Temporal Causal Disentangled Factorization

• Temporal causal disentangled factorization is a framework that separates the underlying, causally related latent factors driving time-evolving data.
• It integrates algebraic methods, information-theoretic decomposition, and mechanism sparsity to address feedback, nonstationarity, and confounding.
• Evaluation metrics like IRS and unconfoundedness reveal challenges in full disentanglement, emphasizing partial identification and residual entanglement.

Disentangled factorization in temporal causal systems refers to the theoretical and methodological process of separating the underlying, causally related factors that drive observed time-dependent data. In temporally evolving systems, factorization is complicated by feedback, causal interplay, nonstationarity, and potential structural or observational confounding. Disentangled factorization thus extends traditional concepts of statistical or independent factorizations by imposing, leveraging, or recovering (possibly partially) the true temporal and causal structure among latent components. Analyses span algebraic, information-theoretic, probabilistic, and deep learning frameworks, often with attention to identifiability, intervention, and model evaluation under domain shifts and complex dependence.

1. Foundational Theory: Temporal Causal Factor Models

Central to the theory of disentangled factorization in temporal causal systems is the dynamic latent factor model of the form

$$Y_t = A n_t + \varepsilon, \qquad n_t = B n_{t-1} + W_t,$$

where $Y_t$ represents observed indicators or measurements, $n_t$ is a vector of latent factors, $A$ is a loading matrix, $B$ encodes temporal and potentially causal relationships between factors, and $W_t$ is a noise or innovation term (VanderWeele et al., 2020). The causal relationships among latent factors are specified in $B$; when $B$ is not the identity, factors causally influence each other over time.

Temporal iteration and Jordan decomposition (writing $B = Q D Q^{-1}$) reveal that persistent causal connections can cause the process to collapse to lower effective dimensionality: unless $B = I$, repeated application causes convergence to a rank-deficient structure. For example, with two factors, any off-diagonal element in $B$ (i.e., any causal effect) leads to asymptotic unidimensionality in the observed space, even if two conceptually distinct factors exist initially.

This result generalizes to $k$-factor models: mutual causal influences within an "equivalence class" can collapse multiple factors into a single effective component, and stationary covariance structure at equilibrium becomes indistinguishable from a lower-rank factor model. Consequently, unless the absence of causal relations is established a priori, single-wave factor analysis cannot rule out collapsed, causally entangled multidimensionality (VanderWeele et al., 2020).

2. Algebraic Disentanglement and Latency Kernels

An important algebraic perspective considers dynamic linear systems as operators over Laurent series modules, distinguishing causal and strictly causal maps by their action on submodules representing the temporal "past" (Hammer et al., 2020). Here, the causal factorization problem is: $\bar{h} = \bar{g} \bar{f},$ where $\bar{f}, \bar{h}$ are Laurent-series linear maps, and $\bar{g}$ is a sought-after causal operator. The necessary and sufficient condition for a causal factorization is nestedness of the "latency kernels" ($\Delta = \ker \pi^- \bar{f}$), where $\pi^-$ is the projection onto the future-quotient module.

Latency indices $\nu_i = -\operatorname{ord}(d_i)-1$ (where $d_i$ are basis elements of $\Delta$) quantify intrinsic system delays, and "proper bases" of the kernel correspond to the irreducible delay factors. This algebraic invariant remains canonical under bicausal (invertible and causal in both directions) transformations and gives a precise characterization of disentanglability: the delayed response structure must be compatible between the system and its possible compensators or factorizations.

3. Information-Theoretic Decomposition of Temporal Causality

Information-theoretic approaches to disentangled factorization frame causal influence in terms of certainty and mutual information, separating out contributions of direct, joint, and higher-order linkages between causes and their temporal targets (Leeuwen et al., 2020). Conditional mutual information is recursively decomposed: $$I(x; y, z) = I(x; y | z) + I(x; z | y) + \text{synergy/2-link terms}$$ This yields a normalized "causal strength" measure (e.g., $cs(x; y_{1:N}) = I(x; y_{1:N}) / W(x|y_{1:N})$), where $W$ is certainty, facilitating the quantification and disentanglement of both direct and joint (polyadic) causal effects, as well as the detection of missing, hidden, or unresolved drivers through residual certainty contributions.

Unlike standard DAG-based frameworks that parse only dyadic relations and often miss joint nonlinearities, this approach generalizes naturally to systems with polyadic, memory-laden, or feedback structures common in temporal causal systems (e.g., ENSO climate indices or the Lorenz attractor), and can assign causal strength even to unresolvable processes.

4. Partial Disentanglement and Mechanism Sparsity

True disentanglement—identification of latent variables up to permutation and scale—can only be guaranteed under specific graphical criteria: the causal graph must allow separation between source mechanisms (Lachapelle et al., 2022). When the ground-truth graph is arbitrary, only partial disentanglement is achievable. This is formalized using a "consistency" equivalence: $L = CP^{\top}$ where $C$ conforms to structure-induced sparsity (S-consistency), and $P$ is a permutation. The learned latent space can be disentangled only to the degree permitted by the graph's sparsity pattern; mechanism sparsity (enforced via constrained $\ell_0$ optimization on the edge structure) enables maximal separation compatible with the ground truth (as measured by block sparsity of $L$), but never more.

Thus, in complex temporal causal systems, partial disentanglement reflects the extent to which factorization is encapsulated by the true temporal/causal connectivity: some factors remain entangled when inseparable according to the system's mechanism graph.

Setting	Identifiability	Relevant Condition
Graphical crit.	Up to permutation/scale	Strict separator for each latent (see item 4 above)
General	Partial (block) disentanglement	S-consistency (zeros in $C$ per causal graph structure)

5. Causal Disentanglement and Confounding

Causally disentangled representations demand not only factorized statistical variation but also robustness under observation of confounders. If latent factors (e.g., color and shape) are statistically correlated due to a hidden common cause (e.g., object class or environment), independence-based disentanglement fails. Conditioning on confounder labels (or partitioning by domain knowledge), and enforcing independence within those partitions, enables "C-disentanglement" (Liu et al., 2023):

$P(Z_i \mid do^c(Z_{-i}), X) = P(Z_i \mid X)$

for all $i$, where $do^c$ denotes an intervention (setting all but $Z_i$ arbitrarily) performed within confounder-defined strata. Utilizing mixture-of-Gaussians latents with diagonal covariance in each subgroup, the method achieves true causal identification and robust generalization across distribution shifts, directly applicable to temporally varying confounding in time-series contexts.

6. Evaluation: Metrics, Simulations, and Practical Implications

Metrics suitable for disentangled factorization in temporal causal systems must account for both structural (graph-aware) properties and intervention behavior. Measures such as Interventional Robustness Score (IRS), Unconfoundedness (UC), and Counterfactual Generativeness (CG) evaluate whether interventions or manipulations in latent factors yield consistent and isolated changes in the output—across domains and under realistic confounding—(Reddy et al., 2021, Liu et al., 2023).

Simulation studies across linear, nonlinear, and time-delayed settings consistently show that:

• When causal interplay presents, traditional methods (e.g., one-wave factor analysis, independence-based VAE) conflate dimension, mask distinct but related factors, or conflate confounded effects.
• Mechanism-sparse/directed methods match (up to consistency) the true causal structure, and clarify precisely which factors can be separated.
• Information-theoretic and algebraic perspectives add complementary insights: the former quantifies both direct and higher-order effects (and unresolved drivers), while the latter canonicalizes structural delays and factor dependencies.

These implications are acutely relevant to practical domains (e.g., psychometrics, genetics, climate, medical longitudinal studies) where time-evolution and feedback pervade.

7. Open Problems and Future Directions

Despite recent advances, major challenges persist:

• Indistinguishability from Cross-sectional Data: As shown in (VanderWeele et al., 2020), equilibrium covariance structures may arise identically from unidimensionality or from collapsed, dynamically entangled factors, rendering cross-sectional analyses unreliable for causal interpretation.
• Non-intervenability/Observational Scarcity: Most identifiability results rely on either repeated temporal structure, mechanism sparsity, or side information. In the absence of interventions or confounder labels, only identification up to layer (upstream) structure is generally possible (Welch et al., 31 Oct 2024).
• Partial Disentanglement as a Theoretical Ceiling: Complete separation of all latent factors is generically impossible except under restrictive causal graphs; progress is now focused on quantifying, rather than eliminating, residual entanglement.

Table: Summary of Key Theoretical Limitations

Limitation	Paper	Scope of Indeterminacy/Ambiguity
Collapsed dimensions under causality	(VanderWeele et al., 2020)	Single-wave factorizations conflate causal factors
Partial identifiability via sparsity	(Lachapelle et al., 2022)	Structure-induced blocks in transformation matrix
Observational-only identifiability	(Welch et al., 31 Oct 2024)	Layer-wise separation, but not within-layers

This suggests that progress in temporal causal disentanglement will require further integration of temporal dynamics, observational integration, mechanism sparsity, and the judicious use of side information or regularization.

• "On the dimensional indeterminacy of one-wave factor analysis under causal effects" (VanderWeele et al., 2020)
• "Causal Factorization and Linear Feedback" (Hammer et al., 2020)
• "A Framework for Causal Discovery in non-intervenable systems" (Leeuwen et al., 2020)
• "Partial Disentanglement via Mechanism Sparsity" (Lachapelle et al., 2022)
• "On Causally Disentangled Representations" (Reddy et al., 2021)
• "C-Disentanglement: Discovering Causally-Independent Generative Factors under an Inductive Bias of Confounder" (Liu et al., 2023)
• "Identifiability Guarantees for Causal Disentanglement from Purely Observational Data" (Welch et al., 31 Oct 2024)