# Functional Causal Discovery via Conditional Covariance Ordering

arXiv: https://arxiv.org/abs/2609.27256 · HTML: https://mlsanity.com/paper/2609.27256/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 43 references that are in the map.
- First posted on arXiv 23 Sep 2026.

## Builds on (most cited first, 14 of 14)

- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [CAM: Causal additive models, high-dimensional order search and penalized regression](https://mlsanity.com/paper/1310.1533/) (2013)
- [Causal Discovery from Heterogeneous/Nonstationary Data with Independent Changes](https://mlsanity.com/paper/1903.01672/) (2019)
- [High-dimensional learning of linear causal networks via inverse covariance estimation](https://mlsanity.com/paper/1311.3492/) (2013)
- [On Causal Discovery with Equal Variance Assumption](https://mlsanity.com/paper/1807.03419/) (2018)
- [Causal Inference via Kernel Deviance Measures](https://mlsanity.com/paper/1804.04622/) (2018)
- [Bayesian Graphical Models for Multivariate Functional Data](https://mlsanity.com/paper/1411.4158/) (2014)
- [Causal inference with a functional outcome](https://mlsanity.com/paper/2304.07113/) (2023)
- [Confidence Sets for Causal Orderings](https://mlsanity.com/paper/2305.14506/) (2023)
- [Functional Bayesian Networks for Discovering Causality from Multivariate Functional Data](https://mlsanity.com/paper/2210.12832/) (2022)
- [Directed Cyclic Graph for Causal Discovery from Multivariate Functional Data](https://mlsanity.com/paper/2310.20537/) (2023)
- [Flexible Functional Treatment Effect Estimation](https://mlsanity.com/paper/2309.08039/) (2023)
- [Causal Effect of Functional Treatment](https://mlsanity.com/paper/2210.00242/) (2022)
- [Functional Linear Non-Gaussian Acyclic Model for Causal Discovery](https://mlsanity.com/paper/2401.09641/) (2024)

## Abstract (arXiv)

We study causal discovery where each node is a random function. Previous studies on this topic rely on structural assumptions, e.g., linearity or non-linearity, and distributional assumptions, e.g., Gaussianity or non-Gaussianity. In contrast, we make use of covariance operators to avoid these assumptions. Under functional additive noise models, we propose a new sufficient condition to identify a valid topological ordering based on comparing norms of conditional covariance operators. Taking advantage of this identifiability condition, we develop a new mixed regression model that subsumes linear and non-linear models. Together with variable selection, our procedure yields an estimation of the causal directed acyclic graph (DAG) for functional variables. In theory, we develop the least-squares-type theory of this regression model, and derive asymptotic consistency of order determination, sparse regression, as well as identifying the DAG. Computational algorithms based on discrete observations are provided. Applied to simulated data, our approach performs satisfactorily among existing approaches. A real data example of brain effective connectivity is also presented.