Multi-granularity Causal Structure Learning (MgCSL)

• Multi-granularity causal structure learning (MgCSL) is a framework that integrates fine-grained micro-variables with coarse-grained abstractions to enhance causal discovery.
• It combines a sparse autoencoder for macro-variable extraction with dedicated multi-input MLPs to capture both micro-to-micro and macro-to-micro causal interactions.
• The method enforces acyclicity via a Schur-decomposition-based penalty, achieving high precision, low structural Hamming distance, and reduced runtime in benchmarks.

Multi-granularity Causal Structure Learning (MgCSL) refers to a class of algorithms designed to uncover the causal relationships between variables, explicitly modeling both the micro-level (fine-grained) variables and their collective macro-level (coarse-grained) abstractions. MgCSL facilitates causal discovery and representation across scales, addressing challenges inherent in high-dimensional datasets and complex systems by combining sparse representation learning, multi-input neural architectures, and scalable acyclicity constraints. It stands as the first end-to-end method to jointly learn micro- and macro-variable causal directed acyclic graphs (DAGs), establishing new benchmarks in both accuracy and runtime on a range of synthetic and real-world benchmarks (Liang et al., 2023).

1. Architectural Foundations of MgCSL

MgCSL integrates two principal modules:

• Sparse Auto-encoder (SAE): The SAE receives $d$ micro-variables $(v_1,\dots,v_d)$ and learns coarse-grained macro-variables $(u_1,\dots,u_q)$, with $q \ll d$. The encoder-decoder pipeline is governed by

$$\mathbf{Z} = \sum_{i=1}^d \sigma\bigl( \sigma(\mathbf{X}_{[:,i]} \mathbf{W}_{\mathrm{enc}^{(1)}[i,:,:}) \mathbf{W}_{\mathrm{enc}^{(2)}[i,:,:]} \bigr) \in \mathbb{R}^{n\times q}$$

$$\mathbf{Y} = \sigma\bigl( \sigma(\mathbf{Z} \mathbf{W}_{\mathrm{dec}^{(1)}}) \mathbf{W}_{\mathrm{dec}^{(2)}} \bigr) \in \mathbb{R}^{n \times d}$$

The SAE includes an $\ell_{1,1}$ penalty on the encoder ($\alpha_1 \|\mathbf{W}_{\mathrm{enc}}\|_{1,1}$) to enforce sparsity, ensuring interpretable macro constructions. The macro–micro contribution matrix

$$\mathbf{A} = |\mathbf{W}_{\mathrm{enc}^{(1)}}||\mathbf{W}_{\mathrm{enc}^{(2)}}| \in \mathbb{R}^{d \times q}$$

encodes the participation of each micro-variable in each macro-variable.

• Multi-granularity Multilayer Perceptrons (MLPs): For each micro-variable $x_i$, a dedicated MLP$(v_1,\dots,v_d)$0 predicts it using $(v_1,\dots,v_d)$1. This dual-input architecture permits both micro-to-micro and macro-to-micro causal pathways to be expressed. The first-layer weights of the $(v_1,\dots,v_d)$2 MLPs assemble:
  • The pure micro-to-micro weighted adjacency matrix (WAM) $(v_1,\dots,v_d)$3, yielding $(v_1,\dots,v_d)$4.
  • The overall multi-granularity WAM $(v_1,\dots,v_d)$5, capturing both micro and macro links.

A redundancy penalty $(v_1,\dots,v_d)$6 discourages redundant explanations of a target variable at both granularity levels.

2. Scalable Acyclicity Constraint

MgCSL introduces a Schur-decomposition-based acyclicity constraint that is computationally more efficient and provides sharper gradients versus prior approaches such as NOTEARS (Liang et al., 2023). Given micro-micro WAM $(v_1,\dots,v_d)$7, set $(v_1,\dots,v_d)$8. The decomposition

$(v_1,\dots,v_d)$9

(where $(u_1,\dots,u_q)$0 is block upper-triangular with the eigenvalues of $(u_1,\dots,u_q)$1 on its diagonal) leads to the acyclicity penalty:

$(u_1,\dots,u_q)$2

Enforcing $(u_1,\dots,u_q)$3 guarantees the resulting DAG is cycle-free. This approach scales more favorably ($(u_1,\dots,u_q)$4 but with lower constants), provides stronger gradients when any cycle appears, and avoids the computational expense of calculating matrix exponentials as required by NOTEARS.

3. Training Objective and Optimization Procedure

The overall training objective is an augmented Lagrangian, summing reconstruction loss, sparsity, redundancy and acyclicity penalties:

$(u_1,\dots,u_q)$5

where $(u_1,\dots,u_q)$6 denotes all learnable weights. The method initializes parameters with fixed hyperparameters ($(u_1,\dots,u_q)$7), employs the L-BFGS-B solver, updates penalties according to constrained optimization rules, and post-processes the DAG (thresholding and pruning until acyclicity holds and binarizing remaining edges). Stopping occurs when $(u_1,\dots,u_q)$8 or iteration/time limits are met.

4. Experimental Evaluation and Benchmarking

MgCSL demonstrates robust performance across a broad spectrum of synthetic and real-world causal discovery tasks (Liang et al., 2023):

• Synthetic Data: On nonlinear Additive Noise Models (ANMs) and fully joint Gaussian-process functions, with random DAGs (Erdős–Rényi and scale-free topologies, $(u_1,\dots,u_q)$9), MgCSL achieves precision often exceeding 95%, best or comparable F$q \ll d$0, and lowest SHD, outperforming PC, GES, DAG-GNN, GAE, GraN-DAG, and NOTEARS-MLP baselines with 5–20× faster runtime than NOTEARS-MLP and 30–100× faster than DAG-GNN under these settings.
• Multi-granularity Synthetic: Under multi-granularity variants (macro decomposition and micro-MLP expansion), MgCSL maintains $q \ll d$195% precision and SHD $q \ll d$217, whereas all baselines degrade below 70% precision and incur much higher SHD. Wilcoxon signed-rank tests show significance ($q \ll d$30.02).
• Real-world Data: On the Sachs protein-signaling dataset ($q \ll d$4), MgCSL attains 100% precision, 30% recall, F$q \ll d$5=46%, and SHD=14, outperforming all evaluated rivals. In fMRI Hippocampus data (six ROIs, $q \ll d$6), edges found by MgCSL—such as {PHC, Sub$q \ll d$7ERC, ERC$q \ll d$8DG, CA1$q \ll d$9Sub, CA1$$\mathbf{Z} = \sum_{i=1}^d \sigma\bigl( \sigma(\mathbf{X}_{[:,i]} \mathbf{W}_{\mathrm{enc}^{(1)}[i,:,:}) \mathbf{W}_{\mathrm{enc}^{(2)}[i,:,:]} \bigr) \in \mathbb{R}^{n\times q}$$0DG—align with established anatomical pathways, with some direction flips attributed to MgCSL’s enforced acyclicity in cyclic ground-truth structures.

5. Theoretical Guarantees and Limitations

MgCSL’s eigenvalue-based acyclicity constraint provides the same “no local nonzero-cycle” guarantee as NOTEARS but with more efficient gradient computation (Theorem 1 & Prop. 2 in the MgCSL paper). Coarse-graining by SAE is regularized for sparsity and interpretability, but the current method fixes macro-variable count $$\mathbf{Z} = \sum_{i=1}^d \sigma\bigl( \sigma(\mathbf{X}_{[:,i]} \mathbf{W}_{\mathrm{enc}^{(1)}[i,:,:}) \mathbf{W}_{\mathrm{enc}^{(2)}[i,:,:]} \bigr) \in \mathbb{R}^{n\times q}$$1; future extensions are proposed for adaptive or Bayesian determination of macro-variable cardinality. Severe cycles in the ground-truth data may result in directionality errors, as hard acyclicity ($$\mathbf{Z} = \sum_{i=1}^d \sigma\bigl( \sigma(\mathbf{X}_{[:,i]} \mathbf{W}_{\mathrm{enc}^{(1)}[i,:,:}) \mathbf{W}_{\mathrm{enc}^{(2)}[i,:,:]} \bigr) \in \mathbb{R}^{n\times q}$$2) cannot faithfully represent feedback loops. SAE-based macro construction could be replaced by more expressive generative models, such as variational autoencoders with sparsity priors, to support richer and potentially domain-aligned macro representations.

6. Connections to Related Multi-Granularity Methods

Alternative approaches to multi-granularity causal structure learning include modular subgraph-integration methods such as VISTA (Voting-based Integration of Subgraph Topologies for Acyclicity) (Sun et al., 28 Jan 2026). VISTA operationalizes multi-granularity via:

• Node-level subgraph localization (Markov Blanket discovery for each variable),
• Local DAG estimation (using arbitrary base learners within small neighborhoods),
• Global integration via weighted voting and feedback arc set (FAS) pruning to guarantee acyclicity,

as summarized in the following table.

Method	Multi-granularity Model	Acyclicity Handling
MgCSL	SAE macros + micro inputs, multi-level WAM	Schur decomposition eigenvalue penalty
VISTA	Markov Blanket subgraphs (local); weighted global voting	Feedback Arc Set (GreedyFAS)

VISTA is model-agnostic regarding base learners and achieves scalability by parallelizing local subgraph inferences, combining results via theoretically grounded voting. Both MgCSL and VISTA address high-dimensionality and automate the synthesis of local (fine-grained) and global (coarse-grained) causal information, albeit using fundamentally different architectures and integration strategies.

7. Practical Impact and Future Directions

MgCSL and similarly motivated frameworks enable causal inference in domains where granularity hierarchy, data dimensionality, and computational efficiency are critical, as illustrated by successful application in neuroscience (fMRI), computational biology (protein-signaling networks), and complex synthetic systems. Key limitations include the rigid enforcement of acyclicity (which may not match all domains), fixed macro dimensionality, and sensitivity to macro variable semantics. Plausible future directions include adaptive and Bayesian macro-layer structures, more powerful disentangling autoencoders, and the integration of interventional data to resolve ambiguities arising from purely observational settings.

In sum, MgCSL establishes a new standard for scalable and accurate multi-granularity causal structure discovery by marrying sparse representation learning, neural estimation, and computationally tractable acyclicity constraints, with extensive quantitative and qualitative validation across data regimes (Liang et al., 2023, Sun et al., 28 Jan 2026).