Topological Causal Effects

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 (12)

• Figure 1: Left: Example of untreated vs. treated macromolecule structures. Right: Corresponding persistence diagrams, highlighting treatment-induced changes in the 1st-order homology features.
• Figure 2: Silhouette functions revealing differences in $1$-dimensional features in the ORBIT dataset adams2017persistence. (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 $D$.
• 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.
• 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.
• 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.

Theorems & Definitions (22)

• Lemma 2.1: Lipschitz stability of the weighted silhouette
• Remark 1: Choice of $r$
• Remark 2: Sample splitting
• Theorem 5.1
• Example 5.1: Functional linear smoother
• Theorem 5.2
• Theorem 5.3
• Corollary 5.4
• proof
• proof