# Beyond Means: Topological Causal Effects under Persistent-Homology Ignorability

arXiv: https://arxiv.org/abs/2603.14169 · HTML: https://mlsanity.com/paper/2603.14169/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 46 references that are in the map.
- First posted on arXiv 15 Mar 2026.

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

- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [Fr'echet Regression for Random Objects with Euclidean Predictors](https://mlsanity.com/paper/1608.03012/) (2016)
- [Robust Topological Inference: Distance To a Measure and Kernel Distance](https://mlsanity.com/paper/1412.7197/) (2014)
- [Kernel method for persistence diagrams via kernel embedding and weight factor](https://mlsanity.com/paper/1706.03472/) (2017)
- [The Weighted Euler Curve Transform for Shape and Image Analysis](https://mlsanity.com/paper/2004.11128/) (2020)
- [A Topological Perspective on Causal Inference](https://mlsanity.com/paper/2107.08558/) (2021)
- [Causal inference with a functional outcome](https://mlsanity.com/paper/2304.07113/) (2023)
- [Induced Matchings and the Algebraic Stability of Persistence Barcodes](https://mlsanity.com/paper/1311.3681/) (2013)
- [Doubly robust estimation of causal effects for random object outcomes with continuous treatments](https://mlsanity.com/paper/2506.22754/) (2025)
- [Causal Homotopy](https://mlsanity.com/paper/2112.01847/) (2021)
- [Topological Causal Effects](https://mlsanity.com/paper/2603.02289/) (2026)
- [Topological Residual Asymmetry for Bivariate Causal Direction](https://mlsanity.com/paper/2602.00427/) (2026)

## Built on by (most cited first, 1 of 1)

- [A Mathematical Framework for Topological Causal Data Analysis](https://mlsanity.com/paper/2607.28161/) (2026)

## Abstract (arXiv)

Average treatment effects (ATE) and conditional average treatment effects (CATE) are foundational causal estimands, but they target changes in expected outcomes and can miss treatment-induced changes in the shape of outcome distributions. A canonical failure mode occurs when control outcomes are unimodal, treated outcomes become bimodal, and both distributions have the same mean. In such cases mean-based causal estimands are zero even though the geometry and topology of the outcome law change substantially. This paper develops a topological causal framework based on persistent homology. We formalize a persistent-homology ignorability condition, define topological analogues of CATE and ATE, and prove that these estimands are identifiable up to an explicit error bound under approximate topological ignorability. We also clarify a subtle but important point: a marginal persistence-diagram effect is not identified from conditional topological ignorability alone because persistent homology does not in general commute with mixtures over covariates. To preserve the original intuition while ensuring scientific correctness, we retain the marginal effect as a motivating quantity, but place the mathematically sound conditional estimands at the center of the theory. A synthetic experiment with mean-preserving topology change shows that mean-based causal estimands remain near zero while the proposed topological effect increases sharply and remains recoverable after adjustment for confounding.