Decomposition-Based Causal Discovery Framework

• Decomposition-based causal discovery frameworks are modular approaches that break complex data into interpretable subproblems to isolate causal effects.
• They leverage techniques like temporal, spectral, and tensor decomposition to manage confounders and spurious associations while ensuring scalability.
• These frameworks provide theoretical guarantees and identifiability by decomposing data into distinct generative components, facilitating precise causal inference.

Decomposition-Based Causal Discovery Frameworks provide a rigorous, modular approach for identifying causal structure in high-dimensional, confounded, non-stationary, or otherwise complex data by breaking down the overall causal discovery task into interpretable subproblems. These frameworks leverage structural decompositions—over variables, time, latent components, or generative sources—to isolate mechanisms, manage confounders and spurious associations, and enhance interpretability, scalability, and theoretical guarantees. Their mathematical and algorithmic underpinnings draw from structural causal models, statistical decompositions, optimization theory, nonparametric identification, and information theory.

1. Theoretical Foundations and Motivation

Decomposition-based causal discovery is grounded in the principle that observed associations typically arise from a mixture of distinct generative components—such as trends, seasonality, latent confounders, or separate algorithmic mechanisms. By explicitly partitioning observed variability, these frameworks enable the isolation and attribution of causal effects, facilitate identification in the presence of complicating factors (e.g., non-stationarity, latent variables), and improve downstream interpretability and robustness.

Divergent strands in this paradigm include:

• Additive decomposition of time series into trend, seasonal, and residual components for targeted causal analysis (Ferdous et al., 1 Feb 2026).
• Spectral or structural decomposition to remove pervasive latent confounders prior to DAG learning (Agrawal et al., 2021, Asiaee et al., 31 Dec 2025).
• Decomposition of the observed association between variables into direct causal and spurious effects (e.g., the experimental spurious effect, Exp–SE) via partial abduction–prediction (Plecko et al., 2023).
• Decomposition via algorithmic information theory, splitting contexts by algorithmic generative sources (Zenil et al., 2018).
• Tensor and matrix factorization to reveal latent clusters or trajectories on which downstream causal discovery is performed (Chen et al., 18 Jul 2025).

Universal to these approaches is the use of domain-relevant decompositions as a preprocessing or integral step, yielding representations that are more amenable to causal structure identification.

2. Structural Decomposition and Identification Strategies

A central innovation is leveraging structural decompositions to regularize or transform the causal discovery problem:

• Temporal additive decomposition separates each series $X^i_t$ into trend $T^i_t$, seasonal $S^i_t$, and residual $R^i_t$ components (using STL/LOESS), followed by component-specific causal analysis. Trends are tested for stationarity, seasonality is assessed by kernel-based dependence measures (e.g., HSIC), and residuals are subjected to constraint-based multivariate causal discovery (e.g., PC, PCMCI+). The resulting component-level graphs are then merged (Ferdous et al., 1 Feb 2026).
• Precision/spectral decomposition identifies pervasive latent confounders in linear/nonlinear SEMs by eigen-decomposing the observed covariance, projecting out leading eigenvectors, and performing causal discovery on the residualized data (Agrawal et al., 2021). Extensions to mixed localized and pervasive confounders use precision (inverse covariance) decomposition into a structured (local) and low-rank (pervasive) component by convex optimization (Asiaee et al., 31 Dec 2025).
• Algorithmic information decomposition segments complex observations (e.g., graphs, time-evolving arrays) by estimating algorithmic complexity changes due to local perturbation, thus identifying the minimal set of independent generative mechanisms (Zenil et al., 2018).
• Ordered group decomposition under measurement error leverages independence-based tests (the Transformed Independent Noise, TIN, criterion) to partition variables into ordered groups corresponding to causal depth in latent DAGs, circumventing the need to directly recover latent variables (Dai et al., 2022).
• Tensor factorization (e.g., PARAFAC2 for irregular tensors) decouples high-dimensional, temporally-structured data into interpretable latent clusters or trajectories and then infers their causal dependencies via dynamic Bayesian network estimation, under acyclicity constraints (Chen et al., 18 Jul 2025).

Identification is generally justified by: uniqueness of decompositions under structural, low-rank, or statistical constraints; invariance properties that relate manipulated or residualized data to causal quantities; or information-theoretic separation of generative sources.

3. Algorithmic Approaches and Complexity

Decomposition-based causal discovery structures the workflow into sequential or block-coordinate steps, often with guarantees of consistency or identifiability under specified assumptions.

Example pipelines include:

• Sequential root-peeling for monotonic SCMs: At each iteration, a set of conditional 1D normalizing flows is trained, one per remaining variable, and the variable that achieves minimal Jacobian (i.e., is most independent of others) is selected as the root. The process repeats, yielding a unique topological order without Markov equivalence ambiguity (Izadi et al., 2024).
• Component-specific tests and graph integration: Following STL decomposition, ADF/KPSS test trends, HSIC analyzes seasonality, and PCMCI+/PC operates on residuals; edge sets from these are merged without conflict due to spectral separation (Ferdous et al., 1 Feb 2026).
• Convex optimization for precision decomposition: The observed precision is decomposed into a sparse (local) plus low-rank (pervasive latent) term by penalized graphical lasso. Subsequent DAG learning is performed on the deconfounded structured component, maintaining identifiability and “bow-freeness” (Asiaee et al., 31 Dec 2025).
• Block-coordinate descent for joint tensor-causal optimization: Tensor factors and network parameters are alternately updated by augmented Lagrangian/ADMM to satisfy decomposition, sparsity, and acyclicity constraints, with convergence to stationary points provable under regularity conditions (Chen et al., 18 Jul 2025).

Complexity for each framework is dominated by the estimation of the decomposition (e.g., $O(n d^2)$ for root-finding (Izadi et al., 2024); $O(n)$ or $O(n^2)$ for STL and HSIC (Ferdous et al., 1 Feb 2026)), with overall computational burden typically mitigated by modular or parallelizable subproblem structure.

4. Theoretical Guarantees and Identifiability

Decomposition-based frameworks offer strong theoretical foundations for identifiability, often exceeding those of monolithic or generic independence-based approaches:

• Uniqueness of waveband-specific graphs: In spectral-separable settings, trend, seasonal, and interaction graphs are provably non-conflicting and jointly informative (Ferdous et al., 1 Feb 2026).
• Identifiability of latent confounding: Under rank and incoherence, the decomposition of covariance or precision into structured and low-rank parts is unique, and subsequent DAG learning on the adjusted representation is consistent (Agrawal et al., 2021, Asiaee et al., 31 Dec 2025).
• Nonparametric identifiability of spurious decompositions: In Markovian and suitably-structured Semi-Markovian models, the Exp–SE decomposition is identifiable using observed data alone under sufficient graphical criteria (Plecko et al., 2023).
• Consistency and robustness: Sequential decomposition methods avoid combinatorial optimization traps, and theoretical analysis confirms that the true topological order or causal graph is uniquely recovered in the infinite-sample limit given model assumptions (Izadi et al., 2024).

5. Empirical Evaluation and Performance

Empirical validation is central to demonstrating the advantages of decomposition-based frameworks:

Framework	Key Setting	Performance Highlights
Sequential root peeling	Nonlinear monotonic	Median CB=0 (Count-Backward) on synthetic/real data
DCD (trend/season/res.)	Nonstationary time	SHD≈10 (synthetic), sparse interpretable climate net.
Precision decomposition	Mixed confounding	Up to ΔF1=0.15 vs. non-deconfounded; robust to U/V.
DeCAMFounder	Pervasive confounding	30% lower SHD, higher TPR than plain CAM/GOLEM
Tensor decomposition	EHR/biomedical	2× lower SHD, better phenotype/network recovery
Exp–SE decomposition	Semi-Markovian, real	85% contribution/effect decomposition via age in COMPAS

Benchmarks consistently show reduction in false discoveries, improved structural recovery, and interpretable outputs compared to classical or monolithic baselines (Izadi et al., 2024, Ferdous et al., 1 Feb 2026, Asiaee et al., 31 Dec 2025, Plecko et al., 2023, Agrawal et al., 2021, Chen et al., 18 Jul 2025).

6. Applications, Extensions, and Limitations

Decomposition-based frameworks have broad impact across scientific fields and data modalities:

• In climate science, DCD-type frameworks disentangle physical forcing scales and feedbacks (Ferdous et al., 1 Feb 2026).
• Biomedical and single-cell applications exploit SEA and tensor-decomposition frameworks for scalable causal graph discovery and latent-phenotype analysis (Wu et al., 2024, Chen et al., 18 Jul 2025).
• Algorithmic deconvolution finds use in biological module identification, neuroimaging segmentation, and initialization of deep models with mechanistic priors (Zenil et al., 2018).
• Exp–SE decomposition informs fair AI by attributing spurious associations to specific confounders, as in the COMPAS recidivism study (Plecko et al., 2023).

Key limitations include dependence on accurate decomposition quality (e.g., spectral gap, model order selection), possible computational overhead for very large sets (e.g., all-subset MI for nonlinear decomposition), and the need for careful tuning or domain knowledge for interpreting multi-component outputs.

Potential directions involve combining kernel-based or higher-order decompositions, enriching component-specific tests, active learning for subproblem selection, and extending frameworks to non-stationary, high-dimensional, or intervention-rich regimes.

7. Comparative Landscape and Conceptual Advances

Relative to traditional constraint-based, score-based, or independence-test-only pipelines, decomposition-based frameworks provide:

• Superior ability to break Markov equivalence classes or orient edges without auxiliary tests, particularly in monotonic or confounded settings (Izadi et al., 2024, Asiaee et al., 31 Dec 2025).
• Clear separation of causal effects across timescales, variable groups, or latent dimensions.
• Robustness to modeling errors, as errors made by submodules remain tractable and can be attributed or debugged.
• Flexibility to incorporate domain-specific decompositions, such as algorithmic, spectral, or tensorial splits.

A central conceptual insight is that causal inference benefits from explicitly encoding structure in the data, and from modular, interpretable solution steps. This modularity aligns with both mathematical identifiability conditions and the practical needs of analysis, debugging, and scientific discovery.