# Calculation of Entailed Rank Constraints in Partially Non-Linear and Cyclic Models

arXiv: https://arxiv.org/abs/1309.7004 · HTML: https://mlsanity.com/paper/1309.7004/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Sep 2013.

## Built on by (most cited first, 12 of 12)

- [Latent Hierarchical Causal Structure Discovery with Rank Constraints](https://mlsanity.com/paper/2210.01798/) (2022)
- [A Versatile Causal Discovery Framework to Allow Causally-Related Hidden Variables](https://mlsanity.com/paper/2312.11001/) (2023)
- [Independence Testing-Based Approach to Causal Discovery under Measurement Error and Linear Non-Gaussian Models](https://mlsanity.com/paper/2210.11021/) (2022)
- [Generalized Independent Noise Condition for Estimating Causal Structure with Latent Variables](https://mlsanity.com/paper/2308.06718/) (2023)
- [Permutation-Based Rank Test in the Presence of Discretization and Application in Causal Discovery with Mixed Data](https://mlsanity.com/paper/2501.18990/) (2025)
- [Distributional Equivalence in Linear Non-Gaussian Latent-Variable Cyclic Causal Models: Characterization and Learning](https://mlsanity.com/paper/2603.04780/) (2026)
- [Automating the Selection of Proxy Variables of Unmeasured Confounders](https://mlsanity.com/paper/2405.16130/) (2024)
- [A Review and Roadmap of Deep Causal Model from Different Causal Structures and Representations](https://mlsanity.com/paper/2311.00923/) (2023)
- [On the Three Demons in Causality in Finance: Time Resolution, Nonstationarity, and Latent Factors](https://mlsanity.com/paper/2401.05414/) (2023)
- [Bounding Causal Effects with Leaky Instruments](https://mlsanity.com/paper/2404.04446/) (2024)
- [Causal Structure Learning in Hawkes Processes with Complex Latent Confounder Networks](https://mlsanity.com/paper/2508.11727/) (2025)
- [Equilibrium Causal Games: Separation, Identification, and the Identifiability of Cyclic Latent States](https://mlsanity.com/paper/2607.19531/) (2026)

## Abstract (arXiv)

The Trek Separation Theorem (Sullivant et al. 2010) states necessary and sufficient conditions for a linear directed acyclic graphical model to entail for all possible values of its linear coefficients that the rank of various sub-matrices of the covariance matrix is less than or equal to n, for any given n. In this paper, I extend the Trek Separation Theorem in two ways: I prove that the same necessary and sufficient conditions apply even when the generating model is partially non-linear and contains some cycles. This justifies application of constraint-based causal search algorithms such as the BuildPureClusters algorithm (Silva et al. 2006) for discovering the causal structure of latent variable models to data generated by a wider class of causal models that may contain non-linear and cyclic relations among the latent variables.