Differentiable Latent-Tree Causal Discovery via Triplet Branch-Point Objectives and Three-Terminal Root Identification

Discovering directed causal structure in the presence of latent variables remains challenging, particularly when identifiability principles must be translated into practical learning objectives. We study latent-tree structural causal models, in which observed variables are leaves and latent variables form the internal structure of a directed tree. We introduce a differentiable framework for recovering both the latent-tree skeleton and its causal orientation from observational data. For skeleton learning, we develop a differentiable triplet branch-point (TBP) formulation for latent-tree completion: using the branch-additivity form of the established median property, simplex-valued triplet-center assignments jointly learn unknown branch-point identities and latent-related distances. For causal orientation, we establish a tree-conditioned three-terminal independence (TTI) characterization of the latent root under linear non-Gaussian LT-SCM assumptions; orienting all edges away from the identified root then recovers the directed latent-tree causal graph. Synthetic experiments across diverse latent-tree structures demonstrate strong structure-recovery performance, while experiments on the White Wine Quality dataset and a VAE-based generative model illustrate the applicability of the proposed framework to real data and end-to-end representation learning.