Dynamic-Static Disentanglement Design

• Dynamic-static disentanglement design is a modeling paradigm that separates time-invariant (static) factors from time-varying (dynamic) components to enhance clarity in sequential data.
• It employs parallel latent branches and specialized regularization techniques to force each branch to model either static or dynamic elements without supervision.
• Practical implementations in robotics, video, audio, and graph domains demonstrate improved reconstruction accuracy and causal identifiability with optimized architectural choices.

Dynamic-Static Disentanglement Design is a representational architecture and modeling paradigm whose goal is the separation (“disentanglement”) of latent factors corresponding to time-invariant (static) and time-varying (dynamic) components in observed data. This dichotomy is fundamental in sequential domains—such as robotics, video, audio, graph evolution, and even parallel programming—where explanatory power, interpretability, and controllability depend upon identifying which elements of a sequence or environment can be controlled or predicted, and which persist as background or invariant structure. The central technical contribution of dynamic-static disentanglement designs is the construction of neural and probabilistic models in which distinct internal representations are forced to specialize either to dynamic or to static explanatory roles, typically in a fully unsupervised manner.

1. Model Architectures for Dynamic-Static Separation

Dynamic-static disentanglement design typically employs parallel latent branches, each dedicated to either static or dynamic factors. In paradigmatic implementations such as that of "Disentangling Controllable and Uncontrollable Factors of Variation by Interacting with the World" (Sawada, 2018), two deep neural networks (DNNs) are jointly trained on raw sequential inputs $x$:

• Controllable branch $(f_c, g_c, \pi_{\psi})$: Encodes and reconstructs only controllable (dynamic) objects. $f_c: x \mapsto \mathbb{R}^K$ where $K$ is the number of atomic actions.
• Uncontrollable branch $(f_u, g_u)$: Encodes and reconstructs all uncontrollable (static) obstacles. $f_u: x \mapsto \mathbb{R}^M$, $M$ is chosen based on scene complexity.

The total reconstruction is additive: $\hat{x} = g_c(f_c(x)) + g_u(f_u(x))$. Policy networks $\pi_{\psi_k}$ are attached to each controllable latent dimension to measure selectivity under their corresponding actions. Static and dynamic encoding branches are encouraged to occupy orthogonal subspaces via architectural separation and regularization.

In sequence VAE methods, e.g., S3VAE (Zhu et al., 2020), the architecture factorizes the latent space into a sequence-constant static code $z_f$ and sequence-varying dynamic codes $z_{1:T}$, with independent inference and prior modules for each factor. This ensures that the static code $z_f$ cannot encode time-localized information, while dynamic codes $z_t$ are tightly bound to the temporal evolution.

2. Optimization Objectives and Disentanglement Constraints

Separation of static and dynamic factors is fundamentally enforced through hybrid reconstruction and regularization objectives. The canonical loss function in dynamic-static disentanglement models, as in (Sawada, 2018), comprises:

• Reconstruction loss:

$$L_{\text{recon}} = \mathbb{E}_{x \sim \mathcal{D}} \| x - [g_c(f_c(x)) + g_u(f_u(x))] \|^2$$

which forces both branches to jointly reconstruct the data.

• Selectivity regularizer:

$$S_{i,k} = \sum_{a=1}^K \pi_{\psi_k}(a|z_c) \sum_{s' \sim P(\cdot|s,a)} \log \left(\frac{1}{K} + \frac{|z_{c,k}(x') - z_{c,k}(x)|}{\sum_{j=1}^K |z_{c,j}(x') - z_{c,j}(x)|} \right)$$

which encourages each dynamic latent dimension to track only its corresponding controllable action.

The full objective is: $$L(\Phi, \Theta, \Psi) = \mathbb{E}_{x \sim D} \bigl[ \| x - (g_c(f_c(x)) + g_u(f_u(x))) \|^2 \bigr] - \lambda \sum_{k=1}^K \mathbb{E}_{x \sim D}[ S_k(f_c(x), \pi_{\psi_k}) ]$$ with $\lambda$ controlling the balance.

S3VAE (Zhu et al., 2020) enhances classical ELBO objectives with triplet consistency for static codes, dynamic factor prediction regularizers, and mutual-information penalties. TS-DSAE (Luo et al., 2022) introduces a two-stage constrained/informed-prior ELBO framework with swap-based KL regularizers to robustly suppress static-dynamic leakage.

These regularization techniques are operative at the behavioral (representation) level and are often supported by explicit architectural decoupling or domain-specific constraints.

3. Identifiability, Causality, and Theoretical Guarantees

Recent advances formalize identifiability conditions for dynamic-static models under realistic generative settings. For instance, in (Simon et al., 10 Aug 2024), the generative model is specified by

$$z_{1:T} = (s, d_{1:T}) \sim p(s) \cdot p(d_1 | s) \cdot \prod_{t=2}^T p(d_t | s, d_{<t}), \quad x_t = g(s, d_t)$$

where $s$ is static and $d_t$ is conditional dynamic.

Comprehensive identifiability results (Def. 1; Props. 1-3) necessitate:

• Explicit modeling of $p(d_t|s, d_{<t})$, capturing dependencies of dynamics on static.
• Sufficient code dimension for latent variables.
• Architectural bijectivity (conditional normalizing flows).

Crucially, a single shuffle constraint in the ELBO, which permutes static code estimates across frames before aggregation, is both necessary and sufficient for disentanglement under these causal models.

In generative adversarial settings, CoVoGAN (Shen et al., 4 Feb 2025) demonstrates provable disentanglement by enforcing minimal change (low dynamic latent dimension, explicit independence between static and dynamic) and sufficient change (component-wise conditional independence via normalizing flows). Identifiability theorems guarantee recovery of ground-truth factorization up to invertible mappings, assuming the specified linear operators over latent transitions are injective.

4. Implementation Strategies and Practical Considerations

A number of practical design choices consistently improve dynamic-static disentanglement stability:

• Pretraining: Train dynamic and static branches separately in environment conditions where each is unambiguous (e.g., dynamic-only in obstacle-free scenes), then use learned parameters to initialize joint training (Sawada, 2018).
• Inductive bias via architecture: Single-sample static code extraction, subtraction-based encoding for dynamics, and decoupled posterior factorization robustly suppress leakage (Berman et al., 26 Jun 2024, Gheisari et al., 18 Jul 2025).
• Temporal regularization: In autoregressive or recurrent models, swap-based KL losses and temporal noise-sharing (for diffusion models) can enforce independence and consistency (Luo et al., 2022, Gheisari et al., 18 Jul 2025).
• Parameter selection: Hyperparameters such as $\lambda$ for selectivity and swap regularizer weights must be carefully tuned (e.g., $\lambda \in [0.01, 0.1]$ in (Sawada, 2018), linear KL weights in (Gheisari et al., 18 Jul 2025)).
• Modality-agnostic encoding: Dynamic-static frameworks generalize naturally across images, video, audio, general time series, and even graph-structured data (D2G2 (Zhang et al., 2020)).

Efficiency gains are often realized by reducing storage required for synthetic data, e.g., distilling video into small static and dynamic memory blocks plus tiny integrator networks (Wang et al., 2023).

5. Empirical Validation and Benchmarks

A range of metrics and evaluation protocols have become standard:

• Correlation metrics: For instance, Corr$(f_{c,k}, \text{true}\;x/y)$ quantifies alignment between latent codes and true dynamic variables (Corr$>$0.9 in (Sawada, 2018)).
• Swap-based accuracy: Swap static and dynamic latent codes between samples and measure classification accuracy on generated sequences (Zhu et al., 2020, Gheisari et al., 18 Jul 2025).
• Mutual Information Gap (MIG): Measures the concentration of each ground-truth factor in one latent code (Yamada et al., 2019).
• Ablation studies: Evaluate the impact of disabling dynamic or static branches, removing regularizers, or omitting pretraining (Sawada, 2018, Wan et al., 4 Dec 2025).
• Storage and robustness: Compare model performance and efficiency when varying static/dynamic memory budgets, or under missing-modality scenarios (Wang et al., 2023, Wan et al., 4 Dec 2025).
• Domain-specific metrics: Fréchet Audio/Video Distance, AUROC/AUPRC for prediction tasks, equal error rate (EER) for speaker/audio identity (Luo et al., 2022, Gheisari et al., 18 Jul 2025, Wan et al., 4 Dec 2025).

Tables in benchmark frameworks, e.g., MSD (Barami et al., 20 Oct 2025), catalogue datasets across modalities and enumerate static/dynamic factors and sequence lengths.

Benchmark Dataset Table

Dataset	Modality	Factors (Type, Classes)
BMS Air Quality	Time series	Station (static,12), Month/Year/Day/Season (static)
dMelodies-WAV	Audio	Instrument (static, 4), Rhythm/Chord/Tonic/Scale (dynamic, multi-class)
dSprites-Static	Video	Color, Shape, Position (static); ScaleSpeed, RotationSpeed (dynamic)
3D Shapes	Video	FloorHue, WallHue, ObjHue, Shape (static); Scale, Orientation (dynamic)

Zero-shot consistency is evaluated via classifiers or Vision-LLMs, which have demonstrated near-perfect ranking alignment with ground-truth labels for multi-factor sequential disentanglement (Barami et al., 20 Oct 2025).

6. Extensions and Generalizations

Dynamic-static disentanglement frameworks are being actively extended in several directions:

• Graph domains: Factorized VAE architectures for dynamic graphs distinguish static (topology) from multiple dynamic factors (node attributes, edge presence, hybrid) (Zhang et al., 2020).
• Multi-modality clinical modeling: Spatiotemporal disentanglement in disease progression uses region-aware encoders, explicit orthogonality constraints, and temporal-consistency losses to separate anatomical from pathologic features (Liu et al., 13 Oct 2025).
• Time-attenuated curve modeling: In biomedical imaging, static-dynamic factorization enables hallucination and interpolation of missing modalities (e.g., contrast phases in CT) while maintaining robust prediction (Wan et al., 4 Dec 2025).
• Diffusion models: Modal-agnostic diffusion autoencoders achieve state-of-the-art sequential disentanglement without complex multi-term objectives (Zisling et al., 7 Oct 2025). Shared-noise schedules and cross-attention are key novel inductive biases (Gheisari et al., 18 Jul 2025).
• Programming language runtime: Static-dynamic disentanglement is recast in parallel systems as "task-locality" (disentanglement), enforced statically by timestamped type systems and dynamic fork-join semantics (Moine et al., 28 Nov 2025).

7. Limitations and Open Challenges

Although dynamic-static disentanglement designs yield interpretable, modular representations, they face challenges:

• Leakage and over-regularization: Information leakage remains possible, particularly with large dynamic latent dimensions or insufficient regularization (Berman et al., 26 Jun 2024).
• Collapse and label-switching: Without proper initialization or swap-based regularizers, networks may swap explanatory roles or collapse codes entirely (Sawada, 2018, Luo et al., 2022).
• Multi-factor and hierarchical extension: Current theory addresses binary static/dynamic splits; practical systems require multi-factor (hierarchical) disentanglement and causal dependency modeling (Barami et al., 20 Oct 2025, Simon et al., 10 Aug 2024).
• Evaluation complexity: Benchmarking requires multi-metric, multi-modal, intervention-based consistency analysis, facilitated recently by automated zero-shot VLMs (Barami et al., 20 Oct 2025).
• Expressivity vs. tractable learning: In TypeDis (Moine et al., 28 Nov 2025), subtiming and timestamp polymorphism challenge decidable type inference and demand explicit annotations.

A plausible implication is that future designs will combine factorized latent architectures, causal modeling, expressive normalizing flows, and diffusive sampling, with automated evaluation frameworks and domain-specific regularizers to address these limitations.

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Dynamic-static disentanglement design thus occupies a central methodological role in sequential modeling, offering principled and practically validated approaches for separating time-invariant structure and temporally evolving phenomena across diverse neuroscientific, engineering, and computational domains (Sawada, 2018, Zhu et al., 2020, Luo et al., 2022, Wan et al., 4 Dec 2025, Gheisari et al., 18 Jul 2025, Barami et al., 20 Oct 2025, Moine et al., 28 Nov 2025).