Direct Linear Latent Causal Structure Learning via Triplet Difference Constraint

Learning causal structures among unobserved latent variables is fundamental yet challenging, especially when latent variables are causally related. Existing non-Gaussian methods rely heavily on repeated, pairwise or subset-wise independence tests, suffering from error accumulation and heavy computational overheads. To tackle this challenge, we propose a direct framework for latent causal discovery based on higher-order statistical information. Specifically, we introduce the the Triplet Difference (TD), a novel algebraic criterion formulated in terms of second- and third-order cumulants, which enables the characterization of latent causal relationships through observable data. Building on this criterion, we establish identifiability conditions for latent causal clusters and causal structures among latent variables. Rather than relying on repeated independence testing, our framework exploits the constraint-satisfaction patterns of TD across variable triplets to distinguish latent causal topologies, thereby enabling latent cluster identification and causal ordering. Complexity analysis confirms that our algorithm avoids exponential evaluation costs over observed variables. Extensive experiments on both synthetic and real-world datasets demonstrate the effectiveness of the proposed approach in recovering latent causal structures, corroborating the theoretical findings and practical utility of the framework.