Jacobian-based Causal Discovery with Nonlinear ICA

Published: 18 Apr 2023, Last Modified: 18 Apr 2023Accepted by TMLREveryoneRevisionsBibTeX
Abstract: Today's methods for uncovering causal relationships from observational data either constrain functional assignments (linearity/additive noise assumptions) or the data generating process (e.g., non-i.i.d. assumptions). Unlike previous works, which use conditional independence tests, we rely on the inference function's Jacobian to determine nonlinear cause-effect relationships. We prove that, under strong identifiability, the inference function's Jacobian captures the sparsity structure of the causal graph; thus, generalizing the classic LiNGAM method to the nonlinear case. We use nonlinear Independent Component Analysis (ICA) to infer the underlying sources from the observed variables and show how nonlinear ICA is compatible with causal discovery via non-i.i.d data. Our approach avoids the cost of exponentially many independence tests and makes our method end-to-end differentiable. We demonstrate that the proposed method can infer the causal graph on multiple synthetic data sets, and in most scenarios outperforms previous work.
License: Creative Commons Attribution 4.0 International (CC BY 4.0)
Submission Length: Regular submission (no more than 12 pages of main content)
Changes Since Last Submission: - **Title:** we emphasize that our method relies on the Jacobian and removed the ambiguous reference to general nonlinear functions; - **Section 3:** we clarify the contributions of our work by restructuring our claims as follows: - We show that causal discovery is possible via the ground-truth inverse generative model's Jacobian (Sec. 3.2); and - We characterize the requirements on the estimated inference model such that our insight can be transferred from the ground-truth inverse model to the estimated inference model (Sec. 3.3); - **Section 4:** we discuss the assumptions for contrastive nonlinear ICA and show how the non-i.i.d. nature of the data arises, enabling causal discovery and point out the differences between identifying the DAG and identifying the function class; furthermore, we discuss how observation/intervention pairs are used for causal discovery; we also added references to the Compatibility of CL and CD paragraph in 4.2; - **Section 6:** we provide context about the use of the Jacobian throughout the literature and make our proposal's novelty explicit in the “Using the Jacobian” paragraph; - **Section 7:** we clarify the relationship between ICA and causal discovery, and update our conclusions to reflect the changes in Sec. 3 and 4; and added a new paragraph CD with identifiability beyond ICA to the Discussion (Sec 7) discussing TCL and sparse mechanism shifts; - **App. B:** we define notions of identifiability; - **App. C:** we discuss the compatibility of assumptions in the ICA and causality literatures; - **App. D:** we summarize the use of the Jacobian in the literature in Tab. 5; - **App. G:** we detail the assumptions for contrastive nonlinear ICA and discuss the testability of these assumptions; - **App. H:** we provide code for the Sinkhorn operator.
Code: https://github.com/rpatrik96/nl-causal-representations
Assigned Action Editor: ~Nishant_A_Mehta1
Submission Number: 673