Topological Causal Effects

Kwangho Kim; Hajin Lee

stat.MEMar 2, 2026

Topological Causal Effects

Authors: Kwangho Kim, Hajin Lee

Organizations: Department of Statistics, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul 02841, Korea

Abstract

Estimating causal effects is particularly challenging when outcomes arise in complex, non-Euclidean spaces, where conventional methods often fail to capture meaningful structural variation. We develop a framework for topological causal inference that defines treatment effects through differences in the topological structure of potential outcomes, summarized by power-weighted silhouette functions of persistence diagrams. We develop an efficient, doubly robust estimator in a fully nonparametric model, establish functional weak convergence, and construct a formal test of the null hypothesis of no topological effect. Empirical studies illustrate that the proposed method reliably quantifies topological treatment effects across diverse complex outcome types.

Figures & tables

Panel of Figure 1 — Figure 1: Left: Example of untreated vs. treated macromolecule structures. Right: Corresponding persistence diagrams, highlighting treatment-induced changes in the 1st-order homology features.

Panel of Figure 2 — Figure 2: Silhouette functions revealing differences in $1$ -dimensional features in the ORBIT dataset ( Adams et al., 2017 ) . (a) Point clouds $A,B,C,D$ ; (b) their power-weighted silhouettes; (c) silhouette contrasts $\phi_{A}-\phi_{B}$ (top) and $\phi_{C}-\phi_{D}$ (bottom). Strong signals in $\phi_{A}-\phi_{B}$ indicate new $1$ -dimensional features in $A$ relative to $B$ , whereas near-zero $\phi_{C}-\phi_{D}$ shows little change between $C$ and $D$ .

Panel of Figure 3 — Figure 3: (a) CT-scan images of non-infected (top) and infected (bottom) patients; (b) corresponding 0-dimensional persistence diagrams; (c) average silhouettes of non-infected and infected patients given $r=0.1$ (top) and difference in average silhouettes between non-infected and infected patients.

Panel of Figure 4 — Figure 4: The true silhouette function and its PI, IPW, AIPW estimates. (a) 0-dimensional silhouettes computed from the SARS-CoV-2 dataset, (b) 0-dimensional silhouettes computed from the GEOM-Drugs dataset, and (c) 1-dimensional silhouettes computed from the GEOM-Drugs dataset.

Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

	SARS-CoV-2		GEOM-Drugs				ORBIT
	$\mathcal{H}_{0}$		$\mathcal{H}_{0}$		$\mathcal{H}_{1}$		$\mathcal{H}_{0}$		$\mathcal{H}_{1}$
	$L_{1}$ dist.	Std.	$L_{1}$ dist.	Std.	$L_{1}$ dist.	Std.	$L_{1}$ dist.	Std.	$L_{1}$ dist.	Std.
PI	4.544	2.712	3.215	0.740	7.041	1.283	0.571	0.530	36.063	37.704
IPW	2.936	9.035	1.670	5.756	18.886	8.337	0.073	3.722	18.210	93.978
AIPW	1.246	3.099	3.296	1.470	2.136	1.553	0.121	0.802	6.920	47.127

Appendix

Table 1:

L_{1}

distance between the average estimate and the true effect, together with the standard deviation of each estimator over 20 repetitions.

\mathcal{H}_{d}

denotes homology dimension

d

, and the scale is in 1e-05 for SARS-CoV-2 and ORBIT and 1e-02 for GEOM-Drugs.

Data (Sample Size)	SARS-CoV-2 (2481)	GEOM-Drugs (30433)	ORBIT (2000)
Runtime	244 seconds	21 second	25 seconds

Appendix

Table 2: The runtime of computing weighted silhouettes for each dataset in seconds.

Panel of Figure 5 — Figure 5: Visualization of the point-wise mean (dotted line) and the point-wise 1-standard deviation error bands (shaded area) of PI, IPW, and AIPW estimators on the SARS-CoV-2 dataset. The true topological causal effect is shown as a blue line.

Panel of Figure 6 — Figure 6: (a) Propensity score distribution by subgroup; (b–c) representative molecular graphs.

Panel of Figure 7 — Figure 7: Visualization of the point-wise mean (dotted line) and the point-wise 1-standard deviation error bands (shaded area) of 0-dimensional (top) and 1-dimensional (bottom) PI, IPW, and AIPW estimators on the GEOM-Drugs dataset. The true topological causal effect is shown as a blue line.

Panel of Figure 8 — Figure 8: 0-dimensional (left) and 1-dimensional (right) true silhouette functions and its PI, IPW, AIPW estimates for the ORBIT dataset.

Panel of Figure 9 — Figure 9: Visualization of the pointwise mean (dotted line) and the pointwise 1-standard deviation error bands (shaded area) of 1-dimensional PI, IPW, and AIPW estimators on the ORBIT dataset. The true topological causal effect is shown as a blue line.

Panel of Figure 10 — Figure 10: True silhouette and its PI, IPW, AIPW estimates for 0-dimensional homological features on the SARS-CoV-2 dataset when one of the nuisance functions is misspecified.

Panel of Figure 11 — Figure 11: True silhouette and its PI, IPW, AIPW estimates for 0- and 1-dimensional homolgical features on the GEOM-Drugs dataset when one of the nuisance functions is misspecified.

Panel of Figure 12 — Figure 12: True silhouette and its PI, IPW, AIPW estimates for 0- and 1-dimensional homolgical features on the ORBIT dataset when one of the nuisance functions is misspecified.

SARS-CoV-2 ( $\mathcal{H}_{0}$ )	$\pi$ is misspecified, $\mu$ is correct		$\mu$ is misspecified, $\pi$ is correct
SARS-CoV-2 ( $\mathcal{H}_{0}$ )	$L_{1}$ dist.	Std.	$L_{1}$ dist.	Std.
PI	1.390	3.132	7.985	3.132
IPW	9.674	4.421	2.243	4.328
AIPW	0.786	2.761	1.529	3.395

Appendix

Table 3:

L_{1}

distance between the average estimate and the true effect, together with the standard deviation of each estimator over 20 repetitions, when either of the nuisance functions is misspecified.

\mathcal{H}_{d}

denotes homology dimension

d

, and the scale is in 1e-05 for SARS-CoV-2 and ORBIT and 1e-02 for GEOM-Drugs.

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Beyond Means: Topological Causal Effects under Persistent-Homology Ignorability

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Open paper AskPDF 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.

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