but Eq. (12) and Algorithm 1 train EPθ(z_t, u, t). The argument order is swapped between training and inference. Since EPθ is a MAVEN-style network that is not symmetric in its inputs, following the pseudocode literally would make the P-CFM prediction step use the network in a regime it was not trained for. Please correct the argument order in Algorithm 2 (or in Eq. (12) and Algorithm 1) and state the expected input–output signature of EPθ unambiguously. This is necessary for reproducibility of the parallel prediction method.
[Preliminaries, 'TMI Maps for Identifiable SCMs'; Method, 'Do operator'] The identifiability guarantee is up to invertible component-wise transformations of the exogenous variables: the recovered noises are h_i(u_i) for unknown component-wise bijections h_i, not the original noises. For counterfactual queries on the observed variables X, this ambiguity is harmless if the learned model is exactly a reparameterization of the true SCM, because the counterfactual value of any endogenous variable is invariant under such transformations. The paper, however, never states this argument; it moves from 'identifiable up to component-wise invertible transformations' directly to reporting counterfactual MSE against ground truth. Please add the missing implication (or, if the authors disagree, state precisely which counterfactual quantity is identified and why the reported MSE is meaningful without it).

minor comments (8)

[Introduction] In the contributions list, 'a identiﬁable ﬂow models' should read 'an identifiable flow model'.
[Preliminaries, Eq. (1)] The phrase 'each f_i : R → R' is incorrect: f_i is a function of x_{1:i}. Please write 'f_i : R^i → R' and define monotonicity in the i-th argument precisely.
[Preliminaries, 'Causal consistency'] The Jacobian identities ∇uT(u) ≡ I + Σ A_i and ∇xT^{-1}(x) ≡ I − A are only valid for linear additive-noise SCMs; for nonlinear SCMs the Jacobian is a triangular matrix with the same sparsity pattern but not a constant matrix. Please reformulate to avoid a false identity.
[Method, 'Do operator'] The paper defines P-CFM's intervention as backtracking and states that it 'modifies Pearl's do-operator.' Since the experimental ground truth is presumably generated from the true SCM, please specify whether the ground-truth interventional and counterfactual distributions use Pearl's do or backtracking semantics, and note that for non-monotone mechanisms these can differ.
[Appendix, Algorithm 5] Line 5 of Algorithm 5 says 'x = Predict(ui)' but should read 'x = Predict(u)' with the updated noise vector; the current line appears to be a typo but changes the described procedure.
[Theorem 1, proof] The Picard–Lindelöf uniqueness argument at the crossing time requires global existence of the solution on [0,1]; the stated conditions (continuity in t, Lipschitz in z) do not in general guarantee global existence. Please add a boundedness or linear-growth condition, or state that the flow is only defined for the time horizon on which solutions exist.
[Appendix, 'Data denoising'] The alternative velocity approximation v_t = (z_t − vθ)/t is undefined at t = 0, and the cap t ∈ [5e−2, 1] is a heuristic not mentioned in the main text. Please state that all main results use the velocity-matching objective of Eq. (7).
[Table 3] The VACA entry for LargeBD NADD reports an observational MMD of 150.46×10^-2 with a standard deviation of 1.11, which is anomalously large relative to other entries; please verify this number and its formatting.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the identifiability proof rests on external theorems and standard flow-matching, not on self-citation or fitted quantities renamed as predictions.

full rationale

The derivation chain is self-contained in the relevant sense. The paper constructs a flow via the IVP in Eq. 5, proves in Theorem 1 that this flow is a triangular monotonically increasing map using the external Picard–Lindelöf theorem, and then invokes the external identifiability result of Xi and Bloem-Reddy (Proposition 5.2) to conclude identifiability up to invertible component-wise transformations. None of these load-bearing steps depends on the authors' own prior work; the only overlapping self-citation (Nguyen et al. 2023) appears as general motivation in the introduction and is not used in the proof or method. The flow-matching objective (Eq. 7) is a standard conditional density estimator from paired noise and data samples, and the counterfactual/interventional metrics are evaluated against ground-truth samples from known SCMs, so no fitted parameter is later renamed as a prediction. The main caveat is an assumption gap rather than circularity: the Method section states only that SCMs are 'Markovian, acyclic with diffeomorphic structural equations,' which does not by itself imply the TMI condition required by Theorem 1 and the Xi–Bloem-Reddy proposition; several non-additive benchmark equations in the appendix are non-monotone or non-invertible in the noise variable. This means the theoretical identifiability guarantee may not cover every reported experiment, but that is a correctness/validity concern, not a case where the paper's result is defined into existence or forced by a self-citation chain.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on external identifiability and ODE theorems, a Markovian/acyclic assumption, a known ordering, and an implicit TMI-representability requirement for the true SCM. No free parameters are introduced ad hoc; the model is learned by standard flow matching. No invented entities.

assumptions (6)

domain assumption True SCM is Markovian and acyclic, with a known valid causal ordering pi.
Stated in the Method section: 'we assume that SCMs are Markovian, acyclic with diffeomorphic structural equations' and the paper assumes a valid causal ordering is available.
domain assumption True SCM is representable as a triangular monotonically increasing (TMI) map of exogenous variables, up to component-wise invertible transformations.
The identifiability proof cites Xi and Bloem-Reddy Proposition 5.2 for TMI maps. The paper does not explicitly flag this as an assumption about the true data-generating process; the appendix's non-additive synthetic datasets can violate monotonicity.
domain assumption Velocity network is continuous in t and Lipschitz in z_i^t.
Required for Theorem 1 via Picard-Lindelöf; the authors argue it holds for MLPs with finite weights and 1-Lipschitz activations.
standard math Xi and Bloem-Reddy (2023) Proposition 5.2: TMI nonlinear ICA is identifiable up to component-wise invertible transformations.
External result underpinning the identifiability claim.
standard math Picard-Lindelöf theorem.
Used in Theorem 1 to rule out crossing of solution trajectories.
standard math Flow matching / stochastic interpolant framework yields a valid probability flow ODE.
Background from Lipman et al. 2023 and Liu et al. 2023 for the training objective.

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Pith. "Pith review of Learning Structural Causal Models from Ordering: Identifiable Flow Models." pith.science (2026). https://pith.science/paper/RQUWOQCL

@misc{pith2026241209843,
  author       = {Pith},
  title        = {Pith review of: Learning Structural Causal Models from Ordering: Identifiable Flow Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RQUWOQCL}},
  note         = {Machine review of arXiv:2412.09843}
}

read the original abstract

In this study, we address causal inference when only observational data and a valid causal ordering from the causal graph are available. We introduce a set of flow models that can recover component-wise, invertible transformation of exogenous variables. Our flow-based methods offer flexible model design while maintaining causal consistency regardless of the number of discretization steps. We propose design improvements that enable simultaneous learning of all causal mechanisms and reduce abduction and prediction complexity to linear O(n) relative to the number of layers, independent of the number of causal variables. Empirically, we demonstrate that our method outperforms previous state-of-the-art approaches and delivers consistent performance across a wide range of structural causal models in answering observational, interventional, and counterfactual questions. Additionally, our method achieves a significant reduction in computational time compared to existing diffusion-based techniques, making it practical for large structural causal models.

Figures

Figures reproduced from arXiv: 2412.09843 by the authors.

**Figure 2.** An illustration of the case where ˆf i [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Inference time (in seconds) against number of node [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (10 more)

**Figure 4.** Figure 4: Graph structures of synthetic datasets strong performance, often achieving the best or competitive results in all metrics compared to CausalNF-NSF and CausalNFMAF [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]

**Figure 5.** Figure 5: Pair plot of true (blue) and P-CFM predicted (orang [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]

**Figure 6.** Figure 6: Pair plot of true (blue) and P-CFM predicted (orang [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]

**Figure 7.** Figure 7: Scatter plots of x 3 and x 4 from the Y, Nonadditive dataset, showing the effects of interventions on x 1 , x2 , are compared between the ground-truth and predictions from different methods Visualization In this section, we present a visualization that demonstrates the…

**Figure 8.** Figure 8: Causal graph or Morpho-MNIST, t is Thickness, i is Intensity, s is Angle, y is Digit class, x is Image t=1.8, i=122, s=-0.7 t=3.4, i=231, s=1.0 t=2.2, i=157, s=-0.2 t=2.6, i=176, s=0.6 t=2.1, i=128, s=-0.7 t=2.0, i=119, s=-0.6 t=2.5, i=149, s=-0.1 t=2.7, i=192, s=-0.4 …

**Figure 9.** Figure 9: Random samples from the Morpho-MNIST dataset [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]

**Figure 10.** Figure 10: Reconstructed images generated by the VAE from th [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]

**Figure 11.** Figure 11: Counterfactual when intervening on Thickness [PITH_FULL_IMAGE:figures/full_fig_p020_11.png]

**Figure 12.** Figure 12: Counterfactual when intervening on Intensity [PITH_FULL_IMAGE:figures/full_fig_p021_12.png]

**Figure 13.** Figure 13: Counterfactual when intervening on Angle [PITH_FULL_IMAGE:figures/full_fig_p022_13.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Triangle, Nonlinear f 1 ( u1) = u1 f 2 ( x1, u2) = 2 · ( x1) 2 + u2 f 3 ( x1, x2, u3) = 20 1 + exp ( − (x2)2 + x1 ) + u3

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[2] [2]

Simpson, Nonlinear f 1 ( u1) = u1 f 2 ( x1, u2) = softplus ( 1 − x1) + √ 3 20 · u2 f 3 ( x1, x2, u3) = tanh ( 2 · x2) + 3 2 · x1 − 1 − tanh ( u3) f 4 ( x3, u4) = ( x3 − 4 ) 5 + 3 + 1√ 10 · u4

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[3] [3]

Simpson, Nonlinear2 f 1 ( u1) = t ( u1) f 2 ( x1, u2) = x1 + t ( u2) f 3 ( x1, x2, u3) = 2 · ( x1) 2 + ( x2) 3 + t ( u3) f 4 ( x3, u4) = x3 + t ( u4)

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[4] [4]

Diamond, Nonlinear f 1 ( u1) = u1 f 2 ( x1, u2) = ( x1) 2 + u2 2 f 3 ( x1, x2, u3) = ( x2) 2 − 2 1 + exp (−x1) + u3 2 f 4 ( x2, x3, u4) = x3 |x2 + 2| + x3 + 0.5 + u4 10

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[5] [5]

Diamond, Nonadditive f 1 ( u1) = u1 f 2 ( x1, u2) = √ |x1| · ⏐ ⏐u2 + 0.1 ⏐ ⏐ 2 + ⏐ ⏐x1⏐ ⏐+ u2 5 f 3 ( x1, x2, u3) = 1 1 + |u3 + 0.5| · exp (x1 − x2) f 4 ( x2, x3, u4) = ( x2 + x3 + u4 4 − 7 ) 2 − 20

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[6] [6]

Y , Nonlinear f 1 ( u1) = u1 f 2 ( u2) = u2 f 3 ( x1, x2, u3) = ( − ( x2) 2 + 4 1 + exp (−x1 − x2) + u3 2 ) /1.83 f 4 ( x2, x3, u4) =   20 1 + exp ( (x3)2 2 − x3 ) + u4   /3.26

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[7] [7]

Y , Nonadditive f 1 ( u1) = u1 f 2 ( u2) = u2 f 3 ( x1, x2, u3) = ( x1 − 2 · x2 − 2 ) · ( ⏐ ⏐u3⏐ ⏐+ 0.2 ) f 4 ( x3, u4) = ( cos ( x3) + u4 2 ) 2

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[8] [8]

CoRR, abs/2109.04173

Relating Graph Neural Networks to Structural Causal Models. CoRR, abs/2109.04173. Missing Details from Method section Identiﬁability of TMI map Below is the proposition for the identiﬁability of TMI maps f rom (Xi and Bloem-Reddy 2023), included herein for the purpo se of comprehensiveness. Proposition 1. Let Z = X = Rd. The nonlinear ICA model where F ar...

arXiv 2023