Causal Domain Clustering Methods

• Causal Domain Clustering is a framework that groups data based on underlying causal mechanisms, enabling the discovery of homogeneous causal domains across environments.
• It leverages joint or alternating optimization of clustering and causal learning using residual discrepancies, latent embeddings, and kernelized causal responses.
• Empirical studies show CDC outperforms traditional clustering by achieving higher accuracy metrics such as ARI and AUC while more effectively recovering causal structures.

Searching arXiv for papers on causal domain clustering and closely related clustered causal inference frameworks. Causal Domain Clustering (CDC) denotes a family of methods in which clustering is driven by causal mechanism, causal structure, or causal response rather than by raw feature-space similarity. In the formulations represented here, the clustered object may be samples from multiple unknown environments, subjects with shared causal dynamics, individuals with similar treatment responsiveness, or recommendation domains with similar transfer effects; the common principle is that the recovered groups are intended to correspond to domains in which the generative causal process is relatively homogeneous (Chen et al., 2021, Liu et al., 29 Jul 2025, Li et al., 4 Sep 2025, Wang et al., 6 Sep 2025, Luo et al., 9 Jul 2025).

1. Definition, scope, and terminological boundaries

Within the CDC literature, the term “domain” usually refers to an environment, subgroup, cluster, or regime that shares a causal mechanism. In the heterogeneous-causality setting of MCVCI and MCVCC, observed data are assumed to arise from multiple environments or mixture components with different causal mechanisms, and clustering is used to reveal that domain structure (Liu et al., 29 Jul 2025). In CCSL, each subject or sample trajectory is generated by a latent cluster $c_s$, and all subjects in the same cluster share the same causal structure $\mathcal G^{(k)}$ (Chen et al., 2021). In HCL, samples are generated by a mixture of $K$ latent SCMs $G_1,G_2,\dots,G_K$ with unknown cluster assignments (Li et al., 4 Sep 2025). In causal subgroup discovery for treatment effects, the “domain” is a treatment-sensitive subpopulation defined by similarity in estimated CATEs rather than by raw covariates (Wang et al., 6 Sep 2025). In multi-domain recommendation, CDC is used in the literal sense of clustering observed recommendation domains according to transfer effects under training (Luo et al., 9 Jul 2025).

This usage is not universal across arXiv. “CDC” can also mean “Domain-Contextualized Concept Graphs,” a knowledge-representation framework in which domain context is explicit in graph relations; that work is not a clustering method (Li et al., 19 Oct 2025). Likewise, Cluster DAGs, C-DAGs, C-DMGs, transit clusters, and clustering for causal data fusion concern clustering variables in causal graphs to support identification, rather than clustering observations, subjects, or application domains (Anand et al., 2022, Ferreira et al., 2 Apr 2025, Tikka et al., 2021, Tabell et al., 21 May 2025, Yvernes et al., 3 Nov 2025). This suggests that “Causal Domain Clustering” is best treated as a methodological umbrella rather than a single standardized formalism.

2. Problem formulations and clustered objects

The central CDC problem is to recover groups that are homogeneous in causal mechanism when the observed data are heterogeneous. The motivating situations are explicitly described as shifts in the distribution of the cause, changes in the functional mechanism, changes in the noise distribution, latent mixture structure, unknown environments, interventions, or multi-domain training interference (Liu et al., 29 Jul 2025, Chen et al., 2021, Luo et al., 9 Jul 2025). Ordinary clustering is regarded as inadequate in these settings because feature similarity or density similarity need not coincide with causal similarity.

A compact view of representative CDC formulations is given below.

Framework	What is clustered	Causal criterion
CCSL (Chen et al., 2021)	subjects or sample trajectories	same causal structure
MCVCC (Liu et al., 29 Jul 2025)	observations	same causal mechanism / environment
HCL (Li et al., 4 Sep 2025)	samples	one latent SCM / DAG per cluster
CKC (Liu et al., 20 Jan 2025)	heterogeneous subgroups	same causal dependency structure
CATE clustering (Wang et al., 6 Sep 2025)	individuals	similar estimated treatment effects
CDC for MDR (Luo et al., 9 Jul 2025)	recommendation domains	beneficial transfer under training

In the bivariate heterogeneous-causality setting, the observed relation is modeled as a mixture of several causal mechanisms: $$Y = \sum_{k=1}^{K} w_k \left(f_k(x_k) + \epsilon_k\right), \qquad x_k \perp\!\!\!\perp \epsilon_k$$ with $K$ mixture components, weights $w_k$, component-specific functions $f_k$, and additive noises $\epsilon_k$ (Liu et al., 29 Jul 2025). In CCSL, the domain label is latent and the assignment is made by posterior fit to a cluster-specific causal model: $$P(c_s=k\mid \mathbf X^s)\propto P(\mathbf X^s\mid c_s=k)\,P(c_s=k)$$ so a subject joins the cluster whose causal model best explains its observed time series (Chen et al., 2021). In heterogeneous-treatment-effect clustering, the causal target is

$\mathcal G^{(k)}$0

and clustering is performed on estimated treatment effects or on a learned similarity derived from them (Wang et al., 6 Sep 2025). In multi-domain recommendation, the domain set $\mathcal G^{(k)}$1 is partitioned into target clusters $\mathcal G^{(k)}$2, while the training source sets $\mathcal G^{(k)}$3 are optimized separately and may satisfy $\mathcal G^{(k)}$4 (Luo et al., 9 Jul 2025).

3. Methodological patterns

A recurring CDC design pattern is joint or alternating optimization of clustering and causal learning. CCSL explicitly rejects the two-stage pipeline of clustering first and causal discovery second, and instead alternates a Causality-related Chinese Restaurant Process with variational-inference-based causal structure learning until convergence (Chen et al., 2021). HCL adopts a bi-directional iterative strategy in which a shared backbone graph is used to compute a causal latent representation $\mathcal G^{(k)}$5, a Bayesian Gaussian mixture model produces soft assignments, cluster-specific DAGs are learned, and clusters are then merged or reorganized according to structural similarity measured by normalized SHD (Li et al., 4 Sep 2025). This suggests that, in much of the CDC literature, domain recovery and mechanism estimation are treated as mutually reinforcing tasks.

A second pattern is the use of residual-like or discrepancy-based causal features rather than raw observations. In MCVCC, after causal direction is selected, the method uses $\mathcal G^{(k)}$6 or $\mathcal G^{(k)}$7 as the causal discrepancy variable $\mathcal G^{(k)}$8, and clustering is performed by

$\mathcal G^{(k)}$9

so clusters are intended to represent different causal mechanisms rather than Euclidean neighborhoods (Liu et al., 29 Jul 2025). HCL similarly defines a structure heterogeneity representation

$K$0

with continuous variables using $K$1 as the reconstruction discrepancy under a shared backbone (Li et al., 4 Sep 2025). CKC transforms each sample into a matrix-valued causal signature $K$2, then compares samples with a Frobenius cosine kernel

$K$3

so the kernel encodes similarity in nonlinear dependence patterns (Liu et al., 20 Jan 2025).

A third pattern is kernelization around causal-response structure. In treatment-effect clustering, causal forests or generalized random forests produce local weights $K$4, from which the learned kernel

$K$5

is constructed; kernelized clustering is then applied to estimated honest CATEs (Wang et al., 6 Sep 2025). In multi-domain recommendation, CDC models transfer with an Isolated Domain Affinity Matrix $K$6 and a Hybrid Domain Affinity Matrix $K$7, then combines them by

$K$8

where the interaction coefficient $K$9 is derived from a causal-distance construction based on treatment effects under random domain sampling (Luo et al., 9 Jul 2025).

4. Identifiability and causal theory

The CDC literature typically justifies clustering through causal asymmetry or causal identifiability rather than through predictive convenience alone. In the Hybrid Additive Noise Model of MCVCI, if the forward relation $G_1,G_2,\dots,G_K$0 satisfies a HANM, then in general the reverse direction $G_1,G_2,\dots,G_K$1 will not also satisfy a HANM except under very restrictive conditions; the paper formalizes this through an ODE-like constraint on $G_1,G_2,\dots,G_K$2 and interprets the reverse HANM as highly non-generic (Liu et al., 29 Jul 2025). The practical implication is that heterogeneous causality remains direction-identifiable when modeled as a mixture of mechanisms rather than a single ANM.

CCSL provides identification results under a linear non-Gaussian structural causal model with instantaneous effects $G_1,G_2,\dots,G_K$3, time-lagged effects $G_1,G_2,\dots,G_K$4, and non-Gaussian independent noise. The paper states that $G_1,G_2,\dots,G_K$5 and $G_1,G_2,\dots,G_K$6 are identifiable as $G_1,G_2,\dots,G_K$7, and that clustering is consistent if the posterior probability of assigning a subject to its true cluster is asymptotically larger than assignment to any other cluster (Chen et al., 2021). HCL sharpens the point that heterogeneity is not identifiable from $G_1,G_2,\dots,G_K$8 alone: it states that distinct heterogeneous SCMs and confounded SCMs can induce the same observational distribution, whereas the proposed latent representation $G_1,G_2,\dots,G_K$9 becomes discriminative, and a shared backbone prior suppresses spurious heterogeneity while accentuating genuine heterogeneity (Li et al., 4 Sep 2025).

CKC supplies a different theoretical route. Its Theorem 4.1 states that

$$Y = \sum_{k=1}^{K} w_k \left(f_k(x_k) + \epsilon_k\right), \qquad x_k \perp\!\!\!\perp \epsilon_k$$0

and Theorem 5.1 presents $$Y = \sum_{k=1}^{K} w_k \left(f_k(x_k) + \epsilon_k\right), \qquad x_k \perp\!\!\!\perp \epsilon_k$$1 as an isomorphic mapping from causal graph space to causal matrix space via sign patterns (Liu et al., 20 Jan 2025). In the treatment-effect setting, the theoretical basis is the standard potential-outcomes identification assumptions of consistency, unconfoundedness, and positivity, combined with orthogonalized estimation via the Robinson decomposition (Wang et al., 6 Sep 2025). A plausible implication is that CDC theory spans several distinct notions of “causal domain”: mechanism invariance, graph invariance, and treatment-response invariance.

5. Representative frameworks and empirical evidence

The empirical literature is heterogeneous but consistently evaluates CDC against non-causal clustering baselines and, where available, against causal baselines. In MCVCC and MCVCI, the simulated datasets SIM, SIM-G, SIM-ln and the real CEP benchmark are used for causal direction identification, with MCVCI reported to achieve the highest accuracy among ANM, PNL, IGCI, LINGAM, RECI, QCCD, CANM, ANM-MM, and others; for clustering, MCVCC is evaluated on synthetic data with multiple functions $$Y = \sum_{k=1}^{K} w_k \left(f_k(x_k) + \epsilon_k\right), \qquad x_k \perp\!\!\!\perp \epsilon_k$$2, $$Y = \sum_{k=1}^{K} w_k \left(f_k(x_k) + \epsilon_k\right), \qquad x_k \perp\!\!\!\perp \epsilon_k$$3, different noise settings, and real BAFU air data, and generally achieves the best ARI/NMI scores against k-means, spectral clustering, GMM, CVAE-km, and ANM-MM (Liu et al., 29 Jul 2025).

CCSL is evaluated on synthetic time-series data generated from Erdős–Rényi graphs with edge parameter $$Y = \sum_{k=1}^{K} w_k \left(f_k(x_k) + \epsilon_k\right), \qquad x_k \perp\!\!\!\perp \epsilon_k$$4, varying the number of variables, sample sizes per subject, number of groups, and number of subjects. The reported findings are that CCSL achieves the best ARI in most settings, the highest AUC often $$Y = \sum_{k=1}^{K} w_k \left(f_k(x_k) + \epsilon_k\right), \qquad x_k \perp\!\!\!\perp \epsilon_k$$5, and convergence in around 18 iterations in the reported example; on resting-state fMRI of 6 brain regions over 84 days it discovers 3 clusters / 3 causal structures, and on Sachs protein-signaling data under two intervention conditions it clusters subjects into 2 groups with ARI $$Y = \sum_{k=1}^{K} w_k \left(f_k(x_k) + \epsilon_k\right), \qquad x_k \perp\!\!\!\perp \epsilon_k$$6, compared with 0.84 for KMeans (DTW), 0.69 for KMeans (Euclidean), 0.03 for DBSCAN (DTW), and 0.34 for OPTICS (DTW) (Chen et al., 2021).

HCL reports superior performance in both clustering and structure learning on mixed observational data. On synthetic data, it outperforms GMM, KMeans, VBGMM, and DP, with ARI around $$Y = \sum_{k=1}^{K} w_k \left(f_k(x_k) + \epsilon_k\right), \qquad x_k \perp\!\!\!\perp \epsilon_k$$7 with only 200 samples per class, improving to $$Y = \sum_{k=1}^{K} w_k \left(f_k(x_k) + \epsilon_k\right), \qquad x_k \perp\!\!\!\perp \epsilon_k$$8 with more data, remaining above $$Y = \sum_{k=1}^{K} w_k \left(f_k(x_k) + \epsilon_k\right), \qquad x_k \perp\!\!\!\perp \epsilon_k$$9 under severe imbalance, and reaching around $K$0 with 7 clusters. On real single-cell protein-signaling perturbation data, it achieves ARI $K$1, compared with $K$2 for DP, and recovers perturbation-specific mechanisms such as MEK suppression under U0126 and PKC downregulation under G06976 with upregulation under PMA (Li et al., 4 Sep 2025).

CKC is evaluated on synthetic heterogeneous subgroups, Indian Ocean Dipole data, and Boston Housing. On six random DAGs with 10 variables, CKC consistently performs best against K-means, polynomial kernel clustering, and RBF kernel clustering; example values include V-measure $K$3 and ARI $K$4 in one linear setting, and V-measure $K$5 and ARI $K$6 in a nonlinear setting. On the IOD benchmark, Table 2 reports TP $K$7, TN $K$8, FP $K$9, FN $w_k$0, Accuracy $w_k$1, Recall $w_k$2, and F1-score $w_k$3 (Liu et al., 20 Jan 2025).

In causal subgroup discovery for CATEs, the IHDP semi-synthetic benchmark, cluster-size simulations, adversarial simulations without true discrete clusters, Synthea EHR, and LaLonde are used. The main reported findings are that Cross Fitted consistently outperformed the feature-space baselines on IHDP, eigengap performed best overall for recovering the true number of clusters, clustering introduced some excess risk when no discrete clusters existed but the penalty shrank with more clusters and higher noise, and on Synthea eigengap selected $w_k$4 with clusters driven by clinically meaningful features while irrelevant covariates had similar distributions across clusters (Wang et al., 6 Sep 2025).

The explicit CDC framework for multi-domain recommendation evaluates Amazon with 25 domains, AliCCP with 50 domains plus 10 merged item categories, industrial datasets MDR-229M and MDR-865M, and an online A/B test. Reported gains include DomainAUC improvements of $w_k$5 on Amazon and $w_k$6 on AliCCP over the best baseline, larger gains for CDC (split) of $w_k$7 on Amazon and $w_k$8 on AliCCP, and on MDR-865M an improvement of $w_k$9 DomainAUC and $f_k$0 TotalAUC over PEPNet. The 14-day online A/B test across 64 domains and 20% user traffic reports a $f_k$1 increase in eCPM, with improvement in 56 of the 64 domains (Luo et al., 9 Jul 2025).

6. Relation to clustered causal abstractions, limitations, and open scope

CDC in the sense of clustering observations or application domains should be distinguished from cluster-level causal abstraction of variables. Cluster DAGs treat groups of variables as macro-nodes while leaving intra-cluster structure unspecified; for these abstractions, d-separation is sound and complete, Pearl’s do-calculus is valid, the ID algorithm is sound and complete, and counterfactual reasoning can be lifted to clusters (Anand et al., 2022). Later work relaxes partition admissibility and allows arbitrary variable clusterings, including cyclic C-DAG representations, with a calculus that is sound and atomically complete with respect to do-calculus (Yvernes et al., 3 Nov 2025). C-DMGs likewise support sound and complete do-calculus for macro causal effects under a cluster-size assumption, with SC-hedges as a sound non-identifiability certificate (Ferreira et al., 2 Apr 2025). Transit clusters provide conditions under which clustering preserves causal identifiability, and these ideas extend to causal data fusion through pruning and clustering as preprocessing operations (Tikka et al., 2021, Tabell et al., 21 May 2025). These papers are closely related to CDC in spirit, but their clustered object is the variable set of a causal graph rather than a collection of samples, subjects, or domains.

Across sample- and domain-level CDC methods, limitations are explicit. The heterogeneous-causality framework of MCVCI and MCVCC is primarily bivariate, depends on hybrid ANM assumptions, requires choosing mixture number $f_k$2 or cluster number $f_k$3, and is designed for observational data with mixture structure rather than general high-dimensional causal graphs (Liu et al., 29 Jul 2025). CCSL is developed under linear non-Gaussian assumptions, focuses on time-series / SVAR-style data with instantaneous and lagged effects, and its performance depends on optimization quality because posterior cluster assignment is derived from an estimated model (Chen et al., 2021). HCL relaxes homogeneity and sufficiency assumptions, but its identifiability results depend on the latent embedding $f_k$4, shared-backbone regularization, and convergence of iterative graph updates (Li et al., 4 Sep 2025). In CATE clustering, the clustering step is a relaxation of a hard combinatorial problem, the final groups depend on the quality of the first-stage CATE estimator, hard clustering may lose fidelity when the true CATE surface is smooth rather than piecewise-constant, and the choice of cluster number $f_k$5 and regularization $f_k$6 matters (Wang et al., 6 Sep 2025). In multi-domain recommendation, the optimization is heuristic because the exact clustering problem is NP-hard, and the use of causal discovery is methodological rather than ontological, since the domains do not have literal physical causal relationships (Luo et al., 9 Jul 2025).

Taken together, these works portray CDC as a broad research direction centered on one organizing claim: clusters are most informative when they correspond to stable causal regimes. Depending on the setting, that regime may be a latent environment, a subject group with a shared graph, a treatment-sensitive subgroup, or a recommendation-domain coalition defined by transfer effects. A plausible implication is that future CDC research will continue to move between these interpretations rather than collapse them into a single universal formalism.