Latent SSM and RSSM Models

• Latent state-space models (SSM/RSSM) are probabilistic frameworks that model time-series data through unobserved, evolving latent states governed by Markovian transitions.
• Modern variants leverage deep neural networks to capture nonlinear, high-dimensional dynamics, enhancing applications in control and forecasting.
• Scalable inference methods, including variational and particle filtering, enable effective system identification and robust performance on complex temporal tasks.

Latent state-space models (SSMs) and recurrent state-space models (RSSMs) are a foundational class of probabilistic dynamical systems modeling frameworks. They posit that observed time series are governed by the evolution of unobserved latent states, linking system identification, time-series analysis, control, and modern deep sequence learning. In SSM/RSSM, at each time step $t$, a typically vector-valued latent state $x_t$ or $z_t$ evolves according to a Markovian transition, possibly conditioned on exogenous inputs, and generates observations via a possibly nonlinear emission mechanism. Technical innovation has enabled the modeling of nonlinear, high-dimensional, and temporally correlated phenomena, with inference and training regimes spanning classical Kalman filtering to doubly stochastic variational Bayesian deep learning approaches, and recent integration as architectural building blocks in foundation-scale neural sequence models.

1. Mathematical Foundations and Model Classes

The archetypal state-space model comprises latent state evolution and emissions:

• Discrete-time latent SSM:
  • State: $x_t \in \mathbb{R}^{D_x}$
  • Input: $u_t \in \mathbb{R}^{D_u}$
  • Observation: $y_t \in \mathbb{R}^{D_y}$
  • Dynamics: $p(x_{t+1} | x_t, u_t)$ (generative transition)
  • Emission: $p(y_t | x_t)$ (observer)
  • Initial: $p(x_1)$

For linear–Gaussian SSMs, transitions and emissions take the form $x_{t+1} = A x_t + B u_t + w_t$, $y_t = C x_t + D u_t + v_t$, with $w_t \sim \mathcal{N}(0,Q)$ and $v_t \sim \mathcal{N}(0,R)$ (Alonso et al., 2024), admitting Kalman filter-based tractable inference. Modern variants parameterize $A,B,C,D$ via neural networks to capture nonlinear dynamics in “deep SSMs” (Lin et al., 2024).

RSSMs extend SSMs with explicit deterministic RNN states $h_t$ and a stochastic latent $z_t$, often intertwined for improved temporal representation (Lin et al., 2024). Hierarchical factorization, global and per-object latent interaction, and normalizing flow-based transition distributions appear in models of complex systems (Yang et al., 2020).

2. Scalable Inference: Variational and Particle-Based Methods

For general nonlinear, high-dimensional or non-Gaussian latent SSMs/RSSMs, closed-form inference is intractable. Scalable inference proceeds by approximate techniques:

• Structured Variational Inference: Exploits block-tridiagonal and Kalman-like structure in the posterior covariance, maintaining temporal correlations and yielding efficient Gaussian smoothing approximations. Both “direct” and “product-of-Gaussians” parameterizations are used to encode posterior mean and covariance with neural nets (Archer et al., 2015). The evidence lower bound (ELBO) is maximized via stochastic gradient backpropagation, leveraging the reparameterization trick for low-variance estimates.
• Doubly Stochastic Variational Inference: In PR-SSM, the true posterior over latent state chains and GP transition functions is approximated with sampling-based Markovian variational chains (over latent state trajectories and GP inducing points), and training is performed using a doubly stochastic ELBO—minibatching both in time (sub-trajectories) and inducing point space (Doerr et al., 2018).
• Particle Filtering and SMC/HMC: For highly nonlinear or non-Gaussian regimes, sequential Monte Carlo (SMC) or SMC augmented with Hamiltonian Monte Carlo (HSMC) techniques are deployed. HSMC integrates Riemannian-manifold HMC steps to directly sample from transition priors with local geometry adaptation, bypassing the need for explicit learned proposals and tightening the ELBO for efficient latent inference (Xu, 2019). Stochastic gradient MCMC with buffered particle filters yields scalable Bayesian learning even under strong temporal correlations (Aicher et al., 2019).
• Deterministic (Sampling-Free) Inference: Recent methods such as ProDSSM push for deterministic layerwise moment-propagation through assumed-dense posteriors, propagating Gaussian beliefs through neural network transitions and emissions without particle sampling, reducing variance and memory magnitude (Look et al., 2023).

3. Model Structure Innovations and Deep Architectures

Emergent deep learning trends position linear and nonlinear SSM blocks as backbone modules in sequence models. Notable architectural instantiations:

• Parameterizations for Efficient Sequence Modeling: S4, S4D, S5, LRU, S6/Mamba, and RG-LRU models deploy various linear and time-varying structures for the state transition matrix $A$ and input/output operators, ensuring spectral placement for long-range dependency, stability, and controllability (Alonso et al., 2024). Foundation models such as GPT-4 leverage SSM scaffolding for improved sequential compression and memory.
• Relational State-Space Modeling: Multi-object and graph-structured domains utilize GNN parameterizations for latent transitions, integrating relational inductive bias and modeling cross-object temporal interaction. Hierarchical global and per-object latent factors, deep GNN flows, and contrastive objectives yield richly coupled systems with scalable inference (Yang et al., 2020).
• Continuous-Time and Mixed-Frequency Extensions: Latent neural ODE/SDE frameworks generalize SSMs to irregular or continuous time, using adjoint backpropagation or Euler–Maruyama integration for learning and inference in latent dynamical systems (Lin et al., 2024).

4. Training Algorithms, Scalability, and Practical Implementation

Efficient training pipelines typically employ:

• Minibatched Stochastic Optimization: Subsequence-based minibatching (in time and/or inducing points for GP transitions) enables tractability on long sequences and large datasets (Doerr et al., 2018). Wall-clock scaling to sequence length is a critical design constraint; parallel scan and prefix computation (in SSM layers, KalMamba, and related architectures) achieves near-logarithmic complexity (Becker et al., 2024).
• Recognition/Inference Networks: Neural networks (MLPs, RNNs, GNNs) parameterize variational posterior moments (mean, covariance) and context encoding—directly outputting kernel parameters for filtering/smoothing (Archer et al., 2015, Pfrommer et al., 2022).
• Exact Gaussian Inference for Changing Dynamics: RSSMs with hidden parameters (HiP-RSSM) exploit global task-specific latent variables for nonstationary systems; all inference remains exact under Gaussian graphical models, no variational approximation required (Shaj et al., 2022).

5. Empirical Performance and Applications

Latent SSM/RSSM frameworks have demonstrated competitive results on diverse tasks:

• System Identification: PR-SSM matches or outperforms Markovian GP-SSM baselines and autoregressive GP methods (REVARB/MSGP) in RMSE on nonlinear real-world physical systems (e.g., SARCOS, ballbeam, furnace, etc.), scaling robustly to thousands of points and high latent dimension (Doerr et al., 2018).
• Multimodal Sequence Forecasting: LLM-integrated Bayesian SSMs (LBS) provide uncertainty-aware numeric and textual forecasts, improving RMSE by up to 13.2% over previous SOTA on the TTC benchmark (Cho et al., 23 Oct 2025).
• Control under Uncertainty: KalMamba integrates scalable SSM-based probabilistic inference with foundation-model sequence modules, achieving near-logarithmic scaling and robust performance on RL benchmarks (Becker et al., 2024).
• Relational and Multi-agent Dynamics: R-SSM yields substantial likelihood, coverage, and rollout gains in synthetic coupled systems and real multi-agent trajectory datasets, outperforming VRNN and GNN-autoregressive baselines (Yang et al., 2020).
• Imputation and Filtering from High-Dimensional Observations: VSSF and L-VSSF filter trajectories far beyond training horizon even under partial or missing data, aligning latent coordinates to interpretable physical quantities (Pfrommer et al., 2022).

6. Theoretical Insights, Limitations, and Future Directions

Recent theoretical work studies learning dynamics for deep SSMs, showing:

• Frequency-Domain Analysis: Diagonalizable SSMs under gradient descent admit closed-form analytical solutions in the frequency domain, with convergence time sensitive to latent dimensionality and covariance spectra (Smékal et al., 2024). Over-parameterization yields linear or quadratic speedups, and deep SSMs mirror deep linear networks mode-wise learning trajectories.
• Control-Theoretic Guarantees: Stability, controllability, and observability are achieved via spectral placement of transition matrices and orthogonal polynomial initialization (HiPPO), ensuring long-term memory and trainability (Alonso et al., 2024).
• Deterministic SSM Layer Scaling: SSM layers now parallel Transformer efficiency on long sequence tasks but with lower memory footprint, although input-adaptive (LTV) variants currently underperform, indicating the need for refined control-theoretic design (Alonso et al., 2024).
• Posterior Collapse, Multi-modality, and Uncertainty: Posterior collapse is mitigated by contrastive regularization and structured flows (Yang et al., 2020); sampling-free deterministic inference offers superior trade-off for predictive performance and budget (Look et al., 2023); integration of GPs and non-diagonal covariances expands modeling power but poses computational challenges (Doerr et al., 2018, Becker et al., 2024).

Limitations persist for highly nonlinear, non-log-concave, or multi-modal latent processes; extensions to rich observation models (images, text), unsupervised clustering of latent task parameters, and foundation-scale robust SSM blocks are active areas of research.

7. Cross-Disciplinary Connections and Interpretability

Latent SSMs unify paradigms from control theory (system identification, filtering), machine learning (VAE, recognition networks, neural ODE/SDE), reinforcement learning (world models, belief-state RL), and now foundation-model architectures. Continuous advances ensure enhanced efficiency, uncertainty quantification, and interpretability—whether through semi-supervised alignment to physical variables (Pfrommer et al., 2022), unsupervised discovery of task parameters (Shaj et al., 2022), or multimodal generation via LLM-SSM interfaces (Cho et al., 23 Oct 2025). The SSM framework remains a cornerstone for rigorous temporal reasoning in both classical and deep learning contexts.