Dual Structural Causal Model Framework

• Dual Structural Causal Model is a framework linking a low-level concrete SCM with a high-level abstract SCM using a surjective linear map.
• It enforces a partition of concrete variables to ensure each abstract variable uniquely summarizes a distinct set of variables.
• The approach enhances causal discovery by leveraging abstract structural priors to reduce computational complexity in large-scale models.

Searching arXiv for relevant papers on dual structural causal models, causal abstraction, and related structural causal formulations. arXiv search query: "dual structural causal model causal abstraction linear structural causal models" Within recent causal-abstraction work, a dual structural causal model is the relationship between a low-level concrete SCM and a high-level abstract SCM connected by a surjective linear transformation $T$, such that the two models describe the same system at different levels of granularity and are interventionally consistent (Massidda et al., 2024). In this formulation, the low-level model provides the detailed causal structure, the high-level model provides a coarser description, and the abstraction map specifies how concrete variables are summarized by abstract variables. Related literature uses nearby ideas in different ways: one line learns paired causal and residual components and is described as “something close to a dual structural causal model” rather than as a formal two-level abstraction (Galanti et al., 2020); another introduces dual-causal interventions in a cross-modal SCM for long-term action recognition rather than a low-level/high-level pair (Shaowu et al., 9 Jul 2025); and work on structural causal dynamical models explicitly states that it does not introduce a separate object called a “Dual Structural Causal Model” (Bongers et al., 2018).

1. Two-level linear SCM formulation

The linear causal-abstraction framework distinguishes a low-level concrete SCM $\mathcal L$ and a high-level abstract SCM $\mathcal H$ (Massidda et al., 2024). The concrete model is

$$\mathcal L = (\mathcal X, \mathcal E, f, \mathcal P_{\mathcal E}),$$

with endogenous concrete variables $\mathcal X = \{X_1,\dots,X_d\}$, exogenous noise variables $\mathcal E = \{E_1,\dots,E_d\}$, structural mechanisms $f_X$, and joint noise distribution $\mathcal P_{\mathcal E}$. For linear SCMs,

$X = W^\top X + E,$

with $W \in \mathbb R^{d\times d}$ upper triangular after a suitable topological ordering, and reduced form

$\mathcal L$0

The abstract model is

$\mathcal L$1

with abstract endogenous variables $\mathcal L$2, abstract exogenous variables $\mathcal L$3, abstract mechanisms $\mathcal L$4, and linear form

$\mathcal L$5

where $\mathcal L$6 is upper triangular in some topological order. Its reduced form is

$\mathcal L$7

The assumptions stated for this setting are DAG structure or acyclicity, causal sufficiency, faithfulness, and, for the learning part, non-Gaussian exogenous noise. The theoretical objective is to characterize when the high-level model correctly summarizes the low-level model under a linear transformation $\mathcal L$8, and how low-level coefficients and the abstraction function determine the high-level coefficients.

2. Abstraction map and interventional consistency

The abstraction between endogenous variables is a surjective linear map

$\mathcal L$9

The high-level SCM is a $\mathcal H$0-abstraction of the low-level SCM if there exists a surjective exogenous map $\mathcal H$1 such that, for every concrete intervention $\mathcal H$2 and every exogenous configuration $\mathcal H$3,

$\mathcal H$4

This is the interventional consistency condition (Massidda et al., 2024).

The same framework also yields observational consistency,

$\mathcal H$5

The dual-SCM relation is therefore not merely a static projection of variables; it is a compatibility condition between interventions, exogenous mappings, and causal responses across levels.

For each abstract variable $\mathcal H$6, the relevant concrete variables are

$\mathcal H$7

These sets are nonempty and mutually disjoint. A central implication is that strong linear abstraction does not allow a concrete variable to be relevant to two different abstract variables. The paper describes this as forcing a partition-like assignment of concrete variables to abstract variables. This addresses a common misconception that an abstract variable may freely overlap with several others at the level of direct relevance: under this definition, such overlap is excluded.

3. Graphical structure, concrete blocks, and coefficient constraints

The graphical characterization introduces $\mathcal H$8-direct paths: directed paths in the low-level graph $\mathcal H$9 between relevant concrete variables whose intermediate nodes are all irrelevant (Massidda et al., 2024). The connectivity condition is stated as

$$\mathcal L = (\mathcal X, \mathcal E, f, \mathcal P_{\mathcal E}),$$0

The sufficient direction is that if there is a $$\mathcal L = (\mathcal X, \mathcal E, f, \mathcal P_{\mathcal E}),$$1-direct path from some $$\mathcal L = (\mathcal X, \mathcal E, f, \mathcal P_{\mathcal E}),$$2 to some $$\mathcal L = (\mathcal X, \mathcal E, f, \mathcal P_{\mathcal E}),$$3, then $$\mathcal L = (\mathcal X, \mathcal E, f, \mathcal P_{\mathcal E}),$$4 in the abstract graph. The paper also emphasizes that faithfulness is needed: without it, paths can cancel and the theorem fails.

The framework defines the concrete block of $$\mathcal L = (\mathcal X, \mathcal E, f, \mathcal P_{\mathcal E}),$$5 as

$$\mathcal L = (\mathcal X, \mathcal E, f, \mathcal P_{\mathcal E}),$$6

where $$\mathcal L = (\mathcal X, \mathcal E, f, \mathcal P_{\mathcal E}),$$7 is the exogenous abstraction matrix from $$\mathcal L = (\mathcal X, \mathcal E, f, \mathcal P_{\mathcal E}),$$8. A block contains all relevant variables for $$\mathcal L = (\mathcal X, \mathcal E, f, \mathcal P_{\mathcal E}),$$9, plus irrelevant variables that are downstream of those relevant ones via $\mathcal X = \{X_1,\dots,X_d\}$0-direct paths. The block composition lemma is

$\mathcal X = \{X_1,\dots,X_d\}$1

The high-level graph induces a coarse causal ordering of concrete variables. If $\mathcal X = \{X_1,\dots,X_d\}$2 is any valid topological order of the abstract graph, then there exists a concrete order $\mathcal X = \{X_1,\dots,X_d\}$3 such that

$\mathcal X = \{X_1,\dots,X_d\}$4

and every block precedes the variables outside all blocks. This result shows that abstraction constrains not only adjacency but also permissible causal orderings at the lower level.

At the parameter level, the exogenous abstraction map is linear: $\mathcal X = \{X_1,\dots,X_d\}$5 The central parameter consistency condition is

$\mathcal X = \{X_1,\dots,X_d\}$6

The block abstraction theorem states that $\mathcal X = \{X_1,\dots,X_d\}$7 is a linear $\mathcal X = \{X_1,\dots,X_d\}$8-abstraction of $\mathcal X = \{X_1,\dots,X_d\}$9 if and only if the concrete blocks respect the abstract topological order and the block coefficients satisfy

$\mathcal E = \{E_1,\dots,E_d\}$0

This gives a complete characterization of valid concretizations of an abstract SCM in the linear setting.

4. Learning the abstraction and the two SCMs

The learning problem assumes abundant concrete observations $\mathcal E = \{E_1,\dots,E_d\}$1 and scarce paired observations $\mathcal E = \{E_1,\dots,E_d\}$2 from the joint concrete-abstract distribution (Massidda et al., 2024). The goal is to learn $\mathcal E = \{E_1,\dots,E_d\}$3, the abstract model $\mathcal E = \{E_1,\dots,E_d\}$4, and the concrete model $\mathcal E = \{E_1,\dots,E_d\}$5.

The procedure is specified as follows. First, estimate $\mathcal E = \{E_1,\dots,E_d\}$6 from paired samples by least squares: $\mathcal E = \{E_1,\dots,E_d\}$7 Second, extract relevant sets by thresholding the learned coefficients: $\mathcal E = \{E_1,\dots,E_d\}$8 Third, create an abstract dataset by projecting concrete samples: $\mathcal E = \{E_1,\dots,E_d\}$9 Fourth, learn the abstract SCM with DirectLiNGAM. Fifth, use the learned abstract graph to generate forbidden paths for the concrete graph. Sixth, learn the concrete SCM with DirectLiNGAM under those constraints.

The resulting method, Abs-LiNGAM, combines abstraction learning with causal discovery. Its stated assumptions are linear SCMs, acyclic graphs, causal sufficiency, faithfulness, non-Gaussian noise, and a small paired dataset from joint observations. The key constraint is that if two abstract variables $f_X$0 and $f_X$1 are not ancestrally related, then no variable in $f_X$2 can causally affect any variable in $f_X$3. The forbidden set is

$f_X$4

5. Computational role and empirical behavior

Abs-LiNGAM uses the abstract model to speed up recovery of the larger low-level model by shrinking the search space considered by DirectLiNGAM (Massidda et al., 2024). The abstract graph supplies structural priors, translated into forbidden concrete edges, before concrete causal discovery is performed. This is especially helpful when the concrete graph is large.

The experiments reported in the paper show several empirical patterns. Abs-LiNGAM can recover the concrete DAG with quality comparable to DirectLiNGAM when enough paired data are available. With too few paired samples, the learned abstraction can be inaccurate and can impose wrong forbidden edges, harming downstream recovery. As the number of paired samples increases, performance approaches the baseline. The method gives substantial speedups, especially for larger concrete graphs. Bootstrapping the abstract causal discovery step improves robustness without major runtime cost.

These findings support a specific interpretation of the dual-SCM perspective: the abstract model is not only interpretive but also computationally beneficial. The high-level model contributes constraints that are unavailable to unconstrained low-level discovery, while the low-level model provides the detailed variables needed for concrete causal reconstruction.

6. Related formulations and conceptual boundaries

A different use of a paired causal structure appears in “A Critical View of the Structural Causal Model” (Galanti et al., 2020). In the univariate case, that paper argues that many benchmark causal-direction tasks can be solved by comparing the individual complexity of cause and effect, without considering their interaction at all. In the multivariate case, it proposes an adversarial framework with networks $f_X$5, $f_X$6, $f_X$7, and discriminator $f_X$8, where $f_X$9 models the causal parent transformation, $\mathcal P_{\mathcal E}$0 models the latent noise or environment component, and $\mathcal P_{\mathcal E}$1 reconstructs the effect through

$\mathcal P_{\mathcal E}$2

Its representation theorem states that, under zero mapping error and independence, $\mathcal P_{\mathcal E}$3 corresponds to the causal component and $\mathcal P_{\mathcal E}$4 corresponds to the residual or noise component up to invertible transformation. The paper describes this as moving toward something close to a dual structural causal model, but not as the same object as the low-level/high-level abstraction framework.

Another nearby but non-equivalent construction is “Cross-Modal Dual-Causal Learning for Long-Term Action Recognition” (Shaowu et al., 9 Jul 2025). That paper defines a cross-modal SCM with variables $\mathcal P_{\mathcal E}$5 for video, $\mathcal P_{\mathcal E}$6 for action text, $\mathcal P_{\mathcal E}$7 for long-term action label, $\mathcal P_{\mathcal E}$8 for cross-modal bias, $\mathcal P_{\mathcal E}$9 for visual confounder, and $X = W^\top X + E,$0 for a causal factor encoding spatial semantics and temporal action correlations. It then applies two interventions: Textual Causal Intervention, based on back-door adjustment, to produce debiased text embeddings $X = W^\top X + E,$1; and Visual Causal Intervention, based on front-door adjustment with mediator $X = W^\top X + E,$2, to produce deconfounded visual embeddings $X = W^\top X + E,$3. This is a dual-causal intervention pipeline within a single cross-modal SCM, not a two-level dual-SCM relation between a concrete model and an abstract model.

A third boundary case is “Causal Modeling of Dynamical Systems” (Bongers et al., 2018). That work develops structural causal dynamical models as time-dependent, stochastic-process extensions of SCMs and shows that certain steady SCDMs equilibrate to SCMs as time tends to infinity. The paper explicitly states that it does not introduce a separate object called a “Dual Structural Causal Model.” Its central correspondence is instead dynamic-to-static: an SCDM induces an equilibrium SCM, and interventions commute with equilibration,

$X = W^\top X + E,$4

This provides a structure-preserving bridge between dynamics and equilibrium, but not a duality in the sense of causal abstraction.

Taken together, these formulations delimit the topic. In the strict sense supplied by recent causal-abstraction theory, a dual structural causal model is the relationship between $X = W^\top X + E,$5, $X = W^\top X + E,$6, and $X = W^\top X + E,$7 under interventional consistency. In broader usage, related work may pair causal and noise representations, or pair two causal interventions, or pair a dynamical model with an equilibrium SCM. These are closely related structural ideas, but they are not identical definitions.