Learning Sparsest Linear Causal DAGs with Latent Confounders via Higher-Order Cumulants

Jul 16, 2026

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Recovering the exact directed acyclic graph (DAG) in linear non-Gaussian acyclic models with latent confounders (LvLiNGAM) is difficult because identification holds only up to an observational equivalence class, even though each class has a unique sparsest DAG. Existing approaches are asymptotically consistent but lack an explicit finite-sample method for recovering that sparsest representative, and they restrict the number of latent confounders. This work proposes a finite-sample procedure without such restrictions, validated by simulations and real-data experiments with improved performance.

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Introduction

Related work and limitations

Proposed finite-sample method and contributions

arXiv :2607 .05984v 1 [ cs .LG] 7 Jul 2026 Learning Sparsest Linear Causal DAGs with Latent Confounders via Higher-Order Cumulants Ming Cai* [cai.ming.52d@st.kyoto-u.ac.jp](cai.ming.52d@st.kyoto-u.ac.jp) Graduate School of Informatics, Kyoto University, Kyoto, Japan Hisayuki Hara [hara.hisayuki.8k@kyoto-u.ac.jp](hara.hisayuki.8k@kyoto-u.ac.jp) Institute for Liberal Arts and Sciences, Kyoto University, Kyoto, Japan Abstract Recovering the exact directed acyclic graph (DAG) in linear non-Gaussian acyclic models with latent confounders (LvLiNGAM) remains a challenging problem. Although LvLiNGAMis identifiable only up to an observational equivalence class, each equivalence class is characterized by a unique sparsest DAG. Recovering the sparsest DAG from finite samples, however, remains difficult. Although existing methods are asymptotically consistent, they do not provide an explicit finite-sample procedure for recovering the unique sparsest DAG, nor do they handle models with an arbitrary number of latent confounders. In this paper, we propose a finite-sample method for recovering the sparsest DAG without imposing any restriction on the number of latent confounders. Simulation studies and real-data analyses demonstrate that the proposed method achieves superior finitesample performance compared with existing approaches. Keywords: Causal discovery; DAG; LiNGAM; Latent confounder; Cumulant; 1. Introduction Linear non-Gaussian acyclic models (LiNGAMs) provide a powerful framework for causal discovery (Shimizu et al. , 2006 , 2011) . In the absence of latent variables, LiNGAM enables complete identification of causal DAGs. In many practical applications, however, latent confounders are unavoidable. Hoyer et al. (2008) introduced LiNGAM with latent variables (LvLiNGAM) and demonstrated that any LvLiNGAM can be transformed into a canonical model in which all latent variables are mutually independent and causally precede the observed ones. They also estimated the mixing matrix using overcomplete independent component analysis (OICA; e.g. , Eriksson and Koivunen, 2004), assuming that the number of latent variables is known a priori. However, because it relies on OICA, this approach is prone to converge to local optima (Shimizu and Bollen, 2014) . To avoid relying on OICA, several methods have been proposed to estimate canonical LvLiNGAMs via residual independence tests, such as Pairwise LvLiNGAM (Entner and Hoyer, 2011), ParceLiNGAM (Tashiro et al. , 2014), Repetitive Causal Discovery (RCD)(Maeda and Shimizu, 2020 , 2022; Maeda, 2022), and BANG (Wang and Drton, 2023) . However, none of these methods can fully identify ancestral relationships or parent-child relationships between observed variables that form the bow structures (Wang and Drton, 2023) . When bow structures are present, Chen et al. (2024) use cumulants of observed variables to identify their ancestral relationships in the bivariate setting with a single latent © M. Cai & H. Hara. Cai Hara confounder. Building on this, Chen et al. (2025) extend the approach to multiple latent confounders in the observed bivariate case. Schkoda et al. (2024) proposed ReLVLiNGAM, a recursive cumulant-based method that accommodates multiple observed variables and latent confounders. Without relying on OICA or requiring prior knowledge of the number of latent variables, ReLVLiNGAM recovers the observational equivalence class of a canonical LvLiNGAM. Under the genericity assumption described below, the sparsest DAG within an observational equivalence class is uniquely determined. The sparsest DAG is generic within its model class, whereas any denser DAG in the same observational equivalence class requires non-generic parameter values. This provides a natural justification for treating the sparsest DAG as the canonical representative of the observational equivalence class. Although ReLVLiNGAM consistently estimates the mixing matrix, it does not provide a procedu

FAQs

What problem does the paper address in LvLiNGAM with latent confounders?

It addresses the challenge of recovering the exact causal DAG from finite samples when latent confounders cause identification to be possible only up to an observational equivalence class. The goal is to recover the unique sparsest DAG representative of that class.

Why is recovering the sparsest DAG from finite samples considered difficult?

Although the sparsest DAG is uniquely determined within an equivalence class, existing methods do not provide an explicit finite-sample procedure to obtain it. They are also limited by assumptions such as known numbers of latent variables or rely on components that can converge to local optima.

What key innovations does the proposed method introduce?

The method adds (1) an update rule that directly residualizes observed variables instead of recursively updating higher-order cumulants, which reduces error propagation and removes the method’s local restriction. It also adds a sequential procedure to identify exact parent–child relationships for sources and their descendants, enabling direct recovery of the sparsest DAG from finite samples.

Jul 16, 2026

Upload by Levi