Nonlinearity, Feedback and Uniform Consistency in Causal Structural Learning
Shuyan Wang
TL;DR
This work advances causal discovery by generalizing the k-Triangle Faithfulness to smooth and non-Gaussian settings, establishing uniform consistency for both structure and causal-effect estimation under relaxed assumptions, and extending latent-variable learning to handle cycles and nonlinearity. It introduces the VCSGS algorithm and a nonparametric edge-estimation framework, grounded in total variation smoothness and NZ conditions, to achieve uniformly consistent structure inference. The latent-structure portion combines GIN, rank constraints, and tensor constraints to identify cycles within latent clusters and across latent blocks, including both Gaussian and non-Gaussian noise regimes, with concrete algorithms and experiments. Collectively, the thesis broadens causal discovery applicability to systems with latent constructs and nonlinear interactions, offering principled probabilistic and algebraic tools for robust causal inference in richer data-generating processes.
Abstract
The goal of Causal Discovery is to find automated search methods for learning causal structures from observational data. In some cases all variables of the interested causal mechanism are measured, and the task is to predict the effects one measured variable has on another. In contrast, sometimes the variables of primary interest are not directly observable but instead inferred from their manifestations in the data. These are referred to as latent variables. One commonly known example is the psychological construct of intelligence, which cannot directly measured so researchers try to assess through various indicators such as IQ tests. In this case, casual discovery algorithms can uncover underlying patterns and structures to reveal the causal connections between the latent variables and between the latent and observed variables. This thesis focuses on two questions in causal discovery: providing an alternative definition of k-Triangle Faithfulness that (i) is weaker than strong faithfulness when applied to the Gaussian family of distributions, (ii) can be applied to non-Gaussian families of distributions, and (iii) under the assumption that the modified version of Strong Faithfulness holds, can be used to show the uniform consistency of a modified causal discovery algorithm; relaxing the sufficiency assumption to learn causal structures with latent variables. Given the importance of inferring cause-and-effect relationships for understanding and forecasting complex systems, the work in this thesis of relaxing various simplification assumptions is expected to extend the causal discovery method to be applicable in a wider range with diversified causal mechanism and statistical phenomena.