Nonparametric Identifiability of Causal Representations from Unknown Interventions

Identifiability

• CRL must identify both latent variables and their graph, while unknown mixing adds ambiguity beyond the Markov equivalence that limits observational structure learning.The paper notes that exogenous variables can also yield a spurious representation.Source: §1 · p. 2

Problem Setting

• The paper studies nonparametric causal mechanisms and mixing, using multiple intervention-generated environments to identify latents up to element-wise nonlinear rescaling and the graph up to isomorphism.The intervention targets are not specified in the problem setting.Source: §1 · p. 2

Contributions

• For two latent variables, an observational environment and one perfect intervention per node suffice under genericity; for arbitrary n, pairs of distinct perfect interventions per node guarantee identifiability.The bivariate condition rules out fine-tuning between intervened and observational mechanisms.Source: §1 · p. 2
• Equivalent solutions preserve the strengths of causal influences among latent variables, supporting causal conclusions from new data.The paper also discusses possible learning objectives and reports that models based on the correct causal structure attain the best fit and identify the ground truth.Source: §1 · p. 2

Related Work on Multi-Environment CRL

• The concurrent work of Buchholz et al. extends the results of Squires et al. from linear mixing to general nonlinear mixings, for linear Gaussian causal models.Source: §1 · p. 3

2.3 Multi-Environment Data

• A single i.i.d. dataset cannot guarantee the target identifiability class, motivating learning from heterogeneous environments linked to interventions.The section frames its task as harder than ordinary observational structure learning and motivates the multi-environment setup.Source: §2 · p. 3, §2.3 · p. 4

2.1 Data Generating Process

• A latent SCM uses causal variables determined by their parents through deterministic functions and mutually independent exogenous variables, with an acyclic graph; observations are generated by mixing the latents.The setup assumes the latent dimension is known and the mixing function is a diffeomorphism onto its image.Source: §2.1 · pp. 3–4
• Faithfulness requires the latent distribution's conditional independences to be exactly those implied by the graph, ruling out cancellations of influences along different paths.The paper describes this as a minimality or genericity assumption.Source: §2.1 · p. 3

2.2 Learning Target: The CRL Identifiability Class ∼CRL

• The learning target is the unmixing function and causal graph, identified up to element-wise diffeomorphisms and graph-isomorphism-preserving permutations.This equivalence captures arbitrary element-wise rescaling and the remaining permutation ambiguity among variables consistent with the causal ordering.Source: §2.2 · p. 4

2.3 Multi-Environment Data

• Across environments, the mixing function and underlying SCM are shared, distribution shifts arise through mechanism interventions, and intervention targets remain unknown.Interventions replace selected structural assignments and their corresponding conditional mechanisms.Source: §2.3 · p. 5
• The main results assume perfect single-node interventions, which remove a variable's parents' influence; imperfect interventions may be insufficient for full identifiability.Perfect interventions replace the affected conditional mechanism with a distribution independent of its parents.Source: §2.3 · p. 5

Identifiability Theory

• Interventional data cannot resolve beyond ∼CRL: equivalent representations can reproduce the same observed distributions under equally sparse interventions.This establishes the irreducible ambiguity of the target equivalence class.Source: §3 · p. 6
• For two latents, an observational environment and one unknown-target perfect intervention per node identify the latents up to ∼CRL and the graph.The result assumes distinct intervened mechanisms and the genericity condition.Source: §3 · p. 6
• The two-variable result requires that base and intervened mechanisms are not fine-tuned to each other, formalized by a φ-based genericity condition.The authors argue this condition should hold for randomly chosen mechanisms; it can fail through fine-tuning.Source: §3 · pp. 6–7
• For arbitrary n, identifiability follows from at least one pair of distinct perfect interventions per node, even when targets are unknown.The pairs concern the same underlying node; the mechanism changes within each pair must satisfy the stated discrepancy condition.Source: §3 · p. 7
• The proof relies on perfect interventions; the cited step does not extend to arbitrary imperfect interventions.With imperfect interventions, the relevant conditional distribution would retain parent dependence.Source: §3 · p. 7
• The arbitrary-variable result uses paired datasets for interventions on the same unknown target, without requiring correspondences between observations or a purely observational environment.The paper distinguishes these datasets from counterfactual pairs of views.Source: §3 · p. 8

4 Interpreting Causal Representations

• Causal influence is invariant to element-wise reparameterization and equivariant to permutations, the two ambiguities allowed by ∼CRL.The paper defines causal influence for an edge using an interventional distribution and KL divergence.Source: §4 · p. 8
• Preserved influence strengths can help uncover changes to latent causal mechanisms across experimental datasets, especially when combined with domain knowledge.The authors present this as a possible scientific use of the inferred representation.Source: §4 · p. 8

5 Learning Objectives

• Both proposed approaches fit multiple environments with a shared mixing function and shared non-intervened mechanisms, preserving intervention sparsity.The theory suggests asymptotic identification up to ∼CRL when the environments are sufficiently diverse and the other assumptions hold.Source: §5 · p. 9
• An autoencoder approach jointly learns an encoder, graph, and intervention targets while reconstructing observations across environments.It can enforce sharing by requiring non-intervened latents to be conditionally independent of the environment given their graph parents.Source: §5 · p. 9
• A generative approach fits base and intervention mechanisms and mixing, then selects the graph and targets with the best multi-environment likelihood.The paper describes this as one possible likelihood-based objective.Source: §5 · p. 9

Setup

• The main experiment uses normalizing flows for a two-latent model with three environments: observational data and one perfect intervention per node.It uses linear Gaussian mechanisms and an MLP mixing function, with observation dimension d = 2 for exact likelihood training.Source: §6 · p. 9

Results

• The correctly specified model usually attains the highest held-out log-likelihood and approximately recovers the ground-truth latents, unlike the misspecified models.The experiment reports these findings for the bivariate setting and treats them as consistent with the identifiability theorem.Source: §6 · p. 9

Additional experiment

• In an additional three-variable experiment with fixed causal order and one intervention environment per node, likelihood-based target selection generally yields approximate latent identification.This tests a setting that violates the paired-intervention assumption of Theorem 3.4.Source: §6 · p. 10

• Interventional data can suffice for causal representation learning without parametric assumptions, supporting expressive methods for high-dimensional experimental data.The paper frames this as a step toward applications where parametric assumptions are undesirable, including complex systems in physics, biology, and medicine.Source: §7 · p. 10
• With soft interventions, identification may be limited to groups of variables and the graph's transitive closure; interventions on untargeted variables may not identify those variables uniquely.A precise characterization of weaker identifiability across intervention types is left for future work.Source: §7 · p. 10
• For arbitrary n, the theorem guarantees identifiability with at least one pair of environments for two distinct perfect interventions on each node.The analysis assumes intervention targets are unknown relative to a pre-imposed causal ordering; the paper contrasts this with the exactly n environments required in the two-variable result.Source: §7 · p. 10
• For n > 2, identifiability from one single-node perfect intervention per node is conjectured under adjusted genericity conditions, but the paper does not claim that condition is necessary.The theorem instead sidesteps the challenge of aligning targets by assuming pairs of environments.Source: §7 · p. 10
• The analysis assumes known n; increasing n to account for unobserved common causes can challenge the invertible-mixing assumption, and unknown n or non-Markovian models are future directions.Estimating intrinsic dimensionality or using Bayesian nonparametrics are suggested as ways to relax the known-n assumption.Source: §7 · p. 10
• The method explored requires searching over graphs and intervention targets, which becomes intractable at moderate n; jointly learning these with mechanisms and the mixing function is challenging.The paper suggests continuous parameterizations of graphs and targets as possible routes toward more practical methods.Source: §7 · p. 10

A Extended Discussion of Related Work

• Some nonlinear causal representation learning approaches use counterfactual views or temporal structure, while others rely on interventions on known targets, observed actions, annotations, or a known causal graph.Counterfactual-pair methods include results for invariant non-descendants under imperfect interventions and latent SCM identification under all single-node perfect stochastic interventions.Source: A Extended Discussion of Related Work · p. 20
• Causal discovery with latent variables from observational i.i.d. data typically requires graph, parametric, or distributional constraints, with several results relying on linear non-Gaussian models and pure observed variables.For discrete variables, a no-twins condition reduces learning the number and cardinality of latents and the graph's Markov-equivalence class to mixture identifiability.Source: A Extended Discussion of Related Work · p. 20
• Other related lines study causal models under coarse-graining or abstraction, or learn invariant representations in supervised settings, often for domain generalization.Source: A Extended Discussion of Related Work · p. 20

• The comparison categorizes prior work by observational, interventional, or counterfactual data and by assumptions on the latent causal model and mixing function.The data categories are color-coded in the table; the caption says all listed works assume invertible or injective mixing and causal sufficiency.Source: §A.1 · p. 20, Table 1 · p. 21
• Observational-only results often require restrictive graphical assumptions, whereas counterfactual data have supported identification with nonparametric causal models and mixing functions.Source: §A.1 · p. 20
• The table is not exhaustive and omits some works that do not fit its categorization or adopted CRL framework, including work using temporal structure or more direct supervision.Source: §A.1 · p. 20

B Proof of Minimality of the CRL Equivalence Class ∼CRL

• CRL-equivalent representations differ by element-wise diffeomorphisms and a graph-isomorphism permutation, and any intervention has an equally sparse counterpart inducing the same observed distribution.Source: B Proof of Minimality of the CRL… · p. 22

C.1 Auxiliary Lemmata

• A measure-preserving map from R × R^n to R, with the same pushforward measure for every fixed second argument, must be constant in that argument.This auxiliary lemma is attributed to Brehmer et al..Source: §C.1 · p. 23
• Conditional independence is preserved when the variables and conditioning vector undergo separate diffeomorphic transformations.The proof uses the change-of-variables formula and the block-diagonal Jacobian to preserve the factorized density.Source: §C.1 · p. 23

C.2 Proof of Thm. 3.2

• The result assumes differentiable, fully supported densities and interventions that change each targeted mechanism everywhere; genericity rules out fine-tuning between base and intervened mechanisms.The setup uses an observational environment and one single-node perfect intervention per node, with unknown targets.Source: §C.2 · p. 24
• The proof’s measure-preservation step relies on perfect interventions; the stated conclusion does not follow for arbitrary imperfect interventions.For imperfect interventions, the relevant expression would involve the intervened conditional mechanism.Source: §C.2 · p. 25
• For an empty ground-truth graph, the inferred graph remains empty, but the proof allows the nontrivial permutation of the two variables.The proof derives independence of the inferred variables before identifying this permutation ambiguity.Source: §C.2 · p. 26
• The proof concludes that the ground truth is identifiable up to the paper’s equivalence class across the considered cases.When intervention targets align, the inferred graph agrees with the ground-truth graph and the latent mapping is element-wise.Source: §C.2 · pp. 24, 25, 29

C.3 Proof of Thm. 3.4

• At least one pair of distinct perfect-intervention environments per node guarantees identifiability up to ∼CRL under the theorem’s assumptions.The targets are unknown; the proof extracts a set with one pair per node and shows the resulting permutation is a graph isomorphism.Source: §C.3 · p. 29

C.4 Proof of Thm. 4.2

• Equivalent representations preserve causal influences for every edge in the ground-truth graph.The theorem considers graph-isomorphic permutations and element-wise diffeomorphisms.Source: §C.4 · p. 31

D.1 Experimental Details for § 6

• Experiments fit separate normalizing-flow models for candidate graphs and intervention targets, selecting by validation log-likelihood and evaluating latent recovery with held-out MCC.Each environment contains 200k data points, and the reported log-likelihood difference compares misspecified models with the correctly specified model.Source: §D.1 · p. 34

D.2 Additional Results: Learning Nonlinear Latent SCMs from Partial Causal Order

• The experiment uses three variables, nonlinear non-additive non-Gaussian latent mechanisms, a fully nonlinear model, and one interventional environment per node despite Theorem 3.4 formally requiring two.It fixes the causal order rather than searching over candidate graphs, fitting the reduced-form SCM with a second normalizing flow.Source: §D.2 · p. 34
• Figure 4 compares latent MCCs and validation log-likelihood differences across target-label permutations, treating the two child-target swap as equivalent to the correct ordering.Each violin plot uses 20 ground-truth data-generating processes; horizontal lines mark minimum, median, and maximum.Source: Figure 4 · p. 35
• A triangular-Jacobian flow encodes causal structure through topological ordering, so candidate models vary intervention-target permutations rather than graph configurations.The flow weights implicitly encode connections among endogenous variables, and permutation layers are omitted to preserve topological order.Source: §D.2 · p. 35
• The nonparametric setting shows less pronounced separation between well-specified and misspecified models than the parametric setting in both identifiability scores and model fits.The comparison is reported as an apparent increase in difficulty for the nonparametric setting.Source: §D.2 · p. 36
• Because only causal order is imposed, the learned SCM may use more or fewer edges than the true graph, and parsing its learned relationships remains future work.The experiment does not search over graph configurations.Source: §D.2 · pp. 34, 36

E Discussion of the Role of Our Assumptions

• The assumptions make causal dependence detectable, specify the number of latent variables, and require an invertible observation map for full recovery.The dependence condition rules out cancellation along paths; latent dimensionality may be estimated by suitable methods, and invertibility is also standard in nonlinear ICA.Source: E Discussion of the Role of Our… · p. 36
• Shared mechanisms and mixing let multiple environments provide additional information, while perfect interventions on all nodes and genericity conditions exclude key non-identifiabilities.The support condition eases the proof, and another condition rules out spurious solutions from measure-preserving transformations on intervention-unaffected domains.Source: E Discussion of the Role of Our… · p. 36