Causal Learning and Geometric Topology

• CLGT is a research theme that formulates causal questions using geometric and topological structures rather than traditional scalar or graph-based approaches.
• It integrates methods from persistent homology, differential geometry, and polyhedral analysis to measure, identify, and control causal effects on complex systems.
• Applications span cross-view geo-localization, spacetime reconstruction, and interpretable neural routing, providing actionable insights in both theoretical and applied settings.

Causal Learning and Geometric Topology (CLGT) denotes a research theme in which causal questions are formulated, identified, or controlled through geometric and topological structure rather than through graph-theoretic or scalar summaries alone. In the literature considered here, CLGT includes outcome-topological causal inference on persistence-diagram summaries, differential-geometric representation learning under interventions, polyhedral and order-topological formulations of causal structure, causal-topological reconstruction of spacetime from null or order data, and application-specific systems that combine causal invariance with geometric priors in tasks such as cross-view geo-localization (Kim et al., 2 Mar 2026, Rathore, 6 Jan 2026, Gogioso et al., 2023, Ouyang et al., 13 Mar 2026). Across these strands, the common move is to treat causality as acting on shape, metric structure, convex-geometric model spaces, or topological equivalence classes, rather than only on Euclidean coordinates or scalar responses.

1. Conceptual scope and internal divisions

CLGT is not a single formalism. The relevant literature spans several mathematically distinct notions of “geometry” and “topology.” One line studies topology of outcomes directly through persistent homology and persistence-diagram functionals. A second line studies local Riemannian geometry of learned latent manifolds, for example through pullback metrics and decoder Hessians under counterfactual interventions. A third line treats causal compatibility and causal discovery as problems in convex and polyhedral geometry, where empirical models or Markov equivalence classes become vertices or slices of polytopes. A fourth line develops causal topology in mathematical relativity, where order, cones, null hypersurfaces, and causal boundaries determine manifold dimension or reconstruct topology. A fifth line uses geometric or topological invariance as an operational learning principle in concrete systems such as knot equivalence learning, cross-view geo-localization, or interpretable routing in sparse Mixture-of-Experts models (Rathore, 6 Jan 2026, Chvykov et al., 2020, Ternovtsii et al., 15 Apr 2026).

The term “topology” is correspondingly heterogeneous. In some works it refers to persistent homology, connected components, loops, voids, or knot classes. In others it refers only indirectly to “routing topology,” “model-space topology,” or topologies on spaces of probability measures and structural causal models. Several papers explicitly distinguish differential geometry from topology proper: “Causal Manifold Fairness” is described as differential-geometric rather than topological; “Greedy Causal Discovery is Geometric” is polyhedral and combinatorial rather than algebraic-topological; “Causal Geometry” is geometric rather than topological in the strict sense; and the MoE routing work uses “topology” informally for routing structure rather than for homology or manifold topology (Rathore, 6 Jan 2026, Linusson et al., 2021, Chvykov et al., 2020, Ternovtsii et al., 15 Apr 2026).

A persistent structural divide therefore runs through CLGT. One branch makes topology itself the target of causal analysis. Another branch uses geometry or topology as an organizing language for causal model spaces, latent spaces, or intervention spaces. A plausible implication is that CLGT is better understood as a family of interoperability results between causal reasoning and geometric/topological representation, rather than as a single unified theory.

2. Topology as causal target and as constraint on causal learnability

The most direct CLGT construction appears in “Topological Causal Effects,” which treats the topology of the outcome itself as the object of causal comparison. The observed data are i.i.d. triples $Z_i=(X_i,A_i,Y_i)$, with binary treatment $A_i$, covariates $X_i$, and complex outcomes $Y_i$ such as images, graphs, point clouds, or other non-Euclidean objects. Instead of targeting a scalar contrast such as $E[Y^1-Y^0]$, the method maps each potential outcome $Y_i^a$ to a filtration, then to a persistence diagram, and then to a power-weighted silhouette function. For homology degree $d$, the topological average treatment effect is defined as

$$\nu_d(t)=\mathbb{E}\{\phi_{1,d}(t)-\phi_{0,d}(t)\}, \qquad t\in T,$$

so the causal effect is a function of filtration scale rather than a scalar (Kim et al., 2 Mar 2026).

This construction is paired with standard causal assumptions—consistency, no unmeasured confounding, and positivity—and with semiparametric estimation machinery. The paper develops plug-in, IPW, and AIPW estimators, proves functional weak convergence in $\ell^\infty(T)$, and states that the AIPW estimator attains the semiparametric efficiency bound at each fixed $t$. It also formulates an omnibus null hypothesis of no topological effect through

$A_i$0

which implies $A_i$1 for all $A_i$2, and tests it with

$A_i$3

The methodological novelty is therefore not merely that TDA is used as preprocessing, but that topology becomes the estimand, identification target, and inferential object (Kim et al., 2 Mar 2026).

A different topological use of causality appears in “A Topological Perspective on Causal Inference,” which topologizes spaces of structural causal models and the three levels of Pearl’s hierarchy. The key learning-theoretic fact is a correspondence between open sets in the weak topology and statistically verifiable hypotheses. On that basis, the paper proves a topological hierarchy theorem: the set $A_i$4 of interventional points from which all Level 3 counterfactual facts are identifiable is meager in $A_i$5, while collapse of Level 2 to Level 1 never occurs, $A_i$6. The stated consequence is that assumptions licensing arbitrary counterfactual inference from observational and experimental data are not statistically verifiable in principle (Ibeling et al., 2021).

Taken together, these two directions define two different roles for topology in causal inference. In the first, topology is the response variable. In the second, topology is imposed on the model space itself to delimit what can be learned. This suggests a broad CLGT distinction between topological estimands and topological impossibility structure.

3. Differential geometry, intervention geometry, and stochastic identifiability

A large part of CLGT is geometric rather than topological in the strict sense. “Causal Manifold Fairness” models a sensitive attribute $A_i$7 as a causal parent that can warp the geometry of the data manifold, not merely shift latent distributions. With encoder $A_i$8 and decoder $A_i$9, it equips latent space with the pullback metric

$X_i$0

and uses coordinate-wise Hessians

$X_i$1

as a practical proxy for second-order geometric distortion. The training objective combines task loss, latent alignment, and a geometric regularizer enforcing

$X_i$2

for factual and counterfactual encodings. The paper explicitly interprets this as requiring that $X_i$3 act approximately as a local isometry, while also noting that it does not derive intrinsic curvature tensors or prove global topological invariance (Rathore, 6 Jan 2026).

“Causal Geometry” offers a complementary abstraction in which a model is equipped with an effect geometry and an intervention geometry on the same parameter manifold $X_i$4. The effect manifold carries Fisher metric $X_i$5, the intervention manifold carries Fisher metric $X_i$6, and causal efficacy is quantified through geometric congruence. The local mismatch term is

$X_i$7

and the geometric effective information approximation is

$X_i$8

The central claim is that, for fixed intervention capability, a causally effective model is one whose intervention geometry matches its effect geometry (Chvykov et al., 2020).

A third geometric strand studies identifiability of causal representations under continuous-time stochastic dynamics. “Causal Representation Meets Stochastic Modeling under Generic Geometry” treats latent variables as continuous-time point processes with Hawkes-type interactions,

$X_i$9

and shifts exact recovery to a weakly convergent equivalent class. Its identifiability arguments are cast in algebraic-geometric terms: ideals, varieties, cumulant tensors, Veronese embeddings, genericity, and zero-dimensionality of the parameter variety. The practical model, MUTATE, is a VAE-style framework with a time-adaptive transition module and frequency-domain factorization

$Y_i$0

This line is relevant to CLGT mainly through geometry of parameter space and identifiability, not through persistent or combinatorial topology (Ren et al., 4 Feb 2026).

Across these works, the central geometric objects differ—pullback metrics, Hessians, Fisher metrics, cumulant varieties, spectral factorizations—but they share a common thesis: causal structure can be exposed or constrained by the geometry of representations, interventions, or parameter spaces.

4. Polyhedral and combinatorial geometries of causal structure

Another major CLGT strand is explicitly convex-geometric. “The Geometry of Causality” introduces causaltopes, defined as polytopes of empirical models cut out by linear causality equations inside ambient products of simplices. For a cover $Y_i$1 and output family $Y_i$2, the causaltope is

$Y_i$3

Empirical models are shown to be exactly the points of these causaltopes, and supported components, causal fractions, and causally separable fractions are computed by linear programs. The framework unifies causality, no-signalling, contextuality, and indefinite causal order within a single polyhedral language. A particularly important observation is that intersections of causaltopes need not correspond to the causaltope of the meet of the underlying spaces, so the incidence geometry of causal model space is richer than the combinatorics of causal orders alone (Gogioso et al., 2023).

“Greedy Causal Discovery is Geometric” gives a related but distinct polyhedral formulation for DAG discovery. Each Markov equivalence class is represented by a characteristic imset vector $Y_i$4, and the characteristic imset polytope

$Y_i$5

has vertices corresponding bijectively to Markov equivalence classes. Restrictions on admissible skeletons define faces of $Y_i$6, while turn pairs and edge pairs define classes of polytope edges. This yields the interpretation that GES, GIES, MMHC, and Greedy SP are greedy edge-walks along convex polytopes. The paper’s new algorithms—greedy CIM and skeletal greedy CIM—exploit this geometric view, and the hybrid variant is reported to outperform current competitors among hybrid and constraint-based algorithms in the reported simulations (Linusson et al., 2021).

The combinatorial lesson is that causal learning can be reformulated as navigation on structured geometric objects: convex slices, faces, adjacency graphs, and decomposition polytopes. The topological content in these papers remains largely standard polytope topology, but their relevance to CLGT lies in treating causal structure as a geometric search space rather than as a purely symbolic object.

5. Causal topology in spacetime, neutral geometry, and simplicial gravity

A mathematically separate CLGT tradition studies how topology is reconstructed from causal order or null geometry in spacetime. “Causal Topology in Future and Past Distinguishing Spacetimes” extends the Malament–Hawking–King–McCarthy framework below strong causality. It proves that if there is a causal bijection between two future- or past-distinguishing spacetimes, then they have the same manifold dimension, and it constructs a topology $Y_i$7 from convergence of sequences of chain-intervals—order-theoretic analogues of null geodesic segments. When the strong-causality-violating set $Y_i$8 is empty or locally manifold achronal, $Y_i$9-convergence is equivalent to manifold convergence, hence $E[Y^1-Y^0]$0 agrees with the manifold topology and is strictly finer than Alexandrov when strong causality fails (Parrikar et al., 2011).

“A New Causal Topology and Why the Universe is Co-compact” develops a different reconstruction route through causal sites, maximal centered sets, and framework duality. In the Minkowski construction, the topology obtained from causal-site data is compact and $E[Y^1-Y^0]$1, and is proved to be homeomorphic not to the Euclidean topology globally but to its de Groot dual, the co-compact topology. On every compact subset, however, the reconstructed topology and the Euclidean topology coincide (Kovár, 2011). The related review “Causal Cones, Cone Preserving Transformations and Causal Structure in Special and General Theory of Relativity” synthesizes cone-preserving maps, Alexandrov topology, Scott and Lawson topologies, and domain-theoretic reconstruction, emphasizing that in suitable settings the interval topology recovers the manifold topology from causal order (Janardhan et al., 2012).

Neutral-signature geometry supplies another route from null structure to topology. “The causal topology of neutral 4-manifolds with null boundary” studies $E[Y^1-Y^0]$2-manifolds with signature $E[Y^1-Y^0]$3, where the pointwise null cone is a cone over $E[Y^1-Y^0]$4 rather than over $E[Y^1-Y^0]$5. In the conformal compactification of flat neutral $E[Y^1-Y^0]$6-space into the $E[Y^1-Y^0]$7-ball, the boundary $E[Y^1-Y^0]$8 is null, the null foliation is the Hopf fibration, and the two totally null plane fields on the boundary are integrable. In tangent hypersurfaces in spaces of oriented geodesics, the corresponding $E[Y^1-Y^0]$9- and $Y_i^a$0-plane fields are contact rather than integrable. The paper proposes possible applications to neutral Kirby calculus, neutral knot invariants, and neutral Casson handles, but these applications are explicitly presented as possible topological applications rather than as completed theories (Georgiou et al., 2016).

Discrete Lorentzian gravity introduces yet another causal-topological invariant. “Causal structure and topology change in (2+1)-dimensional simplicial gravity” classifies 13 causal types of Lorentzian tetrahedra, excluding configurations with null faces, by how the lightcone at a vertex intersects the opposite face. For a bulk vertex, the null intersection curves on the triangulated link $Y_i^a$1 partition the sphere into connected spacelike and timelike regions, summarized by

$Y_i^a$2

Regular vertex causality is defined by $Y_i^a$3, and changes in this local causal-topological class coincide with singularities in deficit angles and the Regge action. The paper also shows that hinge causality and vertex causality are generally independent (Asante et al., 14 Jul 2025).

This spacetime branch is not causal learning in the statistical sense. Its relevance to CLGT is structural: it demonstrates that causal order, null cones, and simplicial lightcone intersections can determine dimension, boundary topology, or local topology-change signatures.

6. Applications, invariance learning, interpretability, and open problems

Task-oriented CLGT systems use these geometric and topological ideas operationally. In cross-view geo-localization, the paper titled “Causal Learning and Geometric Topology” introduces a framework with three modules: a Causal Feature Extractor, Geometric Topology Fusion, and Data-Adaptive Pooling. The CFE uses a DCT-based content-aware mask to perturb presumed non-causal low- and high-frequency components while preserving mid-frequency structural content; GT Fusion injects Bird’s Eye View road topology into street features via cross-attention and Dual Dynamic Fusion; and DA Pooling combines max, average, and GeM pooling through a learned gate. The total loss is

$Y_i^a$4

On CVUSA, CVACT, and corruption benchmarks, the paper reports higher Recall@1 than the EP-BEV baseline, with especially strong gains under corruptions and negligible computational overhead relative to EP-BEV (Ouyang et al., 13 Mar 2026).

Topological invariance learning appears in a different form in knot theory. “Learning Topological Invariance” trains contrastive and generative models so that different representatives of the same knot class map to the same embedding point,

$Y_i^a$5

or decode to a fixed representative of that class. The paper’s centroid loss and autoregressive decoder are designed to quotient out ambient isotopy as represented through braid relations and Markov moves. Its student–teacher interpretation experiments report a strong correlation between the learned embedding and the Goeritz matrix across all tested setups (Halverson et al., 16 Apr 2025).

Causal control in internal geometric representations also appears in sparse MoE LLMs. “Geometric Routing Enables Causal Expert Control in Mixture of Experts” studies cosine-similarity routing in a low-dimensional metric space and reports four main findings: 15% of experts are monosemantic specialists spanning 10 categories; routing exhibits a frequency-to-syntax gradient with all reported $Y_i^a$6; steering toward a temporal expert’s centroid increases $Y_i^a$7 by $Y_i^a$8 median across 44 prompts; and comparable steering is possible in linear routers, but only cosine routing exposes specialization directly through the centroid geometry (Ternovtsii et al., 15 Apr 2026). Here again, the geometric organization is not claimed to be topology in the strict sense; its value is that it makes causal intervention targets directly readable.

The limitations across CLGT are correspondingly diverse. In topological causal inference on persistence summaries, silhouettes aggregate persistence features and may obscure the exact number of changing homological features, while persistent homology remains computationally expensive and the choice of $Y_i^a$9, filtration, $d$0, and truncation rules remains open (Kim et al., 2 Mar 2026). In CMF, the method is local rather than global, uses Hessian discrepancy as a proxy for curvature, and is validated only on a synthetic SCM with known counterfactual structure (Rathore, 6 Jan 2026). In continuous-time causal representation learning, identifiability is framed for a weakly convergent equivalent class rather than exact recovery, and the fully nonparametric kernel case remains unresolved (Ren et al., 4 Feb 2026). In polyhedral discovery, only subsets of polytope edges are characterized, and performance depends on search policy (Linusson et al., 2021). In topological learning theory for SCMs, the strongest positive message is also a negative one: broad collapse-enabling assumptions are topologically meager and statistically unverifiable (Ibeling et al., 2021).

A recurring misconception is that all CLGT work is “topological” in the same sense. The surveyed literature does not support that claim. Some contributions are genuinely topological, such as causal effects on persistent-homology summaries, topology recovery from causal order, knot-class quotient learning, or simplicial link classifications. Others are more accurately described as differential-geometric, algebraic-geometric, polyhedral, or metric-geometric. The unifying feature is not a single topology, but the systematic use of non-Euclidean structure to define, identify, inspect, or intervene on causal phenomena.