Topological Causal Effects
Authors: Kwangho Kim
Hajin Lee
Venue: ICLR 2026 Poster (2026)
Keywords: topological data analysis
causal inference
doubly robust estimator
persistence landscape
Silhouettes
persistent homology
Primary area: causal reasoning

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.

Published as a conference paper at ICLR 2026
TOPOLOGICAL CAUSAL EFFECTS
Kwangho Kim, Hajin Lee
Department of Statisties, Korea University 145 Anam-ro, Seongbuk-gu, Seoul 02841, Korea kwanghk, hlee2745] ( @korea. ac.kr
ABSTRACT
Estimating causal effects is particularly challenging when outcomes arise in com- plex, non-Euclidean spaces, where conventional methods often fail to capture meaningful structural variation. We develop framework for topological causal inference that defines treatment effects through differences in the topological struc- ture 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.
INTRODUCTION
Causal inference has become central tool across many disciplines, providing statistical framework for learning intervention effects beyond mere association. In the potential-outcome framework (Imbens & Rubin 2015), causal effects are defined by contrasting counterfactual outcomes, i.e., what would have happened under alternative treatment assignments. As modern scientific outcomes become increasingly complex, however, standard causal estimands and methods that rely on Euclidean summaries can fail to detect meaningful intervention induced changes in the underlying structure of the outcome. Despite this practical relevance, comparatively little methodological work targets causal inference for outcomes that are intrinsically non-Euclidean high-dimensional, and unstructured.
In this work, we focus on settings where scientifically salient intervention effects are ex expressed through changes in the outcome's 's topological structure, rather than through shifts in simple Euclidean summarics. Such effects arise naturally in the biomedical sciences, where treatments can change macromolecular conformations folding patterns (Kovacev-Nikolic al.) [016 Cang & Wei 2018 (002),, in neuroscience, where stimuli can reshape brain connectivity (Sizemore et al. 2019), and in signal processing and medical imaging, where the goal is to detect structural changes in dynamical systems or diagnostic images (Kim et al.) 2018 Gholizadeh & Zadrozny 2018).
Axelrod &
To formalize and estimate such effects, we introduce a class of causal estimands that quantify intervention-induceed changes in the topological structure of complex outcomes using tools from topological data analysis (TDA). TDA extracts robust, multi-scale geometric and topological descrip- tors from complex data (Carlsson 2020). We build on persistent homology, which summarizes how topological features (for example, connected components, loops, and voids) appear and disappear as the resolution parameter varies (Chazal & Michel] 2021). While TDA has been used improve predictive robustness under distribution shift and perturbations (e.g., Carrière et al., (2200] 2021 Kim (et al. (020), and has been incorporated as regularizer conventional average treatment effect pipelines (Farzam et al., (025). to our knowledge there is no existing framework that (i) defines causal estimand directly in terms of topological summaries and (ii) provides corresponding nonpara- metric estimation and inference. Our work fills this gap by defining topology-awar causal estimand and developing a corresponding efficient, doubly robust estimator with valid inference.
Illustrative example. We illustrate the proposed methodology using macromolecule dataset (xel rod & Gomez-Boon Bombarelli 2022), shown in Figure [ and revisited in Section 6 Each molecule is
"Corresponding author. Both authors contributed equally this work.

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Unt soted
Treated
Untroated
Treated
12:13
30r3
Figure 1: Left: Example of untreated vs. treated macromolecule structures. Right: Corresponding persistence diagrams, highlighting treatment-induced changes in the Ist-order homology features.
represented as a simplified graph, with graphs varying in size across samples. We consider hypothet- ical chemical treatment that induces additional loop-like structure the molecular geometry. Such changes can be difficult to detect using conventional Fuclidean summaries, but they appear clearly in firs-dimensional persistent homology, as evidenced by shifts in the corresponding persistence diagrams. We then convert these diagrams into functional summaries that are amenable to estimation and inference (Chazal ct al. 2014, Bubenik et al., 2013). This perspective connects our approach to recent work causal inference with functional outcomes (Bcker et al.) 2024 Testa et al. 2025). line of research that remains the early stages of methodological development, particularly for fully nonparametric estimation and inference.
Contributions. We introduce a new class of topological causal effects, defined through intervention- induced changes in persistent homology summaries and, in particular, the expected contrast of power-weighted silhouette functions under the potential outcomes. This target is designed to capture structural effects that are invisible scalar or Euclidean summaries, while retaining the stability properties of topological descriptors under perturbations. We develop an efficient, doubly robust, fully nonparametric estimator for the resulting functional estimand, obtain fast / rates under standard product-rate condition on nuisance errors, and establish weak convergence, enabling valid and tractable inference in fully nonparametric settings. We also derive new stability bounds for weighted silhouettes under Wasserstein perturbations of persistence diagrams and, by combining these bounds with our weak convergence result, construct a formal test of the null hypothesis of no topological effect with asymptotically valid size and consistency. Together, our approach broadens the scope of causal inference by enabling rigorous analysis of structural effects in complex systems and supporting causal investigations in diverse modern data modalities, illustrated in empirical studies.
2 PRELIMINARIES: TOPOLOGICAL TOOLS FOR MACHINE LEARNING
This section provides brief overview of foundational concepts in topological data analysis and introduces notation used throughout the paper. For further details, see Hatcher (202): Edelsbrunner
Harer (010): Chazal et al.) (2016) well as Appendix A
Simplicial complexes. To represent the shape of complex data from finite sample, we approximate the underlying object by simplicial complex, higher-dimensional generalization of graph. Given vertex set V, an (abstract) simplicial complex is collection K of finite subsets of V such that a E K and 7 C o, then т K. Each a E K called a simplex, its dimension is dim(r)= |a| 1, and the dimension of K is the maximum simplex dimension among its simplices. In particular, a 1-dimensional simplicial complex corresponds to a graph (possibly with isolated vertices). In upplications, simplicial complexes typically constructed from observed point clouds using standard rules such as the Victoris-Ripp complex the Čech complex.
Persistent homology and diagrams. A filtration is a nested family of simplicial complexes F (! t R} such that t < t' implies K(t) C K(t).. Filtrations are often generated by monotone filtration function f K - R satisfying f(T) < f(a) whenever т C a, by setting ((t) f ' (o), Persistent homology summarizes how topological features emerge and disappear as the filtration parameter t increases, separately for each homology degree 0, 2, (for example, d = 0 connected components,
loops, = 2 voids). We write H. ([()) for the
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d-th homology group at level t, and represent each feature by its birth time and death time b a; the resulting birth barth-death pairs (a. 6) form the persistence diagram. Formally, the d-th persistence diagram induced by IF is denoted DF(F) and is multiset of points in
R 2+ = {(a, b) E (R (RU (oo))) : :a< <).
When the filtration clear from context, we write Da for Da(F), and we use D to denote a generic persistence diagram.
Weighted silhouettes. Because is multiset, is often convenient to embed it into a function space. Given D, define for each p (ap. bp) E D the tent function Ap RR R.
Ap(t) (0, min ap. by +}}, E
(1)
where T C R is fixed compact interval. The weighted silhouette is the normalized weighted average of these tents,
(1; D) = EpeD up Apt) EPED Wp
1 LET,
where wp 2 0 is the weight corresponding to point and E EEED up > 0. A common choice is the power weight up (6p ap)" with r >> 0, yielding the power-weighted silhouette
(t; Dr = Eper(bp ap)' Ap(() Eped(b, - ap)
& E
(2)
Larger emphasizes longer- lived features, while smaller r assigns relatively more weight to short- lived ones. Since only controls which features are emphasized and does not affect identification, estimation, or inference, we suppress r the notation when is fixed.
The following lemma records basic regularity property of silhouettes that will be used in Section 5
Lemma 2.1 (Lipschitz stability of the weighted silhouette). For any 5 > 0,
(b D) (bt:D) s 6,
sup 56
and hence
jó(s; D) D) [ Sd.
E
sup S
Choice of filtration. Computing persistent-homolopg descriptors requires specifying filtration F (K(t)= t E RR. typically obtained by choosing a simplicial (or cubical) complex together with monotone filtration function f K - IR and setting K(t) = ((-oo f)). The appropriate construction depends on the data modality and the scientific notion of scale one wishes to probe. For point-cloud data, Vietoris-Rips and a-filtrations such as digital images, sublevel or superlevel filtrations on cubical complexes are natural. For graph data, one use clique filtrations, the persistent homology transform (PHT) (Turner et al., 2014)). These standard options allow our framework to accommodate a broad range of complex outcome types by pairing each modality with an appropriate filtration. Concrete definitions of several standard constructions (Vietoris-Rije Rips, a-, and cubical filtrations) are collected in Appendix
common choices; for grid-structured data
3 FRAMEWORK
Let (Z, = (X,, A,, denote an i.i.d. observed sample. Here, A; A {0,1} is binary treatment (intervention) indicator and X; E X C R is an l-dimensional covariate vector. Let F, denote the filtration of simplicial complexes constructed from Y,, and let F7" denote the corresponding potential (counterfactual) filtration that would be constructed from the potential outcome Y," under treatment A; = a. For each homology degree d, define the observed and potential persistence diagrams by D,, == Da(Fi) and Dr, Da(F.". Also, let (t)
De.a) and (g,a(t) = @tt; D for E T. denote the corresponding power-weighted silhouettes in equation the domain of definition of Ø.
where T is

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Weighted Silhouette
Difference between Silhouettes
A
Silhoutte
Silhouette B
Point cloud
Point cloud
Silhouette
Silhorette
Silhowette
Point cloud
Point cloud
70
(b)
(c)
Figure2 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 A &g (top) and Φc &D (bottom). Strong signals in Φ 0g indicate new 1-dimensional features in A relative to B, whereas near-zero de ΦD shows little change between C' and D.
Our target parameter of interest is the lopological average treatment effect (TATE), For each fixed d, define the functional causal effect
(t)()) = E Eio -o0 dga(t)}} E
Equivalently, for a chosen maximal homology degree dmaxx we may stack these components into the Raauut--valued curve (t) (1ho (t), ddm... (t)). V4 T, that ria(t) is the dth coordinate of (t). Each function vaa captures the treatment-inducen change in d-dimensional topological features at filtration scale (ta > indicates that, on average, treated units exhibit more or stronger d-dimensional features at scale t, while cia(t) < 0 indicates attenuation under treatment. Thus, the mapping & > v/((t) is functional causal effect in a Hilbert space, tracing how treatment alters topology across filtration scales.
There are several compelling advantages interpreting equation 3 as greament-induceed topological effects. First, (ta is inherently scale-aware: the silhouette is indexed by the filtration parameter I, thereby retaining information about topology across geometric scales, thus allowing (t((t) localize reatment-induuee changes in topology across geometric scales. Second, it is robust to noise: power- weighted silhouettes downweight short-lived, potentially spurious features, so (a((t emphasizes persistent structural differences that are more likely to be meaningful. Third, unlike raw persistence diagrams, (t) is vectorizable, in the sense that it lives in a separable Hilbert space, which facilitates theoretical analysis and integration into gradient-base based machine learning pipelines. Fourth, Va(t) naturally fits within recent advances in functional causal inference. Collectively, tha(t) can be viewed topology-aware analogue of the average treatment effect, capturing how intervention reshapes the d-dimensional homological structure of the underlying outcome across treatment groups
One limitation that silhouettes can obscure the exact number of changing homological features, since they aggregate multiple tent functions into single weighted summary. In principle, however, the TATE can also be defined using individual persistence landscape functions across homology degrees up to chosen level, which may provide finer topological resolution. Nevertheless, representing each diagram by a single silhouette curve offers a simple, interpretable summary of overall topological variation within a given homology degree, and the power parameter r gives transparent way to emphasize the most persistent features when desired.
Although one could apply standard average treatment effect estimators after vectorizing each persis- tence diagram into an ad hoc Euclidean feature vector, such heuristics generally lack principled link to the underlying topological object. Our approach is fundamentally different: we define the causal estimand directly in terms of persistence- -iagram geometry and use the silhouette as functional embedding obtain well-posed target in Hilbert space. This yields coherent notion of topological causal effect, along with doubly robust estimation and valid inference that are explicitly aligned with this target.
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Figure 2 illustrates how the target parameter equation captures structural variation. The point-cloud pairs (A, B) and (C, D) in Figure 2a represeent potential outcomes under treatment and control. The treatment induces loop in A relative B, producing pronounced silhouette contrast in Figure 2c whereas the contrast for (C, D) is close to zero, indicating negligible structural change. The sign of the silhouette difference curve encodes the direction of topological change: positive values correspond to features that emerge under treatment, while negative values correspond to features that disappear relative to control.
For identification, we invoke the standard causal assumptions: (C1) Consistency, Fi Fr whenever A, a; (C2) No unmeasured confounding. A, 1 (t) X;; and (C3) Positivity, P(A, X,) > almost surely, for all T,, E A, and homology degree d. Despite the increased complexity of our target parameter, the identification conditions closely parallel those in standard causal inference settings (e.g., [imeens & Rubin 2015). We assume that Assumptions (CI) hold throughout the paper. Note that component-wis identification is sufficient for identifying the functional effect (t))) at each filtration scale t. Then, for each d, the target (t((t (t) is identified as
(ta ([E(or,(()) X1, A; = 1 B(dra(!) |,, A, 0}] (1- (r.((t 1 (X,)
(4)
A (i,(t)
E
(5)
(X,))
The identifying expressions above motivate the use of plug-in and inverse probability weighting estimators, which will be discussed in greater detail in the following section.
Remark (Choice of r). The TATE (b((t) depends the power parameter r, which controls how strongly the silhouette emphasizes long-lived versus short-lived persistence pairs. We view as pooblem-specifee tuning parameter: practitioners may select it using domain knowledge (e.g., larger r to upweight more persistent features), or by simple data-drives criterion, such as checking the stability of Va over candidate values of (for example, via validation step within treatment arm or via cross-ved validatii... In our experiments, the qualitative shape and statistical significance of the estimated effects remained stable across modest range of r values, suggesting limited sensitivity to its precise choice.
ESTIMATION
In this section, we develop an estimation strategy for vaa defined in equation 3 Hereafter, when no ambiguity arises, we omit the subscript indexing the i-th observation and write generic random variables in place of their sample realizations. For convenience, define the propensity score and the conditional silhouette regression functions by
" (④x = P(A= 1 x x),
Ma (t, = (da(t)) x -
Let âे and Ao denote estimators of π and Max respectively. We write P for the empirical distribu- tion used to fit the nuisance estimators (#, Po. (1), and we assume access to separate empirical distribution P., independent of P. used to construct the final estimator.
Motivated by the identifying expressions in equation I and equation 5 we propose the plug-in (PI) regression estimator and the pworre--robabiiitty-- weighttng (IPW) estimator
(P), Pr. Pr -000))Aoof) -oo X: d)}
(6)
VIrw.a (t)) = =Pr { A A) (b(l))
*(X) 1 1- #(X)
The estimators VPL. and VIPw.d inherit their convergence rates from those of f. and म. respectively. In many observational studies, the IPW estimators can appealing because modeling the treatment assignment mechanism is often more feasible than modeling the full conditional mean of a functional outcome (Imbens & Rubin 2015). In our setting, VIPw.a a be particularly convenient because estimating the functional regression Pl (a(:s (+s:3d) may be substantially more challenging than estimating the scalar propensity score π.
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A more efficient estimator can be constructed using semiparametric efficiency theory. Define the uncentered efficient influence function (EIF) by
A
1 1-A
(p,lt 2;n) = pn (1, x; - poll, x;d) +
1 x(x)
(da(1)))a(( -M X;d); (8)
((X)
1-
where 1) = { Mo, /1 collects the nuisance functions. By construction, [E(4a((, 7; (ta(t) for each & E T. The influence-function representation yields augmented IPW estimator,
T AIPw (.,(t 7) =P.((a(tt);
where 7 = nP) {ft Pa. } is estimated using the nuisance- -rrannnnn sample P.. This construction yields the usual second-order remainder structure and the familiar double-robust behavior, allow- ing VAIPw d to attain fast rates under weak nonparametric conditions on the nuisance estimators (Kennedy 2016 2024).
There are two standard approaches for establishing asymptotic properties of augmented inverse- estimators. One approach to impose empirical-proces, conditions (e.g.. Donsker or suitable entropy conditions) on the indexed class of influence functions (tpa(t, (;)) t tE T). along with corresponding regularity of the nuisance estimators, but such conditions can be overly restrictive for modern, highly adaptive nuisance learning. An alternative is sample splitting, which allows arbitrarily complex nuisance estimators by separating nuisance fitting from the final debiasing step, thereby mitigating overfitting concerns. In principle, we remain agnostic about how the nuisance functions are learned. For clarity, however, we adopt a sample-splitiin setup throughout: the nuisance estimators are fitted on P, and the final estimator is constructed on an independent sample Pn, (See Assumption (A2))
probability-weighee
Remark 2 (Sample splitting). Independent samples Pn, and P can be obtained random data splitting. Full-sample efficiency can be recovered via cross-fitting (e.g.. Chernozhukov al. 2018 Newey & Robins| [018)) For clarity of exposition, we work with a single split, though extending to Kennedy et al. 2023).
multiple splits is straightforward (e.g.. Kennedy 2019 2023
5 INFERENCE
Conditions guaranteeing desirable large-sample properties, such as /n-raae convergence and asymp- totic normality, well understood for average treatment effect estimation with scalar outcomes (e.g., Kennedy (0016) 2024).. In our setting, the outcome is a curve indexed by the filtration scale, so valid inference requires additional structure control the behavior of the resulting stochastic process.
We begin with simplified setting in which the treatment assignment mechanism is fully known, as in randomized experiments. In particular, we assume that the true propensity score i is known almost surely. In this case, the IPW estimator in equation 7 is unbiased.
To obtain weak convergence of the entire estimated effect curve in (TT, however, is not enough to assume only pointwise second moments; one also needs a functional central limit theorem for the indexed empirical process (Ea(-.t) E T}. This can be ensured, for example, by assuming that the indexed class is P-Donsker (or by standard intropy-and-enved measurability). In our setting, these conditions are plausible because the silhouette (a((t) is Lipschitz in the index t (see Remark in Appendix C.2). The following theorem shows that, when is known, d converges weakly to Gaussian process in 8or (00
conditions with suitable
(T) under mild conditions.
Theorem 5.1. Fort E T, define Ea(Z,t) =
-(x] I-x(XT] Φa(t), where π is known. Assume that suptet var(Ex (2, t)) < oc and that the class (Ea(:.t=: : E T} is F-Donsker (or satisfies any other standard sufficient condition for a functional central limit theorem go0 (T)). Then
Vn (iPP,,..()) - va(tt}} ~ Ga(t) in (T).
where Ga is a mean-zero Gaussian process with covariance function cor(Ga(s),Gate cov cov(Ea(Z, s). t)), Vs, e T.
In observational settings, it is often preferable to use the AIPW estimator defined in equation 9 Hereafter, we use the notation || |r.g to denote the (IP))mom. To analyze the asymptotic properties, introduce the following set of assumptions.

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(A1) ||1/ < 0o and |1/(1 - < oc.
(A2) Sample splitting: Nuisance estimates îj are fit on P. and the final estimator is computed on an independent sample P., (I,, P).
(A3) Cross-sectional consistency: l1 - x(())lp; = (I).
(A4) Rate condition on nuisance estimation: (|t (e((X)))(()(M))))) (X)]lpp { [EEE (laal,, Pa(txX:d)leea)
X:
on(n Vt E T.
(A5) Uniform consistency: For all x E X, Ma (t, xz d) is uniformly Lipschitz in on T, and following conditions hold:
sup Pa (a 1; - pa(t, o1(,), sup [(x) x(x) oz TTT
sEA
(A5')
Lipschitz modulus for Pa: For 8 and a E A, (;(;;;d) satisfies E sup e Ma(s, X;d) (allt x; Lppd, for some positive constant T.g.
Assumptions (AT) (A4) are standard in semiparametric causal inference (e.g.. Kennedy 2019
Kennedy et al. [2023] Kennedy (024). (AI) is a mild boundedness condition, while (A2) separates
nuisance fitting from the final empirical average and thereby avoids empirical-pren proccsss restrictions on
the nuisance learners. [A3)enforcee enforces L2 (IP) consistency of the estimated influence function at each f,
and (A4) ensures that the second -oder von Mises remainder is on 11/2).
For the functional nuisance Ag ( d), we consider two alternative regularity conditions. Assumption (A5) adopted in Testa et al. (2025). imposes a Lipschitz modulus directly on the random sample paths of Na by requiring an L,1 bound on sup) 56 Ma (s, d; - (t, X; d)/. In contrast, Assumption (A5) requires only that the target regression (a(:; dd) is uniformly Lipschitz / and that Aa converges uniformly to Ma. In fact, under a mild integrability condition such as [sSupreT (t, X;d)
Ho (t, o(1,, I we have
(x, X: - Aall, (x(l,,xd)) x:di]] s Iy 8 o(1)
T
sup 18-1/28
so Ar need not be smooth itself, provided converges uniformly to Lipschitz limit.
Depending how iag is learned, the regularity requirements in Assumptions ( (A5)andd (A5((A))) can often be relaxed, and in some cases can be completely avoided, as illustrated by the following example.
Example 5.1 (Functional linear smoother). Consider a functional linear smoother for pa (e.g., Reiss
al. 2017): suppose that, for each d and E {0, 1}. the fitted regression admits the representation [t, X;d) = Σÿ=1 (X) @,,,(t), t T, where Φ,,((t) = @t;; D are the training silhouettes and La.g5( X) are (possibly data-aulptive) weights determined by the smoother, evaluated at X. Then,
for any 5 >
sup
Pa (: - aaa x:d] $
La (X) sup (0),((s) - )5,((t]]
555
10-1/56
< 8 EE
Las (ay(
==1
< co, then Assumption (A5') holds with La = E{E"_ La,; (X))). This verification relies only on moment control of the smoothing weights and the intrinsic Lipschütz property of silhouettes. Consequently, for this class of estimators the much simpler primitive condition E(Σ;-, [Xa]] < 0o. Ordinary least squares is a notable special case.
where the last inequality follows from Lemma 2.1 In particular,
may dispense with Assumption [A5] replace Assumption (A5') by
The following weak convergence result is the main technical ingredient for our inferential framework.
In Appendix C.3 we establish the result under Assumption (A5') Appendix C.4 then shows that the same conclusion holds under Assumption (A5)
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Theorem 5.2. Under Assumptions (AI)
A4)
and either (AS') or (A5) we have
/ ((1) va(()) ~ Ga(l) in 8x (T),
where Ga is a mean-zero Gaussian process with for all s,t T.
Ga(t)) = corlpals, Z:n), palt, 2;n)}
Pointwise vít-consisennyy and asymptotic normality are immediate corollaries of Theorem 5.2
In particular, for each fixed T. (AIPw,a(t) attains the semiparametric efficiency bound with asymptotic variance (ar(+ n:n) }. and achieves yn rates whenever the nuisance errors satisfy a product-rate condition (for example, each converging at 0re(1 -1/4) in (P))).. In contrast, the PI and IPW estimators typically incur first-order bias terms driven by jaa Na and # R, respectively, and thus generally require substantially stronger nuisance rates to achieve v /nn-consistent inference under flexible nonparametric learning (Kennedy 2024). In Appendix B we further provide an explicit bias-variance analysis of the AIPW, IPW, and Pl estimators under lixed deviations of fta and क from their true values, demonstrating the doubly robust behavior of the AIPW estimator under nuisance misspecification.
Building on Theorem 5.2 we can construct simultancous confidence bands over T. There are several approaches to constructing asymptotically valid 1 confidence band, depending on the strength of additional assumptions one willing impose or the level of computational complexity one is prepared to accommodate. For instance, one may apply the pivotal method proposed by Liebl & Reimberr (023), or adopt the parametric bootstrap approach developed by Pini & Vantini (2017).
For more detailed discussion and comparison, see Testa et al. (2025).
For many, if not most, practitioners, the primary inferential goal is to determine whether there is any topological effect all, as captured by persistent homology. Hypothesis tests based on estimated, vectorized topological summaries (c.g., silhouettes, persistence images, or landscapes) generally do not yield valid inference in the metric space of persistence diagrams. In contrast, our framework furnishes a formal test of the null of 'no topological effect'. To this end, we first establish stability bounds for weighted silhouettes, which, our knowledge, have not previously appeared in the literature.
Theorem 5.3. Let (D, D') denote the q- (-Wasserstein distance between two persistence diagrams D and DD' and let m² denote the corresponding optimal W1 matching. Assume that
2 its corresponding persistence diagram 'D is bounded such that for all (ap. bp) E 'D -o < ap S bp < co. Moreover, there exists
(A6) For the power-weighted silhouette defined in equation
global constant L such that, for all p E D and given weighting exponent T, Tbp-ap) The same holds for D' as well.
by by- S L.
Consider weighted silhouette functions & and &' corresponding D and D'. Under Assumption (A6)
| 0'11x0 S (1 2Lrcc=")) (D,,,'),
for some constant e that depends only on r and an upper bound on the persistences in the diagrams D, D'.
Appendix C.5 provides the formal definitions of the Wasserstein distance, the associated optimal matching m". and the constant c. Building upon the stability result above and the Gaussian weak convergence of our estimator, our framework provides, to our knowledge, the first formal test of the null hypothesis of no topological effect, with asymptotically correct size and consistency against fixed alternatives, as formally stated below.
Corollary 5.4. Fix dl No and consider the null hypothesis
Ho
Wi WD) D2) a.s.,
under which va(()= = for all t E T. Let Tn := /n TiAAIw,allos Under Assumptions (AI) (A6) we have Tn ~* ||Gall where Ga the Gaussian limit in 7heorem|5.2 Assume moreover that Gaussian- or Rademacher-multiplhe consistently estimates the law of ||Gal conditionally on the data. Let ci-a be the conditional (1 - a) qu-quantile of ||Gn,allo. Then the test that rejects Hg when T,, > Ci-a has asymptotic size a and is consistent against any fixed alternative with l/walloc dc >> O.
function bootstrap process Gn,d based on the estimated function
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Noninfected
Noninfected
Infected
(b)
Figure 3: (a) CT-scan images of non-infected (top) and infected (bottom) patients; (b) corresponding O-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.
Under Ho. T, = www.wairrw.al behaves like the supremum norm of the centered empirical- process fluctuation and converges |/Galloo. Under any fixed alternative with ||va|] T, diverges to +o0, so the test rejects with probability tending to one. Because the limit law of Tn /
depends on the unknown distribution of the Gaussian process Ga, assume a valid multiplier bootstrap so that the conditional (1 a -qqanttee of ||Gal can consistently approximated from the data, yielding asymptotically correct critical values.
EXPERIMENTS
To demonstrate the effectiveness of our method, carry experiments two semi-synthetic datasets and one synthetic dataset; results for the synthetic (ORBIT) data are deferred to Appendix [D.4 for completeness. For all experiments, potential outcome pairs are designed to exhibit distinct topological contrasts across treatment groups. We generate the covariates X R from multivariate Gaussian distribution, imposing a subgroup structure by specifying different mcan vectors for each subgroup. Given X, treatment A is assigned with probability л (X) = c This treatment mechanism is designed such that one subgroup has higher probability of receiving treatment than the other. We model the silhouette regression function Mh using function-on. on-scalar regression with a Fourier basis expansion, while the propensity score is estimated via random forest classifier. Our goal is to estimate the true topological causal effect based on the observable data. All experiments are repeated over 20 simulations. For complete details of the experimental setup, see Appendix D
construct a hypothetical dataset (X, A, Y yt), where the
0. 1.2 0.6 O 0.1
0.7.X, X3).
SARS-CoV-2 Dataset. The first semi-synthetic experiment uses image, possibly in different sizes, with synthesized covariates and treatment assignments while retaining real outcomes, allowing controlled yet realistic evaluation of causal estimators. We employ the BARS-CoV-2 dataset (Soares) et al. 2020), which contains CT-scans collected from real patients who are infected or non-infected by COVID-19. Infected patients exhibit high rates of ground-glass opacities and consolidations that appear as isolated regions in CT-scans, which can be captured by O-dimensional persistence diagrams of Lower-star filtration (Igbal al.) (025). In Figure and 3b] the difference between an infected and non-infected CT scan image is reflected in the associated persistence diagrams. Figure 3c (top) illustrates the average silhouettes infected and non-infected patients, where non-infected group exhibits higher values in [0.2, 0.4] and the infected group exhibits higher values in [0.7, 0.9]. Thus, treatment can assumed be more effective when TATE exhibits larger magnitudes over the interval (0.2, 0.4] (positive direction) and |0.7, 0.9] (negative direction), illustrated in Figure 3c (bottom). In this experiment, the true TATE known, as we manually construct (Y²9, Y by assigning 500 infected samples to y and subsequently pairing with Y which consist of 75% non-infected and 25% infected samples. We compute the PI. IPW, and AIPW estimators from the observed data and compare their estimates with the true effect.

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Silhoustte of Kn
Silhoustte of
Silhaustte of Hi
PW
UPN
AFN
4PM
(b)
(c)
Figure 4: The true silhouette function and its PI, IPW, AIPW estimates, (a) O-dimensional silhouettes computed from the SARS-CoV-2 dataset, (b) O-dimensional silhouettes computed from the GEOM- Drugs dataset, and (c) 1-dimensional silhouettes computed from the GEOM-Drugh dataset.
The blue curve Figure 4a shows the true silhouette, which by design clearly reflects a causal effect. Although all three estimators reasonably capture the overall shape of the true target, the IPW estimator systematically overestimates the true treatment effect, whereas the PI estimator underestimates it. In contrast, the AIPW estimator provides accurate reconstruction of the true silhouette function, achieving minimal bias and closely matching the exact shape the ground truth.
Results.
GEOM-Drugs Dataset. Next, evaluate framework graph data through semi-synthette experiment with the GEOM-Drugs dataset (Axelrod & Gomez- Bombarellij 2022), which provides graph-structured representations of molecular compounds. To analyze graph-structured data, first assign each graph node a scalar weight via a weighted sum of its features, and subsequently define each edge weight as the maximum of the weights of its adjacent nodes; we then apply a sublevel set filtration. Adopting procedures analogous to the previous experiment, we construct (Y"), y by taking y to consist of 1000 single-loop graphs and pairing it with Y which is mixture comprising 75% two-loop graphs and 25% single-loop graphs. As before, we compute the PI, IPW, and AIPW estimators from observed data and compare their estimates to the true effect.
Results. All three estimators closely align with the true O-dimensional silhouette (Figures 4b which exhibits a negative effect, implying that some connected components were merged under treatment. The true I-dimensional silhouette (Figure 4c) shows pronounced positive effect, indicating that treatment induces the formation of new loops, consistent with our experimental design. However, the IPW estimator overestimates the true I-dimensional treatment effect, while the PI estimator fails to capture the relatively complex curvature. Overall, the AIPW estimator delivers the most accurate and reliable approximation of the true silhouettes, consistent with the theoretical results in Section 5
DISCUSSION
We introduce a novel connection between causal inference and TDA, enabling estimation of causal effects that capture not only shifts in mean variance but also changes in the meaningful intrinsic topological structure the outcome space. This broadens the scope of causal inference and opens new avenues for investigating causal mechanisms complex, high-dimensional settings, including those involving unstructured data.
Several caveats and prospective solutions deserve attention, highlighting fruitful directions for future investigation. First, the proposed TATE framework designed to capture macroscopic topological shifts and may be less informative when the primary interest lies in detecting fine-grained, local changes. In such cases, standard causal estimands could be estimated in parallel if possible, potentially after appropriate preprocessing. Relatedly, as discussed Section 3 silhouette functions do not exactly quantify the number of changing homological features. Nevertheless, the proposed estimators and the analyses in Sections 4 and 5 exteend naturally to individual persistence landscape functions, offering finer topological resolution when required. Lastly, the construction of our estimators can be computationally intensive due to the use of persistent homology. This cost may be mitigated by adopting more efficient topological summaries, such Euler characteristic curves (Turner et al.) 2014). Additional extensions include adapting the framework to more complex causal settings, such continuous treatments, instrumental variable designs, longitudinal exposures.
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CODE AVAILABILITY
Python code implementing the proposed methodology is available at htttps://github.cm com/ Kwangho-
ACKNOWLEDGEMENTS
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (Mo. RS-2022-NR068754, RS- AS-2024-00335008, RS-2025-245344590 by Korea University Grant (K2304801), and by Samsung Science and Technology Foundation under Project Number SSTF-BA2502-01.
REFERENCES
Henry Adams, Tegan Emerson, Michael Kirby. Rachel Neville, Chris Peterson, Patrick Shipman, Sofya Chepushtanova, Eric Hanson, Francis Motta, and Lori Ziegelmeier. Persistence images: A stable vector representation of persistent homology. Journal of Machine Learning Research, 18(8):
1-35, 2017.
Simon Axelrod and Rafael Gomez-Bombarelli Geom, energy-annotated molecular conformations for property prediction and molecular generation. Scientific Data, 9(1):185, 2022.
Patrick Billingsley. Convergence of probability measures john wiley & sons. INC, New York, 2(2.4), 1999.
Peter Bubenik et al. Statistical topological data analysis using persistence landscapes. J. Mach. Learn. Res., 16(1):77-102, 2015.
Zixuan Cang and Guo- Wei Wei. Integration of element-specific persistent homology and machine learning for protein-ligame binding affinity prediction. International Journal for Numerical Methods in Biomedical Engineering, 34(2)=2914, 2018.
Gunnar Carlsson. Topological methods for data modelling. Nature Reviews Physics, 2(12):697-708, 2020.
Mathieu Carrière, Marco Cuturi, and Steve Oudot. Perslay: A neural network layer for persistence di- agrams and new graph topological signatures. In International Conference on Artificial Intelligence and Statistics, pp. 8796, 2020.
Mathieu Carrière, Ulrich Bauer, and Steve Oudot. Optimizing persistence diagrams using differen- tiable topological layers. International Conference on Machine Learning, 2021.
Frédéric Chazal and Bertrand Michel. An introduction to topological data analysis: fundamental and practical aspects for data scientists. Frontiers in artificial intelligence, 4:108, 2021.
Frédéric Chazal, Brittany Terese Fasy, Fabrizio Lecci, Alessandro Rinaldo, and Larry Wasserman. Stochastic convergence of persistence landscapes and silhouettes. In Proceedings of the thirtieth annual symposium on Computational geometry, geonmttrry pp. 474 483, 2014.
Frédéric Chazal, Steve Y. Oudot, Mare Glisse, and Vin De Silva. The Structure and Stability of Persistence Modules. Springer Verlag, 2016.
Chao Chen and Michael Kerber. An output-sensitive algorithm for persistent homology. In Pro- ceedings of the twenty-sevnnth annual symposium on Computational geometry, pp. 207-216. 2011.
Victor Chernozhukov, Denis Chetverikov, Mert Demirer, Esther Duflo, Christian Hansen, Whit- ney Newey, and James Robins. Double/debiased machine learning for treatment and structural parameters. The Econometrics Journal, 21(1)CC1-668 01 2018.
11

---

Published as a conference paper at ICLR 2026
Miroslav Dudík, John Langford, and Lihong Li. Doubly robust policy evaluation and learning. In Proceedings of the 28th International Conference on International Conference on Machine Learning, pp. 1097-1104, 2011.
Kreske Ecker, Xavier de Luna, and Lina Schelin. Causal inference with functional outcome. Journal of the Royal Statistical Society Series C: Applied Statistics, 73(1):221-24) 2024.
H. Edelsbrunner and J. Harer. Computational Topology: An Introduction. Applied Mathematics. American Mathematical Society, 2010. ISBN 978082184925.
Herbert Edelsbrunner and Dmitriy Morozov. 26 persistent homology.
Amirhossein Farzam, Ahmed Aloui, Vahid Tarokh, and Guillermo Sapiro. Topology-awar robust representation balancing for estimating causal effects. In High-dimensional Learning Dynamics 2025, 2025.
Seyedehsamaneh Gholizadeh and Bianca Zadrozny. short survey of topological data analysis in time series and systems analysis. arXiv preprint arXiv:1809.10445 2018.
A. Hatcher. Algebraic Topology. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401.
Guido W Imbens and Donald B Rubin. Causal inference in statistics, social, and biomedical sciences. Cambridge University Press, 2015.
Sohail Iqbal, Hafiz Fareed Ahmed, Talha Qaiser, Muhammad Imran Qureshi, and Nasir Rajpoot. Classification of 19 via homology of ct-scan. Computers in Biology and Medicine, 193: 110226, 2025.
Edward H Kennedy. Semiparametric theory and empirical processes in causal inference. Statistical causal inferences and their applications in public health research, pp. 141-167, 2016.
Edward H Kennedy. Nonparametric causal effects hased on incremental propensity score interventions. Journal of the American Statistical Association, 1 114(526):645-536 2019.
Edward H Kennedy. Towards optimal doubly robust estimation of heterogeneous causal effects.
Electronic Journal of Statistics, 17(2):3088-309) 2023.
Edward H Kennedy. Semiparametric doubly robust targeted double machine learning: a review. Handbook of Statistical Methods for Precision Medicine, pp. 207-236, 2024.
Edward H Kennedy, Sivaraman Balakrishnan, and Max G'sell. Sharp instruments for classifying compliers and generalizing causal effects. 2020.
Edward Kennedy, Sivaraman Balakrishnan, and LA Wasserman. Semiparametric counterfactual density estimation. Biometrika, 1 110(4):875889) 2023.
Kwangho Kim, Jisu Kim, and Alessandro Rinaldo. Time series featurization via topological data analysis. arXiv preprint arXiv: / 812.02987, 2018.
Kwangho Kim, Jisu Kim, Manzil Zaheer, Joon Kim, Frédéric Chazal, and Larry Wasserman. Pllay: Efficient topological layer based on persistent landscapes. Advances in Neural Information Processing Systems, 33:15965-1559 15977,, 2020.
Violeta Kovacev-Nikolic, Peter Bubenik, Dragan Nikolić, and Giseon Heo. Using persistent homology and dynamical distances to analyze protein binding. Statistical applications in genetics and molecular biology, 15(1):19-38, 2016.
Dominik Liebl and Matthew Reimhen. Fast and fair simultaneous confidence bands for functional parameters. Journal of the Royal Statistical Society Series B: Statistical Methodology, 85(3): 842-868. 2023.
Whitney K Newey and James R Robins. Cross- Cross-fitting and fast remainder rates for semiparametric estimation. arXiv preprint XXiv:1801.00 2018.
12

---

Published as a conference paper at ICLR 2026
Alessia Pini and Simone Vantini. Interval-wise testing for functional data. Journal of Nonparametric Statistics, 29(2):407-422 2017.
Philip T Reiss, Jeff' Goldsmith, Han Lin Shang, and R Todd Ogden. Methods for scalar-on-function regression. International Statistical Review, 85(2):228-22 2292, 2017.
Ann E. Sizemore, Chad Giusti, and Danielle Bassett. Importance of the whole: Topological data analysis for the network neuroscientist. Network Neuroscience, 3(3):656 673, 2019.
Eduardo Soares, Plamen Angelov, Sarah Biaso, Michele Higa Froes, and Daniel Kanda Abe. Sars-cov- ct-scan dataset: A large dataset real patients ct scans for sars-cov-2 identification. MedRxiv, pp. 2020-04, 2020.
Lorenzo Testa, Tobia Boschi, Francesca Chiaromonte, Edward H Kennedy, and Matthew Reimherr. Doubly-robust functional average treatment effect estimation. arXiv preprint arXiv:2501.06024, 2025.
The GUDHI Project. GUDHI User and Reference Manual, GUDHI Editorial Board, 3.4.1 edition, 2021. URL http:t:::///udh inria...i...........i....i.. f 2t/doo/3..4.1
Katharine Tumner, Sayan Mukherjee, and Doug M Boyer. Persistent homology transform for modeling shapes and surfaces. Information and Inference: A Journal of the IMA, 3(4):310-34 344,, 2014.
Aad W van der Vaart. Semiparametric statistics. In Lectures on probability theory and statistics (Saint-Flour, 1999), pp. 331-457. Springer, 2002.
Aad W Van Der Vaart and Jon Wellner. Weak convergence. In Weak convergence and empirical processes, pp. 16-28. Springer, 1996.
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APPENDIX
A STANDARD FILTRATION CONSTRUCTIONS AND RELATED DESCRIPTORS
This appendix provides concrete definitions and examples of the filtration constructions, as well as related topologiaal summaries referenced in Section 2
(Choice of filtration).
Vietoris-Rips complex. Let X be finite set of points equipped with distance d (for example, the Euclidean distance when X C !R", For r > 0, the Vietoris-Ripp complex at scale r is the simplicial complex
Rips(r) = {o x:: d(u, u') S 2r for all u, u' E o).
If r1 S 12, then (rps(r1) S Rips(r2), so (Rips(r) : 2 0} forms a natural filtration. a-complex. Let X C R be finite point set. For each X, let V, denote the Voronoi cell of u,
Vu = {a R= ||2 ull S || u' for all u' E x
and let (r) for E R! ||- ull s r be the closed ball of radius centered at Define the truncated ball (r.(r) = (r) n V. The a-complex scale is
Alpha(r) { fo c (R.(+)
uEO
As increases, Alpha(r) is nested, yielding a filtration (Alpha(r) : 2 0}.
Cubical complexes. A cubical complex is an analogue of a simplicial complex built from axis-aligned cubes, and well suited for grid-structured data such as digital images. An elementary interval is an interval of the form [] [1, 1] for some / E 2, called degenerate and nondegenerate, respectively. An elementary cube is a finite product Q = Iı x ... X In of elementary intervals, and its dimension is the number of nondegenerate factors. A cubical complex K is finite collection of elementary cubes such that every face of cube in is also in K. A filtered cubical complex can be constructed by assigning filtration values to cubes and taking sublevel (or superlevel) subcomplexes.
Persistence landscapes. Persistence landscapes (Bubenik et al. 2015) embed persistence diagrams into functional Hilbert space. Given persistence diagram D, its persistence landscape is the collection (Ax(; D))keN defined by
A (1; D) kmax (t). PED
REN,
where Ap is the tent function in equation T kmax denotes the k-th largest value (counting multiplici- ties), and T C IR is a fixed compact interval. When fixed, we may write A(t; D, k) for (x D), and X(t; Da, k) for the landscape associated with the d-th diagram. If D has N off-diagonal points, then (=(: (:; D) = O for all k: > N.
B BIAS AND VARIANCE ANALYSIS
In this section, we explore how our estimators VrI,G. firww4: and Aarrw perform as the nuisance estimates (t, a;; d) and ft. (x:) deviate from the truth. To this end, we analyze the bias and variance of our estimators in a similar manner to Dudik et al. (201I). Henceforth, we omit the homology dimension d from all notations,
the subsequent results do not depend d. Let (t x) denote additive deviation of Aa (r) from Ha (a(1, = (6;(1 D) X x, a). and 6x(x) the X z). such that At, (t,(t. (t(t,
multiplicative deviation of ft (z) from (x(x) IPA
(x( na (z)
(z)))
As discussed in Remark 2 we assume that the estimates (t, and (x))) are fitted from an independent sample via sample splitting, and are thus fixed. For notational convenience, we the shorthand pa for (t x),, faa for at (t, n. for T. (x), n, for a, (x), 1, for 1(A = a), o for (tt; D), A. for (t,,,))) and 6, for da (z).
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B.1 BIAS ANALYSIS
We first analyze the bias of MAIPW. Let Pa == Ma + 1e (Ø pa), and let Pa be its estimate using iro and a.. Then, the expectation of Pa is
La _ (Ø Pu))
B(Fa)= = Pla +
Ta
1a (0-
E
Ma + Du +
Ha - Au)
Ra
1.
-((aa) + Ex
E
Da
+
( - Ha Da) X
ta
=B(0") + Ex
Au
Ta
+ Ta
(Ma
Hay
= B(d") + (l)
where the fourth equality follows from the law of total expectation, coupled with the (C1) consistency and (C2) unconfoundedness assumptions from Section From the above formulation, can derive the bias of V'IPW as follows:
E (DAIP)) - 1( Pu) &
E(O'' -00) - - +++E(A E( A0do)
B(A,di - Ando).
The biases of WPI and View naturally follow from the observation that the PI and IPW estimators are special cases of the AIPW estimator such that L. Hu
(aa) = and ja O, respectively.
E pPI (IPW)
+ E(AI Ao)
E
e -(8111 5o
In general, none of the estimators uniformly dominates the others in terms of bias. However, the bias of the AIPW estimator will be close to O if either Pa se or म. ss Tas whereas the PI estimator generally requires Åa = Pa and the IPW estimator generally requires ita a Ta. Thus, the AIPW estimator can effectively integrate both sources of information for better estimation.
B.2 VARIANCE ANALYSIS
We now analyze the variance of the estimators. For convenience, we let Var(.) = Cov(ib(s) (())) for any s,t E and the subsequent results hold for arbitrary choices of and We define Ea (6- Ma) for notational simplicity. Note that from the law of total expectation, B(ca X) 0.
AIPW. Given that Var ( = 1 Var( - Go). it suffices to analyze Var(ş1 (p). Since Var(ş; - Po) = (($) - )n) - -E( 20)2 and E(G, - Go) has already been derived in the previous section, analyzing (( ) will give us the desired result.
B((21 - { = E
iu +
1 (0 -p2) - -u
1u
(Φ
(io)
2
R
no
E
(M) mo) + (El 60) + AI
(1-
1i
Do (1 (1-
(1-
to }
1o
27
E (w [= + EE [E((e1 -en) +
A 1 क
Do
1o
}
27
noo
+ 2E
(mI -o) { (1-
1i
An
(1-
1
Lo
}
foo
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where the expectation of cross products of 61 is 0 consequence of iterated expectation and E(ca X) O. Next, we further expand the third term and the last term. The third term can be expanded as
11
10 T0
E
A1
1
Au
1
1
11
10
11
10
E
A
1
1
24
1
11
#1
To
11
11
10
10
(1
11
1g
E
A
2
#1
+
#1
+ 2
1
2
TT
+
"0
22
24,
#1
fr
Ex
A
1
2
+ A
1 2
TO
2A,
1
TT
#1
77
MI
TO
00
TT
Ex
A + A
T1 111
+ 100
+ A
Tu
NO)
24 A00000200 80 + 22
TT
#1
#0
#1
Ex
(A161 Aodo)² +
T1T0 #1
A
To mO
+ 2 A
T TO
Exx
(Aid A + - 61)²
+ A
50)²
A1- - 50)
T
Exx ((A181 - A060) Ex [(A/ 81)
To/ Ao 80) V T1 no
where the third equality follows by iterated expectation. The last term can be expanded as
E (p) mo) { (1- 11 Do #1
(1-
(1-3)
10 to
H
Ex (P1 - Mo) { A ( #1 Ex ((41 - (A1)) Anōo)}
An
To (1
}
To
where the first equality follows by iterated expectation. Combining the above results, we have
({(( {0))} {(1 - 110 } + 2Ex ((u1 - ( 2gdn) + Ex {A101 A0d0) )²
E ((EL - =0 + Ex { 11 - 81) TO//TO/TI/1 + Do - 60) 60) Vo)
27
=E + 1111 - - -
}3] +
(c1 - =
+ Ex | - 61) @ + Ao 5n) v / To }
27
Having derived ([ - }. we can now compute the variance of the AIPW estimator. From the bias analysis in the previous section, we know that E(q1 - (0) = B(u1 + o :4 - Ando). Thus, the variance of the AIPW estimator is
Var(&AIPW)
Var( (80)
72
E[((1 pu) - Ell1 - (0)))
Var (mu A1di -40 Ando) + E [((E =o)2]
+ Ex (1511-81V70/F3 Ao(1 5u) /2/ To "))
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IPW. To derive the variance of WIPw, we proceed in an analogous fashion by first analyzing the second moment.
10_ Φ
11 1o
2
Φ = E
1i co
+
14
1
(11 60) + (/1 - vo) - 141 (1 (1-
11
1o
}
27
= E
+
140
$1
no
11
1l
}
E {(e co)2] + { -Ha)² +
- puo
to
11 2E (11 mo) { 11 #1 )
(1 1o to
}
- 140
where the expectation of cross products of 61 6n is consequence of iterated expectation and B(Ea X)) = O. Note that the third and the last terms have the same structure as the corresponding terms that appeared in the derivation of the AIPW estimator. Thus, the previous results can be directly applied, and we have
11 $1
- 140 (1 (1-
1u
)
[[rlL 51)
27
= Ex {1111 Ho
To// m1 + 1o 60) 60) /
p1
and
E
[1111 (ul - mo)
{ { (1 (1- 11
(1(1
ni
(1
10 10 no
}}
1x (((((m (1) -o)(midi-pooo)
#
Combining the above results, we obtain
11 Φ
10 Φ
= E {(11 - - - -2Ex (111 - -o)(4rd -
+ Ex {1u10) - )
11
E {(1 - =)) + Ex { [11(1 - -81)
+ 1o(1 50) // /1/TT {
To/ -8:)VoO/1T
E [(1 - 51)111 - - )n) }²] +E (E) - =o)
- 61) - Ex { [ui(1-6i) 6.) Ti + 1o 50) V क / )z
27
From the bias analysis in the previous section, we know that [(1 - 6,)01 - - 00) Thus, the variance the IPW estimator is
E
1i Ø
10
1u
#o
1 Φ
1.
Var(Sipw)
Var
n
no
1
10 ④
2
1
10 To
E
Φ
0
n
11
to
#1
Var {(1 - 81)m1 - (1 do) + +E (EI - -60)2}
n
+ Ex (1i(1-6i)/ 8i) + p0 56) / / 2 )1)
We observe that Var(&AIPW) and Var(&rpw) have similar structures. The first term will generally be of similar magnitude for both estimator when á ss w, but it may be smaller for AIPW if fi as ju is simultaneously satisfied. The second term is identical. In the third term, the only difference is that AIPW involves A, whereas IPW replaces it with μ. In general, A will of smaller magnitude than 14, and IPW is therefore more likely to exhibit larger variance than AIPW.
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PI. The variance of dpr is straightforward:
Var(śpr) = Var(fi - Pa))
Var(u1 + +11 - Po Ao)-
72
Unlike AIPW and IPW estimators, the variance of contains only a single term, which corresponds to the first component in the AIPW and IPW variance decompositions, Therefore, Pf estimator will generally have smaller variance compared to AIPW and IPW estimators. However, as discussed in the bias analysis, Pl estimators are susceptible to high bias if the regression model is misspecified.
C PROOFS
Notation. We let |احاا denote Lg norm for any fixed vector x. For a given function f. we use the notation Il.lIr., = (P(()1))) = [[] 1/a as the Lgg (no) of f. Also, we let denote the conditional expectation given the sample operator f. as in P(f) {())d(((2). Notice that IFf) is random only if f depends on samples, in which case P(f) + B(f). Otherwise and E can be used exchangeably. For example, if f constructed on a separate (training) sample B f(Z) D* for new observation Z P.. We let F,
D (Z., Zn), then {f(Z) denote the empirical measure as in P,,(f) P. ((f()) E.= f(Zi). Lastly, we use the shorthand bn to denote On I cb, for some universal constant > 0.
C.I PROOF OF LEMMA 211
Proof. For notational convenience, we denote ((t) := <tt; D, for any fixed persistence diagram D and the power parameter r. Fix p = (a, b) E The tent function Ap is linear on [a, (a b) with slope +1 and on [aa b)/2, bj with slope -1. Hence, for all 8, L RR
[Ap(s) At) < 1s - E/.
Namely, each Ap is 1 1-Lipsshitz.
Rewrite the weighted silhouette as p(l) = Ey=1 Ap, (1) with weights a, -w Wn/E. uh 10,
satisfying E, 03j 1. Then follows that
l - <(t)
App (t) S a t/.
which shows the silhouette functions is also Lipschitz with constant L 1.
Even when the persistence diagram random, the inequality holds pathwise; thus, taking expecta- tions yields
B sup lo(s) - o(0)] s 6.
56
C.2 PROOF OF THEOREM 5.1
Proof. Because is known, we have
(I) = (Z,)))
tET. (a(t) P(Ea(Z,t)
Hence,
/n (t) - va((t} = - G. Ea(:,),
where G, == (T( PP) denotes the empirical process.
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By the given condition that the indexed class (Ea(:,t) t): E T) is P-Donsker (or satisfies any other standard sufficient condition for a functional central limit theorem (T)), the functional central limit theorem yields
(G,, Ea(t, IET ~ (Ga(t) ET in e (T),
where Gu is tight, mean-zero Gaussian process indexed by T. On the other hand, for any s, E T, we have
cov cov(G)(s), Ga(()} (5)
which follows from the standard covariance characterization of the Gaussian limit in the Donsker theorem, together with supreT (ar (Z. t)} oo (Van Der Vaart & Wellner, 1996 Theorem 2.5.2).
Combining the display above with the empirical cuppricaa-processs representation give the desired result:
V ((t) (ba(t)} Ga(t) in (T).
Remark 3. A sufficient condition for applying a functional central limit theorem to the empirical process indexed by (Ea(-,t) E T) is that this class is P-Donsker, which is implied by standard bracketing entirop bounds under an L2 (IP envelope and suitable measurability, In our setting, these conditions are plausible because
(i)a(!) is Lipschitz, which transfers Lipschitz, control of 1 Ed(Z, 1) and thus yields small bracketing entropy, as formalized in the following lemma.
Lemma C.1 (A bracketing eeatre sufficient condition for {Ea(-, t))( Assume T C R compact with diameter (TT < co. Then, for all 8, T,
s)) - Ex( - ( + ++
2
1| -- a.s.
Moreover, letting F(Z)= == supieT IZ, we have L2(F) provided suPieT var (E, oo and suptez IP t)) < co. In that case, the bracketing numbers satisfy
(1 (E)().) : E T), (2(P)) S CeT 0<851,
for a constant depending only on dTam(T) and k. Consequently.
log (FellFlle. (Eal (2(P)) de < co,
so (Eal;tt) :t Т} is P- P-Dossker by standard barcketing-entropp criterion.
Proof. Fix s,t E By definition,
A A ((X)
Ca(
1- xx=(xx) (t.)().
Under the positivity condition λ S x 1/k when A = 1 and by 1/18 when = 0, hence by 2/k uniformly. By virtue of Lemma 2.1 obtain
< 1 - K a.s., the prefactor is bounded in absolute value by
A
1 A
2
(z( Ea(,tt)
(oi.d(s) oi.)(t)
t).
(X)
ग (X)
Now cover T by a grid {t,} with mesh size at most 8 E F||FFll 2 so that m S 1 diam (T)/8.
For each & choose j(t) with It Escerl s 8 and define brackets
2
2
u u()) = Ea 4()) 2 K
4(Z) Ea(Z (00)
n
Then (2) < ta(2,t) < (2) a.s. and ||u 6elpp.2 < (4/K)6 2s F1ll. Thus (=] Fl L2())
$ m S E proving the entropy bound and hence the finite bracketing
integral.
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C.3 PROOF OF THEOREM 5.2 UNDER ASSUMPTION (A5')
Recall that is the uncentered EIF for the target functional in equation 3 expansion:
satisfying a Von Mises
(tt - -t; - p(,, d( + PP),
(10)
where the aecond-order remainder term is specified
/ T(X) x(x)] (ui(t, (x(xxxx))xxx
R2 (PP,
/ (-) (X)) 1-# #(X) {uoll, - j x;)) 11-11 (1- dP, VI T.
1
(11)
First, we prove the asymptotic normality of miiitedimensional causal effect projections as stated in the following lemma. The proof follows the argument of Theorem 5.31 in van der Vaart (2022);
similar techniques to those have also been employed in Kennedy (019)).
Lemma C.2 (Asymptotic Normality for Discrete- Time Estimators). Ler k N and 81 ....... tx E TT be fixed. Throughout in the proof, we write Va = VAIPW. Define
D (Fa(li) Fa((x)) Dr (va(l)))...., va((w)))
Under Assumptions (Al) (AS) we have
Vn Pti..... sbe....ts) d (t).tn).))))
where the covariance matrix is given by
Eep...t = EE (D))
(D)
where we write Pr (D) = (p((1: D)... ((<x: ((xx
Proof. For notational simplicity, let "0 Pt...... . (D) and & (t (G(t1; D).... "(t,:.,((t;:)) Recall that @(t: D) defined by
1-A } (o(t.Da) pt I 1- =X))
((;; = (t, X: Ao(t,
#(X)
where all the nuisance estimators are constructed on separate, independent sample (Assumption (A2)). Then, consider the following decomposition:
Vn Tn(Pe........
(n( F) + Vn(P, -P)( (p) + PP( sp)
(12)
By the multivariate central limit theorem, the first term in equation 12 converges to a multivariate Nor- mal distribution with mean and covariance matrix equal to Ex.....f The third term in equation T2 is simply VnR2 where Rs is the remainder term specified in equation II Under the rate condition in Assumption (A4) it follows that vnR2 op(1). Thus, suffices show that the second empirical process term is negligible, i.e., of order (1). By the consistency condition (A3) and the triangle inequality,
p| 2plp,22 =0
Σ (m(t,) ((,)) (1).
Applying the multidimensional Chebyshev's inequality to the proof of Lemma 2 in Kennedy et al. (2020), it is immediate to get
(P, P) -)) (
vplp.22 (10-v))1e2)
or(nn1/22)
Vn
Putting all the pieces together into equation
12 the result follows by Slutsky's theorem.
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We now in position to prove Theorem 5.2 Here, we impose Assumption (A5 and follow
the proof strategy of Testa et al. (2025). We then show that the same result holds under the weaker conditions from Assumption (A5) in Section
Proof. First, we show the stochastic equicontinuity of sh:
(x8)) ((1)| 28 =0,
lim limsup P 6-20 n-000
sup |0-1|56
for every Fix 8 and write
w(v:0)
sup (s)) - ( |s-1|56
For a E 10,1] set (s, (x; (t, d: and R(s,t) ((s;Da) btt;
(Ma(s,
Xx:) X; Pa(t, X;d) Then
(s)) T(1) (i(s,t) 1o(s, +
A [(X) -- R(s,t). t).
}
By Assumption (A5) TP su (s, Lnd, Ya E A. Since persistent pairs with
infinite lifespan are excluded from consideration, have ||0| co. Further, by Lemma 2.1 o() is I-Lipschitz continuous in t, yielding [ø(s; Da) - o(t; Da)l S Is - t/. Thus, it follows that
sup R(s,t))
S (Lg + 1)6. Therefore, by the Markov inequality, for any fixed e >
we have
P (()
(3Lp + 1)6
PP f 8)
8
where the second inequality follows by the triangle inequality and Assumption (A)) Finally, taking 5 - O and then lim sup yields lims--o lim sup [[([;;; 8) 2 s)
0 for every >> 0,
as desired. Having established the stochastic equicontinuity of s. the result follows directly from Lemma C.2 and Theorem 7.5 of Billingsley (1999).
C.4 PROOF OF THEOREM 5.2 UNDER ASSUMPTION (A5)
Proof. The finite-dimensional convergence argument in Section C.3 is unchanged, so it suffices to establish stochastic equicontinuity.
Fix 8 > 0 and write
sup (s)) - ( s Pn
[aa(s 2) - Gall, 2)
sup |5-1/56
15-
Decompose (a(t, palt, Z:n) + (ta(t, Z) (t, Z:n)) obtain
sup 18a(s, 2) (a 2) < sup 1
lpp(s, Z:n) (a Z: +up (6,(u, Z) paluu, Z:n) wET
For the first term, Assumption (AS) implies that - (t, d) is Lipschitz uniformly in and Lemma 2.1 gives Lipschitz continuity of t -> (t.a Together with the boundedness 1/x(X) and 1/(1 mT {)) implied by positivity, this yields constant Lap such that
[pa[s, Z;n) Pa Z: m)) < Li 8 a.s. 58
sup
For the second term, uniform consistency in Assumption (A5) and bounded denominators (Assump- tion (AI) imply
An := sup palu 2) - palu, 2;n) = (1). weT
Therefore,
sup |
pa(ss Z) Pa(t, 2)1 < Lg 5 + 2A
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Choose 8 5n 0 such that L5dn S E/2. Then
(ss) -00--(1)) > S P(2A, e/2) +0
P
sup [5-t|56
which establishes stochastic equicontinuity. The remaining tightness and conclusion follow as in completing the proof under Assumption (A5)
Section C.3
The
convergence argument in Section C.3 is unchanged, so it suffices to establish stochastic equicontinuity of the process vT)
(a)) in 1 (T).
Throughout, recall that the proposed estimator is
P., P.. (f,(t, Z)),
Pa Z) sta(t, Z:
where fj (ft, Au. (n)) is fit on an independent sample from P. where P,, f by Assumption (A2) We also write (t, 2) = pa v)) for the true EIF with v (T,, Ma, Mi).
Fix > Since (t) p., (t, 2)}, we have
sup (s)) o()) sup
Pn [Paaa 2) Fa(t. 2)}
18-1/50
1
Pn 2) - Ea(1, 2) s Pn
[a((, Ea(l, 2)|
sup
sup
1e-t/56
18-1/58
(13)
Next, for any T,
Ga(t, 2) = pa(t, + [Fa(t, Z) pa(t,
and thereby, Vs, E T.
[aa(s, 2) - ()(
< ppals, Z) -plll 2) + (s, 2) - pa(s, 2)| + 1 2) - spa(l, 2)
$ pals, Z) (at(t, 2)|
+ sup
(u, Z) Ga(u, Z)|.
(14)
TET
Recall that
= p1 - pol1, x:d) + A 1-A
}
(X) 1- xX)) (dia(()-Ma(e (A X;d))-
By Assumption (A5) for each a E (0, 10,11.40 1 + (a(1, x;; d) is Lipschitz on T. uniformly over x, so there exists Lu such that
sup (s x; Ha x; d) S L - t). =EX
By Lemma 2.1 t + (t) is 1-Lipschitz, so (dia(s) - a()) S |s 15-1, t). Moreover, the positivity condition stated elsewhere implies that || 00 and 11 =)] oc; set
Cx
-
+
<0.
Then for any s,t E T.
19a(s, 2) - palt, 2)1 S [81 X;d) - (1 X; d)) + (o0ssxx;) - -oo(,, X;d)
A.
1- (X) (105,a(s) (16 (rra(1)) + (ua(s, X; - (l()))
1-A
+
< 2L is - t| + C+ f ss t| + Lpls - t/} = Lo 1s - t)
where Li 24,, Cx [1 L.,) xc. Consequently,
sup |
[sp[(s Z) pttt, 2)1 < L. 8 a.s.
(15)
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From equation 14. define
An
sup ueT
supl Fa 2) palu, 2)1-
Note that
Fa(t, 2) - -pa(t, (t; (t; - (d;
(dra((t) - (1;
+(X)
x(X)
1- A
1- A
(0i,((1) (o;
((X)
1-+(X)
A
1
(1x(l, nA X;d)).
T (X)
1-x(X)
Now, take sup:et . and use the triangle incquality. Assumption (A1) gives bounded denominators for â. and the positivity condition gives bounded denominators for æ. Also, we have || llo < co.
Therefore there exists an almost surely finite random constant ((Z) (depending only on |
|||/ [11(11 = >o ||1/#| co. and || =o) such that
Az = sup Pa 2) (t. Z) = (
max sup (t, X; Ma xxd)+ + (x() - x (x))
1ET
(E EE
(16)
Now Assumption (A5) implics the uniform-in-(t.a) x)) consistency sup,.z |i (t, paa (t
(1) and sup, ( (x)] (1()), hence the right-hand side of equation 16 is op (1). Therefore,
An (Z) (1)
(17)
Combine equation 14 equation
15
and equation 17 to obtain
sup 15
Ga(s, Z) (a 2) S Li 5 (Z).
Plugging into equation 13 yields
(5s) - +(t) s L. 6 + 2 (Z.).
sup
Since P,, ( (A,, (2)) - in probability by equation 17 for any 0 choose a deterministic sequence
5, + such that L, I e/2. Then
P
sup |1-1/5
(5)) - ((t) >E)
PAn x/22) -,
which verifies stochastic equicontinuity, thus completing the proof of Theorem 5.2 under Assump- tion (A5)
C.5 PROOF THEOREM 5.3
We first introduce several definitions that will be used in the development of our stability results. Definition C.1 (Matching). Let l denote the diagonal of persistence diagram. A matching between two persistence diagrams and DD' is defined as a subset m C x DD' such that every point in D A and DD' A appears exactly once in
Definition C.2 (Wasserstein distance). The g-th Wasserstein distance between two persistence diagrams D and D' is
1/9
W WD(DDD' = inf
Σ 1p p' 11%
matching
(p.p')em
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We denote the matching that satisfies Definition C.2 as the optimal Wg matching.
Let Φ = ott; r) and o' = o(t; be the corresponding persistence diagrams D and D', respectively. Given p (ap. bp) D and p' (ap, byy) D', we denote the power weights corresponding to and p' as Wp = bp apl" and wy = 16,' Ogt l' respectively, where the absolute value in the power weights may be omitted. Hence, we assume Wep = (b, ap) and
power-weighted silhouettes for
0. Note that under the diagram boundedness condition of Assumption (A6)
Wp' (by (bp - ary)" throughout this section.
We now establish the following lemma, which serves as preliminary result essential for the development of the main theorem.
Lemma C.3. Given a matching m C x for any (p. (p.p') E m,
11 w I 2rd llp - p'lloo
wpr
where Cpp E (min by - ap: by aw max Up ap: by ap }) is some positive constant that satisfies the Mean Value Theorem for the function g(x) = x², I (0), > 0, given points p and p'. Consequentl.
[p wppl S. 2rc-11 ||p p' pol
for
max max bp - Ops by- (pm
an -ay
max { by - O₂ E DUD'),
so e depends only on the weighling exponent (through the bound) and on the largest lifetime across D and D'.
Proof.
10p wpp = (bb - ap)" - (bp) - ap V
|r pp {(bp ap) - (bp Op H
ppp
1(bp - ap) (by ag') )
I r yp
{aap aprl + 16p bp
2r Eip
max {app - arl 1b, byr
2r Epp
l1p - p' 1lo0
where the second equality uses the Mean Value Theorem.
Notice that when r = 1 the bound does not involve e (cf. the special case in the theorem).
Now we give the proof of Theorem 5.3
Proof. Let Wpp = (bp
ap)", ,", Wpp = (bpp "rr)" )", and define
S
Wp:
S'
Σ Wp'
Σ
WppAp
1
Σ
Wp Ap's
S'
PED
FED
PED
D'ED'
where Ap denotes the tent function centered at p such that ||Ap (6, ap) Augmenting the diagrams with the diagonal if needed, the optimal W1 matching mn" is a bijection between D and D', hence
10 10-0'1l-
Σ
Wpdp S
Wy Ap
Wp S
Wp Ap
St
S
(ppr))m
(n.p")mm
Split each summand
Wp Ap
Ap
Wp s
(Ap A
+ -1App 1oc
Up
Wp
S
St
S
St
Summing over (p.p') m² yields
| 201100 S Ti+ Ti+T2,
Ti 5
Σ
- Aprloos
T2 =
Σ
||
Up
S
S
(p.P")mm
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Bound for Ti. Therefore,
By the 1-Lipschitz property of the tent functions, we have ||Ap -o- - 11p-p' ll
T <
wplly p' 00
S
1
wp
Σ
llp - z' r'llss) Wi(D,D),
5
(p-P")Em*
PED
(p-r")mm
where the second inequality follows by a simple corollary of Hölder's inequality:
E S | |o
||
||=|||
( (E..) ( x.)
for 3r, 24 2 O, and the last equality is the definition of the W cost of m
Bound for T2.
Observe the algebraic identity
w
Wp
(S' S) wp + S(wp wp
S - S Wp
Wp wprl
5
5
SS
SS
Hence T2 < T2.
T20, where
T2a
Σ
|| 1lx
S - S) Spupp
T25 :=
Σ
||
Wp -
wpr
S.S'
5
(p.r")m=
(p-p')Em*
We first tackle the term T2a- Using Hölder's inequality to separate the sums and cancel S, we have
Tza S
(
11
110)
SS' -S-
S'
P'ED'
Since S E(ppp)em (wy Wp). the triangle inequality and Lemma C.3 yield
SS" S1 S Σ
|wp -p--u wpr1 S
Σ
2r
"pp r-1 ||p - p'loo0 S 2r 2- (p.p))mm
Σ ||p - p' ooo
(p.p')m=
(p.p))m=
Moreover, EpeDD' || Imx 1 Ep' (bp boundedness condition from Assumption (A6)
apr) and S EpeD ap."" so by the
EVED || loc
Ep ap)
4
Cbp - apr)) S 2
5'
Therefore
T2a S 2
2rc r-1
Σ
lp pm = -Lrer -r
(p.p")m"*
The analysis of T25 follows the same steps as above. Again by Lemma condition from Assumption (A6)
C.3 and the ratio boundedness
T2e < EpeED ||Ap
lloc
Σ
[. w
Σ
r-1 cr
11e-v'loo S Lrc Lrc W((,D')
2
(p.p/ Emm-
(n.p))m"
Combining the bounds for Tea and T25 gives
T2 "Lrr"-WWi(D.
Putting together the bounds for Ti and Tg. we obtain
| - 6111 < (1 (1+ (D)
Special case
1. When r= 1, Lemma C.3 gives Wp Up 21[p
and
EVED ||Ap/ Il En (br ag') 1/2
S'
Ep (by 4p') 2'
so each of T2n and Ta, bounded by WD,D,D"). while Ti S WD, D'), which yields
|| - -dl-ello (D,'
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Table 1: L1 distance between the average estimate and the true effect, together with the standard deviation of each estimator over 20 repetitions. Hd denotes homology dimension d, and the scale is in le-05 for SARS-CoV-2 and ORBIT and le-02 for GEOM-Drugs.
SARS-CoV-2
GEOM-Drugs
ORBIT
Ho
Ho
H1
Ho
H1
L1 dist.
Std.
L1 dist.
Std.
L1 dist.
Std.
L1 dist.
Std.
L1 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
C.6
PROOF OF COROLLARY 5.4
We provide only brief sketch, since the argument follows from standard empirical-proces machinery (e.g.. Van Der Vaart & Wellner 1996). By Theorem 5.2 vn (WAIPW.. - 4/a) ~ G2 in 8 (T), and the continuous mapping theorem applied to ʃ + I implies Tn ||Gal|>> under Ho. If the multiplicr bootstrap consistently approximates, conditionally on the data, the law of ||Gal| then its (1 - a) --uunnnti C1-a yields asymptotically valid level-a test; under any fixed alternative with || so > 0, we have Tn 2 Valla + or or(vn) - oo, so the power converges to one.
D EXPERIMENT DETAILS
We perform experiments on three benchmark datasets: the SARS-CoV-2 CT-scan image dataset, the GEOM-Drugs molecular graph dataset, and the ORBIT point cloud dataset. In all experiments, we construct a synthetic counterfactual dataset {(X,, Y,", )=) to facilitate the evaluation of estimators against known true effect. We begin by randomly selecting and pairing two data samples to form each potential outcome pair, assigning one to y and the other to Y' in manner that induces a clear topological contrast. Next, we generate the covariates X and treatment according a stochastic data-generating process and treatment assignment mechanism. The same procedure is identically applied to all experiments, with details outlined below.
Data-generating process. We assume setting in which a subgroup structure is imposed on the covariates X E IRs. The covariates are generated from two subgroups, each governed by a multivariate Gaussian distribution with distinct mean vectors Pl, M2, and a common covariance matrix Σ. The covariance matrix Σ is set as a diagonal matrix with diagonal values 0.5, and the mean vector of each subgroup is specified Covariates for half of the samples are generated from N N(m, Σ), while the remaining half are generated from N(22 E).
μr = [1, 0.6, -0.7, 2.2, -1]" and #2 = [0.4, -0.4, -0.6, 3.3, 3)".
Treatment mechanism. Given the covariates X, treatment A E 1) is assigned with probability (XX erpit(-0.5.
0.1X, + 0.6X3 + 0.1X, + 0.1X; 0.5.X2X3 0.7X, This treatment mechanism is designed such that one subgroup has a higher probability of receiving treatment than the other (see Figure 6a). Upon treatment assignment, the observed outcome given by Y AY + (1 A)Y
The aforementioned procedure repeated 20 times to generate multiple datasets with different realizations of X and A, for each of which the PI, IPW, and AIPW estimators are computed. The estimators are constructed by modeling the silhouette regression function Pla with function-on-see regression employing a Fourier basis expansion, and estimating the propensity score using random forest classifier. To assess the performance each estimator, we examine the pointwise mean and the pointwise 1-standard deviation error bands computed across the 20 runs. In addition, we report the L1 distance between the average estimated function and the true topological effect function as single number summary of estimator accuracy in Table [ For completeness, we also provide the standard deviation of each estimator in Table [ obtained by first averaging the empirical covariance matrix to produce scalar summary of variability, and then taking its square root.
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Table 2: The runtime of computing weighted silhouettes for each dataset in seconds.
Data (Sample Size) SARS-CoV-2 (2481)
GEOM-Drugs (30433)
ORBIT (2000)
Runtime
244 seconds
21 second
25 seconds
Sihouuete
Silhouette Mo
Silhouette Ho
Toe
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. dataset. The true topological causal effect is shown as a blue line.
D.1 IMPLEMENTATION CONSIDERATIONS
Before discussing the details of each experiment, we first outline several practical considerations relevant implementing our method.
Sample Splitting, As discussed in Section 4 sample splitting enables the use of arbitrary complex nuisance estimators. Thus, the use of sample splitting is crucial for proper implementation, and we illustrate how it can be carried out in practice. Let D = {Z = (X,, A,, Y.) denote an i.i.d. observed sample of size n. We randomly split into two independent samples Dj and D2, each of size n/2. First, we use Dj to fit the nuisance estimators s), and use D2 to compute vù for the estimation of v. Then, we swap the samples and use D2 for training and D₃ for estimation. If we average the resulting estimators vi, recover full-sample efficiency.
Computational Cost. Our framework relies on persistent homology, which incurs substantial computational cost: the standard algorithm for persistent homology induces O(N3) complexity (Edelsbrunner & Morozov), with N denoting the number of simplices. To partially mitigate this computational overhead and improve scalability, several algorithmic techniques have been developed. For instance, Chen & Kerber (201T) proposed an output-sensitive algorithm for persistent homology with ( log N log log N) complexity. For many applications where O-dimensional persistent homology suffices, such as our SARS-CoV-2 experiment, additional scalability can be achieved with efficient algorithms that run at O(N log N) time (Edelsbrunner & Morozov). Nevertheless, even with improved algorithms, our approach utilizing persistent homology may be computationally demanding and provide restricted scalability on large-scale datasets, as this limitation is intrinsic to persistent homology itself. To achieve enhanced scalability, one may consider extending our framework to topological descriptors based on the efficient Fuler characteristic curve, although Fuler characteristic curve-based descriptors generally lack the expressivity of persistent homology-based methods.
To illustrate that our method is entirely feasible for datasets that are not excessively large, we provide empirical runtime measurements for computing weighted silhouettes across all datasets used in our study. For the SARS-COV-2 dataset, we compute weighted silhouettes on CT images of 1252 infected and 1229 non-infecte non-infected patients. The GEOM-Drugs is computed on the full dataset of 30422 graphs. For the ORBIT dataset, weighted silhouettes computed for 1000 pairs of (Y', YY)), amounting to 2,000 silhouettes overall. The GUDHI (The GUDHI Project (02I) library is used for silhouette computation Apple M2 processor, and the runtime results presented in Table 2
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group1 group?
Proporsity
(a) Density of propensity score by subgroup (Group more likely be trcated).
(b) Sample molecular graph
(c) Sample molecular graph
A (Axelrod & GGomez-Bombarde 2022).
B Axelrod & -ombaz-Bommarelli
2022
Figure 6: (a) Propensity score distribution by subgroup; (b-c) representative molecular graphs.
D.2 SARS-COV-2
The SARS-CoV-2 dataset contains CT-scans collected from real patients Brazil, who are infected or non-infected by COVID-19. In our experiment, 500 potential outcomes pairs (Y Y1) are constructed according to the following procedure: (i) 500 infected images are sampled and assigned to y (ii) 375 non-infected and 125 infected images sampled and assigned to Y (iii) y and Y are randomly paired. By experimental design, TATE exhibits treatment effect since 75% of y1 is non-infected where every individual in y is infected. Silhouettes sublevel set filtration on a filtered cubical complex (see Appendix with r 0.1. For the nuisance estimators, function-on-scalar regression implemented with 10 basis functions, while random forest is constructed with 100 trees. The point-wise 1-standard deviation error bands of the respective estimators are demonstrated in Figure 5 and see Table T for numerical summaries of estimator performance.
computed using
D.3 GEOM-
The GEOM-Drugs dataset consists of graph-structured representations of molecular compounds, as shown in Figure 66 and bc All graph nodes have six features, and we assign each node a scalar weight as a randomly sampled convex combination of these feature values. Each edge is assigned weight given by the maximum its adjacent node weights. Since 1-dimensional homology features have infinite death times, we replace them with values sampled uniformly from [6.5,, 8.5). In our experiment, 1000 potential outcomes pairs (Y"0, Y1) are constructed according to the following procedure: (i) randomly sample 1000 graphs with loop and assign them to y (ii) randomly sample 750 graphs with two loops and 250 graphs with one loop and assign them to Y This allocation scheme gives rise to significant topological differences between the counterfactual groups We compute silhouettes with r = 1, while the nuisance estimators for μla and π are specified with basis functions and 100 trees, respectively. Figures provides visualization of the point-wite wise mean and the point-wise 1-standard deviation error bands of the estimators per homology dimension, and Table for numerical summaries of estimator performance.
D.4 ORBIT
The ORBIT dataset (Adams al.
2017) is synthetic point cloud dataset that is generated by simulating different dynamical systems characterized by a parameter 8. Given a random initial point yo) [0, 1]2 and s >> O, generate point clouds consisting of 1000 points as follows:
En+1 In + sy, (1 ya) mod!
Da+1 = yn + sr, (1 - x. mod 1,
In this experiment, we use s = 3.5, 4, 4.1 to generate 1000 samples for each value of s, with the resulting point clouds illustrated in Figure 2a From each triplet of point clouds generated by s = 3.5, 4, 4.1, we randomly select two point clouds and assign the corresponding to the higher value y with probability 0.7. This procedure
3.5 (top right), s 4 (bottom), 4.1 (top left).
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Silhouette of
Silnoustte of
Silhouette of Ho.
Silhuuette M.
Silhouette of
Silhcuetee of
Tax Fffert
Figure 7: Visualization of the point-wise mean (dotted line) and the point-wise 1 1-standard deviation error bands (shaded area) of O-dimensional (top) and 1-dimensional (bottom) Pl, IPW, and AIPW estimators on the GEOM-Drugs dataset. The true topological causal effect is shown as blue line.
yields pairs of matched potential outcomes (Y"O, ,Y!) for all 1000 samples, where the treated potential outcome intentionally designed to possess more pronounced topological features. Here, we use bases to model the silhouette regression function and 100 trees to estimate the propensity score, and the silhouettes are computed using Alpha filtration (see Appendix A) with power weight
3.
Results. Figure g illustrates the true target silhouettes in homology dimensions 0 and 1, which clearly demonstrate causal treatment effect on first-order homological features of point clouds. In particular, the positive values of the 1 I-dimensional silhouette indicate the emergence of holes in the treated point cloud, Our aim to recover the magnitude of this silhouette function, larger values correspond to more substantial structural changes. Consistent with previous results, the IPW estimator tends to overestimate the treatment effect, whereas the plug-in estimator underestimates The AIPW estimator, by contrast, yields substantially more accurate estimate of the true silhouette function. Moreover, the ncar-zero silhouette for O-dimensional homology features suggests that the data's O-dimensional homology. including connected components, remained largely unchanged after treatment. Figure iillustrates the pointwise mean and the pointwise 1-standard deviation error bands for each of the Pl, IPW, and AIPW estimators on the ORBIT dataset, and see Table [ for numerical summaries of estimator performance.
Hypothesis Testing. The ORBIT dataset provides appropriate setting for empirically evaluating the hypothesis test presented in Corollary 5.4 given that the O-dimensional silhouette exhibits no discernible topological effect, whereas the 1-dimensional silhouette displays significant topological effect (Figure 8). Accordingly, we conducted the hypothesis test of null Ho W W1(D", "D)) 0 examine whether the test outcomes are consistent with our visual interpretation of Figure We generate 1000 bootstrap samples G,,, and use 0.05. For O-dimensional homology, the test statistic T, 0.0037, whereas the a quantile of ||&,,lloo 0.0277. Thus, consistent with our expectation, cannot reject the null hypothesis. For I-dimensional homology, the test statistic T,, = 0.4242, whereas the quantile of ||G, oo is 0.1107. As expected, the test rejects the null hypothesis, reatmenn-induced induced topological shift in the I-dimensional homology of the
indicating a significant treatment-
data.
29

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Published as a conference paper at ICLR 2026
Silhouette Ho
Silhouette of H/
400
Tue Effect
Truc Effect
IPW
IPW
009
000
6.50
Figure 8: O-dimensional (left) and I-dimensional (right) true silhouette functions and its PI, IPW, AIPW estimates for the ORBIT dataset.
Silhouette of
Silhouette of
Silhouette
Tru Offict Play in
Tra Ffbet
Tue Ffte
144
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 blue line.
Figure 9:
D.5 ESTIMATION UNDER MODEL MISSPECITICATION
In previous experiments, complex nonparametric models were used to flexibly estimate the nuisance functions. In this section, investigate how the estimators perform under severe model misspecifi- cation of nuisance functions. Specifically, we are interested in observing how the performance of estimators varies across the following scenarios: (i) is severely misspecified and f is the correct. model, and (ii) is severely misspecified and is the correct model.
Since we have knowledge of the true treatment process, the correct is modeled as #(X) (B X +2 83 +
X3),, whereas मे (X) erpit x 82Xx3) is used for the misspecified model. For the outcome regression 14, whose true form unknown, we retain the function-on on-scalar regression using Fourier basis expansion. We choose sufficiently large number of basis represent the correctly specified model and a deliberately reduced number of basis to construct the misspecified model. Specifically, for the SARS-CoV-2 GEOM-Drugs, and ORBIT datasets, we respectively use 30, 10, and 5 bases for the correct regression model, and 7, 2. and 2 bases for the misspecified regression model. Figures 10 II and 12 illustrate the visualized results, and Table provides quantitative summaries of estimator performance. We can observe that across all datasets and misspecification settings, the AIPW estimator generally has the best performance.
30

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Published as a conference paper at ICLR 2026
Silhouette of Ho
Silhouette of Hp
3001
4.00
Thuo Effect
Tue Effect
IPW
IFW
(a) π is misspecified, y is correct
(b) is misspecified, is correct
Figure 10: True silhouette and its PI, IPW, AIPW estimates for O-dimensional homological features on the SARS-CoVE dataset when one of the nuisance functions is misspecified.
(a) misspecified, μ is correct
(b) j is misspecified, is correct
Figure 1I: True silhouette and its PI, IPW, AIPW estimates for O- and I-dimensional homolgical features on the GEOM-Drugs dataset when one of the nuisance functions misspecified.
Gihouette
Shouett
Showette cfikk
(a) x misspocified, / is correct
(b) μ is misspecified, и is correct
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.
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Published as a conference paper at ICLR 2026
Table 3: L1 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. Ha denotes homology dimension and the scale is in le-05 for SARS-CoV-2 and ORBIT and le-02 for GEOM-Drugs.
π is misspecified, I is correct
l is misspecified, π is correct
SARS-CoV-2 (Ho)
I dist.
Std.
L1 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
π misspecified, is correct
is misspecified,
is correct
GEOM-Drugs
Ho
H1
Ho
H4
Li dist.
Std.
L, dist.
Std.
L dist.
Std.
L dist.
Std.
PI
2.006
1.133
4.022
0.865
4.376
1.133
6.879
0.865
IPW
61.796
3.602
77.963
4.426
10.299
2.831
16.245
3.564
AIPW
2.343
1.168
3,432
0.897
3.380
1.390
6.842
1.112
π is misspecified, # is correct
is misspecified, is correct
ORBIT
Ho
H1
Ho
H4
L1 dist.
Std.
L1 dist.
Std.
L dist.
Std.
L1 dist.
Std.
PI
0.514
0.688
7.298
37.179
0.600
0.688
36.323
37.179
IPW
0.766
2.300
17.576
45.287
0.156
1.482
7.092
55.763
AIPW
1.347
0.681
2.605
39.582
0.284
0.748
2.354
40.813
32


Reviews:
[{"review_id": "EBxGrC6KRK", "reviewer": "Reviewer_R7QM", "summary": "This paper bridges topological data analysis and causal inference to estimate causal effects for non-Euclidean outcomes, like molecular graphs and point clouds. Core contributions include defining the Topological Average Treatment Effect as the expected difference in power-weighted silhouettes of persistence diagrams between treatment groups, proposing plug-in, inverse probability weighting, and augmented IPW estimators, deriving theoretical guarantees, including asymptotic distribution, Lipschitz stability, convergence, and hypothesis testing. The performances of estimators are validated on three semi-synthetic/synthetic datasets with visual evidence of AIPW outperforming alternative estimators.", "questions": "Q1: Can you provide explicit mathematical formulations for the bias and variance of your proposed estimators? How do misspecifications of propensity score or outcome prediction models, or skewed propensity scores, affect these quantities? Then how do you rigorously verify AIPW’s DR property? Do extreme predictions or skewed propensity scores erode double robustness?\n\nQ2: Will you provide tabular experimental results with quantitative metrics for your estimators across datasets? Such as estimated bias, variance, or AUC based on your outcome type? Can you confirm that AIPW’s performance gains over the others are statistically significant? Are your experimental results seed-specific? Please provide pairwise t-test results across multiple random seeds to demonstrate reproducibility.\n\nQ3: How is the comparison between your proposed methods and other existing functional causal inference baselines (such as Testa et al., 2025) in terms of accuracy and robustness?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "The paper’s primary strength lies in its innovative problem formulation: reframing causal effects as treatment-induced topological changes (rather than scalar/vector shifts) and integrating topological data analysis tools to quantify these changes. This opens a new direction for causal inference on non-Euclidean data, with clear relevance to biomedicine, neuroscience, and other fields where structural outcomes matter. Experimental design uses well-characterized semi-synthetic datasets with known true topological effects, providing a controlled setting to test proposed estimators. The paper also effectively breaks down complex topological and causal concepts, with concrete examples for basic definitions and straightforward interpretation of the topological average treatment effect, enhancing accessibility for readers unfamiliar with topological data analysis.", "weaknesses": "W1: The paper suffers from inadequate theoretical rigor. It does not provide explicit bias and variance formulations for IPW and AIPW estimators. While claiming AIPW some kind of double robustness benefit? But it offers no verification, for instance, analyses of how misspecified propensity score or outcome prediction models, extreme predictions, or skewed propensity scores impact estimation performance are entirely absent. There is also no discussion of whether robustness holds under realistic conditions like moderate nuisance model misspecification or unmeasured confounders.\n\nW2: Experimental validation is insufficient. Results rely solely on visual comparisons of silhouette curves, with no tabular reporting of quantitative metrics (such as estimated bias, mean squared error, variance, confidence interval, or coverage etc.) for estimation accuracy. Statistical tests to confirm AIPW’s performance gains over other estimators are not discussed, leaving uncertainty about reproducibility or whether results are specific to chosen random seeds.\n\nW3: Critical robustness analyses are omitted. The paper does not evaluate how misspecifying propensity score models (e.g., linear models for non-linear true scores) or outcome prediction models (e.g., incorrect Fourier basis sizes) affects estimator bias and variance. It also fails to test performance under skewed or extreme propensity score distributions to contextualize the proposed framework’s advantages.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "W1Oz9G6JB3", "reviewer": "Reviewer_ybeB", "summary": "The paper introduces a framework for topological causal inference that defines treatment effects through changes in the topology of subjects. The authors utilize persistent homology from TDA to capture structural shifts due to intervention by a treatment, proposing topological average treatment effect defined as expected differences in power-weighted silhouettes of persistence diagrams between treated and control groups. They develop estimators, derive asymptotic properties, establish stability bounds for weighted silhouettes, and provide a formal hypothesis test for the null of no topological effect. Experiments on semi-synthetic datasets demonstrate the method's effectiveness in capturing topological causal effects.", "questions": "**Q1.** What is the computational complexity of the proposed method? Can you provide empirical runtime comparisons or discuss scalability strategies for large datasets?\n\n**Q2.** How sensitive are the results to the choice of power parameter $r$ in the weighted silhouette? Are there specific guidelines for selecting this parameter (e.g., for different applications)?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "**S1.** The work is novel and makes a foundational contribution that meaningfully extends causal inference to the topology of treatment units, addressing an important gap where structural changes encode scientifically relevant effects.\n\n**S2.** The authors propose a solid theoretical framework with comprehensive analysis.\n\n**S3.** The paper introduces a hypothesis test for no topological effect with consistency guarantees.\n\n**S4.** Experimental validation demonstrates effectiveness on semi-synthetic data.", "weaknesses": "**W1.** The authors do not discuss scalability and computational complexity in the main paper. This is mentioned briefly in the Discussion section as a limitation, but given the claim for practical relevance, I believe this needs to be discussed in the paper.  \n\n**W2.** The silhouette is an aggregated representation of topological features, which could potentially obscure the change in each homological feature. Ideally this should also be discussed further in the paper.\n\n**W3.** The weighted silhouette depends on the hyperparameter $r$. There is lack of guidance in the paper on how this is/should be selected, and how it influences the results.\n\n**W4.** While the paper is indeed novel, it is not the first bridge TDA and causal inference. To my knowledge, a recent work [1] connects the two. [1] is fairly recent and it is understandable if the authors have missed it in their literature review, but the claim in being the first to bridge TDA with causal inference should be corrected. That being said, the paper is still novel in characterizing topological causal effect.\n\n[1] Farzam, A., Aloui, A., Tarokh, V., & Sapiro, G. Topology-Aware Robust Representation Balancing for Estimating Causal Effects. In High-dimensional Learning Dynamics 2025.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "jNzFsexARG", "reviewer": "Reviewer_Zney", "summary": "This work proposes a novel notion of treatment effect for causal inference, defined through algebraic topology.\nGiven unstructured or high-dimensional treatment outcomes, the idea is to apply a dimensionality reduction approach from \ntopological data analysis to extract homology features that represent the underlying topological structure of the data.\nThis allows comparing potential outcomes before and after an intervention from differences in these features. \nTo this end, the approach constructs persistence diagrams from the homology features (e.g., Fig 1 right) and \ncompares their expected difference in silhouette score, resulting in the topological treatment effect measure (Eq. 3).  \n\nThe remainder of the work considers estimation, asymptotic properties, and empirical study of the topological TE. They study the plug-in (PI) and inverse-probability-weighted estimators (IPW and AIPW), show under which conditions they are unbiased \nand how to construct confidence bands, and derive a test for the null hypothesis of no topological TE.\nEvaluations cover three benchmarks derived from real-world datasets, and show that in particular the AIPW gives a close approximation of the true silhouette function.", "questions": "- Related to the above weakness, can an existing TE be included in the experiments, for example to the ORBIT benchmark? \n- Given your claim in ln. 354-355 that the hypothesis test (Cor 5.5) is of primary interest to most practitioners, it would be interesting to study it empirically on the datasets.\n- Can you comment on the connection between the functional ATE (Testa et al., 24) and the topological one, as alluded in line 192?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "The connection between algebraic topology and causal inference is original and has a clear practical value, \ngiven that it allows addressing high-dimensional and/or structurally complex outcomes, for which few methods exist.\nThe presentation is clear and provides example illustrations of the important concepts. There is a careful theoretical analysis with independently interesting results, such as the stability bounds for weighted silhouettes, and the experiments match the theoretical results and are convincing. \nOverall, studying a topology-aware variant of treatment effect is a worthwhile contribution and could open an interesting future work direction integrating insights from TDA to causal inference.", "weaknesses": "The experimental evaluation would be stronger if a comparison to existing TE estimators were given if applicable, and the related work could be expanded with references to previous methods (esp. addressing a high-dimensional setting). Furthermore, while some comments on practical considerations are given (ln. 256-), the authors might consider including a section on implementation considerations and details in the main text or at least Appendix.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "SfIjRwlcL7", "reviewer": "Reviewer_XnWi", "summary": "This work introduces a novel framework for topological causal inference, combining the Potential Outcome framework with Topological Data Analysis tools in order to estimate causal effects when the outcome lies outside Euclidean spaces. The authors detail the identification and estimation steps and provides interesting real-world experiments.", "questions": "* I believe the assumptions to be quite strong. Could you explain how one can test these assumptions. Are these assumptions satisfied in the real-world examples provided?\n * There exists two main causal frameworks: The potential outcome framework and Pearl's DAG framework. I wonder what arguments motivated the authors to chose the potential outcome framework? Is this work compatible with the DAG framework?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "This paper is strong, all claims are well explained and supported by theoretical and experimental results. All proofs are left in the appendix which allows for an easy read. This work seems quite novel and very useful as multiple real world examples are provided.", "weaknesses": "The biggest weakness in my opinion lies in the strength of the assumptions required.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)