# A Mathematical Framework for Topological Causal Data Analysis

arXiv: https://arxiv.org/abs/2607.28161 · HTML: https://mlsanity.com/paper/2607.28161/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 40 references that are in the map.
- First posted on arXiv 30 Jul 2026.

## Builds on (most cited first, 13 of 13)

- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists](https://mlsanity.com/paper/1710.04019/) (2017)
- [Semiparametric doubly robust targeted double machine learning: a review](https://mlsanity.com/paper/2203.06469/) (2022)
- [Sliced Wasserstein Kernel for Persistence Diagrams](https://mlsanity.com/paper/1706.03358/) (2017)
- [Stochastic Convergence of Persistence Landscapes and Silhouettes](https://mlsanity.com/paper/1312.0308/) (2013)
- [Persistence Fisher Kernel: A Riemannian Manifold Kernel for Persistence Diagrams](https://mlsanity.com/paper/1802.03569/) (2018)
- [Kernel method for persistence diagrams via kernel embedding and weight factor](https://mlsanity.com/paper/1706.03472/) (2017)
- [A persistence landscapes toolbox for topological statistics](https://mlsanity.com/paper/1501.00179/) (2014)
- [A Topological Perspective on Causal Inference](https://mlsanity.com/paper/2107.08558/) (2021)
- [Topological Causal Effects](https://mlsanity.com/paper/2603.02289/) (2026)
- [Absolute average and median treatment effects as causal estimands on metric spaces](https://mlsanity.com/paper/2407.03726/) (2024)
- [Topological Residual Asymmetry for Bivariate Causal Direction](https://mlsanity.com/paper/2602.00427/) (2026)
- [Beyond Means: Topological Causal Effects under Persistent-Homology Ignorability](https://mlsanity.com/paper/2603.14169/) (2026)

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

- [Wasserstein Causal Forests for Distribution-Valued Outcomes](https://mlsanity.com/paper/2609.35898/) (2026)

## Abstract (arXiv)

Many modern outcomes, including images, point clouds, networks, and spatial fields, are structured objects for which (Y1-Y0) may be undefined or scientifically inadequate. We introduce Topological Causal Data Analysis (TCDA), a framework separating the observation space, causal-model class, topological representation, and causal query. Topology does not define interventions; it supplies stable, shape-sensitive summaries after causal assumptions have been specified. We distinguish outcome-level TCDA, which transforms individual potential outcomes, from distribution-level TCDA, which transforms interventional outcome laws, and characterize when outcome and distribution level contrasts agree. Building on recent outcome-level theory, we formulate identification and doubly robust representations for Banach-space-valued summaries. At the distribution level, we identify targets through the standard causal (g)-formula and derive stability-transfer bounds and plug-in consistency. We also place target-specific topological ignorability within the framework, clarifying when a covariate-standardized coarse effect can be identified without identifying the full interventional laws. Finally, we delimit the role of observational topology in causal discovery: it can assist diagnosis on restricted model classes but cannot by itself identify causal structure.