DirectLiNGAM: Causal Discovery Algorithm

Updated 30 December 2025

DirectLiNGAM is a causal discovery algorithm that recovers DAG structures and edge strengths in linear non-Gaussian acyclic models using the independence of disturbances.
It iteratively identifies exogenous variables by assessing residual independence, establishing a finite-step causal order and estimating coefficients via OLS regression.
Extensions like TSLiNGAM, ParaLiNGAM, and qLiNGAM enhance robustness, scalability, and even quantum kernel integration for improved performance in high-dimensional settings.

DirectLiNGAM is a causal discovery algorithm for identifying the directed acyclic graph (DAG) structure and connection strengths in linear non-Gaussian acyclic models (LiNGAM). It leverages non-Gaussianity and mutual independence in model disturbances to achieve identifiability from observational data alone, circumventing limitations of conventional approaches such as iterative search or Markov equivalence class constraints. DirectLiNGAM provides a finite-step, parameter-free procedure for causal ordering and estimation, and has served as the foundation for numerous extensions in robustness, parallelization, and quantum enhancement.

1. LiNGAM Model Foundations and Identifiability

The underlying LiNGAM model posits that the observed vector $x = (x_1,\dots,x_p)^{\top} \in \mathbb{R}^p$ is generated according to a linear structural equation system:

$x_i = \sum_{j: k(j) < k(i)} b_{ij} x_j + e_i, \quad i=1,\dots, p,$

where $k$ denotes an unknown causal ordering, $B = [b_{ij}]$ is a strictly lower-triangular matrix (after permutation), and $e = (e_1,\dots,e_p)^{\top}$ comprises mutually independent, non-Gaussian disturbances (Shimizu et al., 2011). This formulation ensures acyclicity—i.e., no later variable influences earlier ones.

Identifiability in LiNGAM is established through the Darmois–Skitovitch theorem and the connection to independent component analysis (ICA). Given the non-Gaussianity and independence of $e$ , the mixing matrix $A = (I-B)^{-1}$ can be uniquely identified (up to permutation/scaling), and the correct causal order $k$ is recoverable subject to a "correlation-faithfulness" condition (Shimizu et al., 2011). This identifiability sharply distinguishes LiNGAM from Gaussian SEMs and conventional DAG approaches.

2. DirectLiNGAM Algorithmic Structure

DirectLiNGAM proceeds via iterative root-finding and edge estimation:

Root Identification: For each unordered variable, regress all others on it and calculate residuals; an exogenous (root) variable is characterized by independence from all its residuals. The independence is quantified typically using kernel-based mutual information estimators or the NOCCO (normalized cross-covariance operator) statistic (Kawaguchi, 2021, Shimizu et al., 2011).
Causal Ordering: At each step, the variable most independent of its residuals is appended to the causal order. Data is updated to reflect the removal.
Structural Coefficient Estimation: Once the order is determined, ordinary least squares (OLS) regression estimates nonzero $b_{ij}$ , constrained by the identified ordering.

Pseudocode is as follows (condensed from (Shimizu et al., 2011, Kawaguchi, 2021)):

Input: Data matrix X (p x n)
Initialize K = []  # causal order
while len(K) < p-1:
    for candidate j not in K:
        for i != j not in K:
            r_i_j = X_i - (cov(X_i, X_j)/var(X_j))*X_j
            # Calculate independence T(j) between X_j and r_i_j
    m = argmin_j T(j)
    append m to K
    update X by residuals w.r.t. m
append last index to K
for each i>1 in K:
    regress X_K[i] on {X_K[1],...,X_K[i-1]} to estimate b_{ij}
Output: estimated B, ordering K

This direct approach replaces earlier ICA-based solutions, offering strict finite-step convergence under model assumptions and no need for iterative search or user-tuned parameters (Shimizu et al., 2011).

3. Extensions: Robustness, High-Dimensionality, and Quantum Kernels

Heavy-Tail Robustness: TSLiNGAM

DirectLiNGAM's OLS regression is not optimal in heavy-tailed, skewed, or contaminated settings. TSLiNGAM replaces OLS with the Theil–Sen estimator for regression slopes, conferring robustness to outliers and efficiency under heavy-tailed distributions. The independence measure can be substituted with distance correlation for computational gains. Theoretically, under LiNGAM model assumptions, TSLiNGAM retains identifiability and finite-step convergence (Leyder et al., 2023). Empirically, TSLiNGAM outperforms DirectLiNGAM in recovery accuracy under heavy tails and contamination.

Parallelization: ParaLiNGAM and AcceleratedLiNGAM

DirectLiNGAM's bottleneck arises from $O(p^3 n)$ sequential comparisons in root-finding. ParaLiNGAM implements GPU-based data parallelism using threshold-based early stopping, inter-worker message-passing (to avoid symmetric re-computation), and mathematical reuse of covariance matrices. This reduces per-round computational cost to $O(p^2)$ and achieves speedups of $10^2$ – $10^4\times$ in large-scale genome and metabolic network data with exact identifiability preserved (Shahbazinia et al., 2021). AcceleratedLiNGAM further refines the GPU implementation, yielding $25$– $32\times$ acceleration and enabling application to thousands of dimensions in gene expression and financial time series (Akinwande et al., 2024).

Quantum Kernel Enhancement: qLiNGAM

The qLiNGAM extension embeds classical data into quantum Hilbert space via IQP feature-map circuits, constructing quantum Gram matrices whose overlaps replace classical kernels in the NOCCO independence test. The quantum kernel enables higher-accuracy independence estimation in small-sample regimes and demonstrates improved recovery of medical causal structure, including cases where classical DirectLiNGAM fails. Empirical deployment on IBMQ hardware validated clinical DAG recovery in real-world datasets (Kawaguchi, 2021).

4. Statistical and Computational Properties

Variant	Independence Test	Regression Slope	Scalability
DirectLiNGAM	Kernel MI/NOCCO	OLS	$O(p^3 n)$
TSLiNGAM	Kernel MI/dCor	Theil–Sen, robust RM	$O(p^3 n \log n)$
ParaLiNGAM	Entropy Proxy	OLS	$O(p^2)$ parallel
AcceleratedLiNGAM	Kernel MI	OLS	$25$– $32\times$ GPU
qLiNGAM	Quantum NOCCO	OLS	NISQ hardware

DirectLiNGAM and its robust/quantum variants retain statistical consistency and identifiability under the model's assumptions, with empirical studies demonstrating superior DAG recovery vs. ICA-based, constraint-based, and continuous optimization alternatives (Shimizu et al., 2011, Leyder et al., 2023, Kawaguchi, 2021, Shahbazinia et al., 2021, Akinwande et al., 2024). Extensions maintain output equivalence when parallelized, ensuring theoretical guarantees are preserved.

5. Empirical Benchmarks and Application Domains

DirectLiNGAM has been evaluated on synthetic and real-world datasets:

Synthetic DAGs: Accurate recovery under diverse non-Gaussian distributions, improved error norms compared to ICA-LiNGAM (Shimizu et al., 2011).
Physics: Double pendulum ARMA data; correct causal directions between angles and angular speeds were recovered (Shimizu et al., 2011).
Sociology: Status attainment models with plausible edge strengths validated against domain knowledge (Shimizu et al., 2011, Leyder et al., 2023).
Medical: Clinical data (UCI Heart, Pima Diabetes) with quantum kernels; clinically valid DAGs identified in low-sample regimes, exceeding classical algorithm performance (Kawaguchi, 2021).
Genomics and Metabolism: GPU extensions enabled analysis of genome-scale metabolic networks ( $p$ up to 2326), completing in seconds with order-equivalent results (Shahbazinia et al., 2021, Akinwande et al., 2024).
Finance: Stock time series DAGs from hundreds of variables accurately inferred (Akinwande et al., 2024).

Robust variants such as TSLiNGAM showed increased accuracy and resilience in heavy-tailed, contaminated, and small-sample settings (Leyder et al., 2023).

6. Practical Guidance and Limitations

DirectLiNGAM requires reliable independence measurement in regression residuals, and its statistical guarantees hinge on non-Gaussianity and mutual independence in model errors. Finite-sample estimation may be sensitive to kernel choices and independence score noise; bootstrap and prior-knowledge constraints can improve performance. Robust and parallel extensions mitigate breakdown and computational issues in, respectively, heavy-tailed and high-dimensional regimes. For very small problems, parallelization incurs overhead; for very large models, tensor-core accelerations and more efficient kernels are active areas of investigation (Shahbazinia et al., 2021, Akinwande et al., 2024).

qLiNGAM empirically suggests that quantum kernel techniques may continue to provide advantages in small- $n$ , large- $p$ clinical studies, including rare diseases and integrative genomics (Kawaguchi, 2021).

Comparative analysis indicates that DirectLiNGAM is preferred over ICA-based, constraint-based or optimization-based methods when the model assumptions are met and full DAG identifiability is essential.

7. Outlook and Research Directions

Continued development is focused on further acceleration (I/O-aware GPU kernels, tensor-core regressions), integration of quantum computing advances (higher-depth IQP circuits, error mitigation), and broader application in genomics, finance, and clinical causal inference. Robustness to nonstandard errors and finite-sample properties remains an active research area. The underlying theoretical structure affords a template for innovation in both algorithmic and hardware-accelerated causal discovery.

DirectLiNGAM and its descendants represent a foundational approach for causal inference in linear non-Gaussian SEMs, bridging the gap between statistical identifiability and computational tractability through principled algorithmic design and continuous methodological evolution (Shimizu et al., 2011, Leyder et al., 2023, Kawaguchi, 2021, Shahbazinia et al., 2021, Akinwande et al., 2024).