Strong and Weak Identifiability of Optimization-based Causal Discovery in Non-linear Additive Noise Models

Causal discovery aims to identify causal relationships from observational data. Recently, optimization-based causal discovery methods have attracted extensive attention in the literature due to their efficiency in handling highdimensional problems. However, we observe that optimization-based methods often perform well on certain problems but struggle with others. This paper identifies a specific characteristic of causal structural equations that determines the difficulty of identification in causal discovery and, in turn, the performance of optimization-based methods. We conduct an in-depth study of the additive noise model (ANM) and propose to further divide identifiable problems into strongly and weakly identifiable types based on the difficulty of identification. We also provide a sufficient condition to distinguish the two categories. Inspired by these findings, this paper further proposes GENE, a generic method for addressing strongly and weakly identifiable problems in a unified way under the ANM assumption. GENE adopts an order-based search framework that incorporates conditional independence tests into order fitness evaluation, ensuring effectiveness on weakly identifiable problems. In addition, GENE restricts the dimensionality of the effect variables to ensure scale invariance, a property crucial for practical applications. Experiments demonstrate that GENE is uniquely effective in addressing weakly identifiable problems while also remaining competitive with state-ofthe-art causal discovery algorithms for strongly identifiable problems.