# Variance Minimization in the Wasserstein Space for Invariant Causal Prediction

arXiv: https://arxiv.org/abs/2110.07064 · HTML: https://mlsanity.com/paper/2110.07064/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 65 references that are in the map.
- First posted on arXiv 13 Oct 2021.

## Builds on (most cited first, 12 of 12)

- [Invariant Risk Minimization](https://mlsanity.com/paper/1907.02893/) (2019)
- [Causal inference using invariant prediction: identification and confidence intervals](https://mlsanity.com/paper/1501.01332/) (2015)
- [Causal Discovery with Continuous Additive Noise Models](https://mlsanity.com/paper/1309.6779/) (2013)
- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [CAM: Causal additive models, high-dimensional order search and penalized regression](https://mlsanity.com/paper/1310.1533/) (2013)
- [Invariant Causal Prediction for Nonlinear Models](https://mlsanity.com/paper/1706.08576/) (2017)
- [Permutation-based Causal Inference Algorithms with Interventions](https://mlsanity.com/paper/1705.10220/) (2017)
- [Permutation-Based Causal Structure Learning with Unknown Intervention Targets](https://mlsanity.com/paper/1910.09007/) (2019)
- [backShift: Learning causal cyclic graphs from unknown shift interventions](https://mlsanity.com/paper/1506.02494/) (2015)
- [Learning Causal Structures Using Regression Invariance](https://mlsanity.com/paper/1705.09644/) (2017)
- [Jointly interventional and observational data: estimation of interventional Markov equivalence classes of directed acyclic graphs](https://mlsanity.com/paper/1303.3216/) (2013)
- [Causal Dantzig: fast inference in linear structural equation models with hidden variables under additive interventions](https://mlsanity.com/paper/1706.06159/) (2017)

## Built on by (most cited first, 5 of 5)

- [Invariant Ancestry Search](https://mlsanity.com/paper/2202.00913/) (2022)
- [Wasserstein Geodesic Generator for Conditional Distributions](https://mlsanity.com/paper/2308.10145/) (2023)
- [Causal Discovery using Model Invariance through Knockoff Interventions](https://mlsanity.com/paper/2207.04055/) (2022)
- [Distributional Matrix Completion via Nearest Neighbors in the Wasserstein Space](https://mlsanity.com/paper/2410.13112/) (2024)
- [Efficient Identification of Direct Causal Parents via Invariance and Minimum Error Testing](https://mlsanity.com/paper/2409.12797/) (2024)

## Abstract (arXiv)

Selecting powerful predictors for an outcome is a cornerstone task for machine learning. However, some types of questions can only be answered by identifying the predictors that causally affect the outcome. A recent approach to this causal inference problem leverages the invariance property of a causal mechanism across differing experimental environments (Peters et al., 2016; Heinze-Deml et al., 2018). This method, invariant causal prediction (ICP), has a substantial computational defect -- the runtime scales exponentially with the number of possible causal variables. In this work, we show that the approach taken in ICP may be reformulated as a series of nonparametric tests that scales linearly in the number of predictors. Each of these tests relies on the minimization of a novel loss function -- the Wasserstein variance -- that is derived from tools in optimal transport theory and is used to quantify distributional variability across environments. We prove under mild assumptions that our method is able to recover the set of identifiable direct causes, and we demonstrate in our experiments that it is competitive with other benchmark causal discovery algorithms.