State-Space Models: A Unified Framework

• State-space models (SSMs) are probabilistic frameworks that describe dynamic systems via an evolving latent state with initializer, transition, and observation distributions.
• They facilitate classical inference tasks such as filtering, smoothing, and parameter estimation, while extending to irregular sampling and nonlinear dynamics.
• Recent advances incorporate deep variational methods and structured recurrent architectures like S4/S5 and Mamba, enhancing long-range sequence modeling and efficiency.

State-space models (SSMs) are models in which an observed sequence is governed by an evolving latent state, with the canonical probabilistic specification given by an initialization distribution, a transition distribution, and an observation distribution. In discrete time this appears as $$p_\theta(x_{0:T}, y_{1:T}) = p_\theta(x_0)\prod_{t=1}^T p_\theta(x_t\mid x_{t-1})p_\theta(y_t\mid x_t)$$, while in continuous time a widely used linear form is $h'(t)=Ah(t)+Bx(t)$, $y(t)=Ch(t)$. Contemporary usage is broader: the same term now covers classical latent-variable time-series models, continuous-time irregularly observed systems, deep variational sequence models, and structured recurrent operators such as S4-, S5-, and Mamba-style sequence layers (Hargreaves et al., 29 May 2025, Liu et al., 2024, Lin et al., 2024).

1. Foundational formulations

The classical SSM assumes that a latent process captures the dynamic information of a system and that observations are generated conditionally from that latent state. In the probabilistic form emphasized in recent software and review work, the model is determined by $p_\theta(x_0)$, $p_\theta(x_t\mid x_{t-1})$, and $p_\theta(y_t\mid x_t)$, together with the standard prediction-update recursion

$$p_\theta(x_t \mid y_{1:t-1}) = \int p_\theta(x_t \mid x_{t-1}) p_\theta(x_{t-1} \mid y_{1:t-1}) dx_{t-1},$$

$$p_\theta(x_t \mid y_{1:t}) \propto p_\theta(y_t \mid x_t) p_\theta(x_t \mid y_{1:t-1}).$$

The discrete-time functional view writes the latent state as $$\boldsymbol z_t = f_t(\boldsymbol z_{t-1},\boldsymbol u_t;\theta)$$ and the observation as $$\boldsymbol x_t = g_t(\boldsymbol z_t,\epsilon_t;\theta)$$, whereas the continuous-time linear view writes $h'(t)=Ah(t)+Bx(t)$0, $h'(t)=Ah(t)+Bx(t)$1 (Hargreaves et al., 29 May 2025, Lin et al., 2024, Liu et al., 2024).

A persistent distinction in the literature is between probabilistic latent-variable SSMs and architectural SSMs. The review literature treats the former as models for filtering, smoothing, and system identification, and the latter as efficient sequence-processing modules. In the architectural setting, the same continuous-time linear system admits the convolutional representation

$h'(t)=Ah(t)+Bx(t)$2

and after zero-order hold discretization becomes

$h'(t)=Ah(t)+Bx(t)$3

with $h'(t)=Ah(t)+Bx(t)$4 and $h'(t)=Ah(t)+Bx(t)$5. This is the formal bridge between recurrence and convolution that underlies S4-style sequence models (Liu et al., 2024).

The term “SSM” therefore names a family rather than a single model. It includes linear Gaussian systems amenable to Kalman filtering, nonlinear and non-Gaussian latent-variable models, continuous-time models for irregular sampling, and structured recurrences used as alternatives to Transformers in long-sequence modeling. This suggests that the modern field is unified more by the state-transition/emission template than by any single inferential or computational technique (Lin et al., 2024).

2. Inference, likelihoods, and irregular time

In classical probabilistic usage, the central inferential tasks are filtering, smoothing, and parameter estimation. The marginal likelihood of an observed trajectory is

$h'(t)=Ah(t)+Bx(t)$6

which is tractable in the linear Gaussian case and intractable in general. The classical solutions are Kalman filtering and smoothing for linear Gaussian systems, EM or direct maximum likelihood for parameter learning, and approximate methods such as particle filters, EKF, Gaussian filters, and UKF in nonlinear or non-Gaussian settings (Lin et al., 2024).

A major extension concerns continuous-time SSMs with observations at discrete but irregularly spaced times. A general formulation observes $h'(t)=Ah(t)+Bx(t)$7 at $h'(t)=Ah(t)+Bx(t)$8, with a latent Markov process $h'(t)=Ah(t)+Bx(t)$9 and conditionally independent observations given $y(t)=Ch(t)$0. When the transition density $y(t)=Ch(t)$1, $y(t)=Ch(t)$2, is available in closed form, the latent continuous state can be finely discretized over intervals $y(t)=Ch(t)$3, converting the model into an $y(t)=Ch(t)$4-state hidden Markov model with time-gap-dependent transition matrix $y(t)=Ch(t)$5. The approximate likelihood then becomes

$y(t)=Ch(t)$6

which can be evaluated by the HMM forward algorithm in $y(t)=Ch(t)$7 instead of $y(t)=Ch(t)$8, and decoded with the Viterbi algorithm (Mews et al., 2020).

The continuous-time framework is deliberately broad: both the latent process and the observation process may be non-linear and non-Gaussian, provided the latent process is one-dimensional and its transition density is analytically available. The paper giving this formulation uses an Ornstein–Uhlenbeck latent process, allows Poisson, negative binomial, categorical, and other observation models, and handles irregular sampling natively through $y(t)=Ch(t)$9 rather than imputation or aggregation. In simulation, $p_\theta(x_0)$0 was often adequate, $p_\theta(x_0)$1 was recommended as a conservative choice if computationally feasible, and a longitudinal delinquency application used $p_\theta(x_0)$2 with latent range $p_\theta(x_0)$3, yielding a strongly preferred continuous-time SSM over a model without a latent state process by $p_\theta(x_0)$4AIC $p_\theta(x_0)$5 (Mews et al., 2020).

3. Deep latent-variable and variational SSMs

Deep learning extends SSMs by retaining latent state evolution while parameterizing transitions, emissions, or inference networks with neural networks. In the VAE-style formulation reviewed for deep SSMs, one introduces an approximate posterior $p_\theta(x_0)$6 and maximizes an ELBO of the form

$p_\theta(x_0)$7

or its time-factorized sequence variant with per-step KL terms. Deep Kalman Filter-style models retain a Markov latent chain but let the Gaussian transition and emission parameters be neural-network outputs, while latent neural ODE and latent neural SDE models move the latent dynamics to continuous time for irregularly sampled data (Lin et al., 2024).

A distinct probabilistic line is the Probabilistic Recurrent State-Space Model, which uses a Gaussian process prior on the transition and a structured variational approximation that preserves temporal correlations in the latent trajectory. Its joint model is

$p_\theta(x_0)$8

and its variational approximation keeps the recurrent latent-state dependence rather than factorizing over time. This is paired with doubly stochastic variational inference and a recognition model for the initial latent state, making the GP-SSM trainable by reparameterized rollouts rather than by mean-field latent approximations that break temporal dependence (Doerr et al., 2018).

A complementary black-box variational approach uses autoregressive density estimation for the joint posterior over latent trajectories and global parameters. The variational family factorizes as

$p_\theta(x_0)$9

with $p_\theta(x_t\mid x_{t-1})$0 parameterized by a local inverse autoregressive flow

$p_\theta(x_t\mid x_{t-1})$1

so that dependence is restricted to a local receptive field of length $p_\theta(x_t\mid x_{t-1})$2. This yields a GPU-friendly approximation aimed at nonlinear SSMs with long latent trajectories, especially when exact inference or bespoke MCMC is too costly (Ryder et al., 2018).

4. Structured, selective, and modern sequence-modeling SSMs

Modern sequence-modeling SSMs inherit the state-transition template but shift emphasis from latent-variable inference to expressivity, efficiency, and long-range memory. The review literature places LSSL, S4, S5, and Mamba in this category. S4 uses structured state matrices and convolutional training; S5 uses diagonal state matrices, a MIMO formulation, and parallel scan in the time domain; and Mamba introduces input-dependent state dynamics and selective scan, making the recurrence context-dependent rather than time-invariant (Lin et al., 2024, Zubić et al., 2024).

Several papers rework the mathematical foundations of such models. One argues that continuous-time LSSL-style parameterizations cannot represent noiseless $p_\theta(x_t\mid x_{t-1})$3 processes exactly, and proposes a discrete-time companion-matrix parameterization instead. In that formulation,

$p_\theta(x_t\mid x_{t-1})$4

with $p_\theta(x_t\mid x_{t-1})$5 in companion form, which can realize autoregressive laws exactly and supports a closed-loop forecasting decoder with

$p_\theta(x_t\mid x_{t-1})$6

The same work gives an $p_\theta(x_t\mid x_{t-1})$7 algorithm for companion-kernel computation and reports best MSE on 14 of 16 Informer forecasting settings and best or second-best AUROC on 6 of 7 ECG and speech tasks (Zhang et al., 2023).

Another line derives structured SSMs directly in discrete time. The lag-operator framework defines a backward lag operator $p_\theta(x_t\mid x_{t-1})$8 and obtains the transition matrix as

$p_\theta(x_t\mid x_{t-1})$9

so that the recurrence $p_\theta(y_t\mid x_t)$0 arises from basis overlap rather than from discretizing an ODE. A related frame-theoretic generalization, SaFARi, replaces the narrow catalog of HiPPO polynomial systems with a construction from arbitrary frames or bases, yielding for the scaled case

$p_\theta(y_t\mid x_t)$1

and analogous translated-window dynamics. Together these works expand the design space of structured SSMs beyond a few special orthogonal polynomial families (Tomonaga et al., 22 Dec 2025, Babaei et al., 13 May 2025).

Optimization theory has also become explicit. A 2024 generalization analysis for continuous-time linear SSMs studies the kernel $p_\theta(y_t\mid x_t)$2 and shows that the relevant complexity couples $p_\theta(y_t\mid x_t)$3 to the time-varying mean and variance of the data rather than only to parameter norms. It proposes a scaling rule for $p_\theta(y_t\mid x_t)$4 based on a data-dependent complexity and a regularizer $p_\theta(y_t\mid x_t)$5, with small but consistent improvements on LRA and strong robustness gains on synthetic data (Liu et al., 2024). At the frontier of expressive selective SSMs, Flash PD-SSM uses a trainable dictionary of structured sparse transition matrices, selecting one per time step. It reduces sequence-dependent memory from $p_\theta(y_t\mid x_t)$6 to $p_\theta(y_t\mid x_t)$7, preserves the proposition that any deterministic FSA with $p_\theta(y_t\mid x_t)$8 states can be exactly represented by a single-layer Flash PD-SSM with state size $p_\theta(y_t\mid x_t)$9, and reports 96.3 average accuracy on synthetic state-tracking tasks versus 98.8 for PD-SSM and 58.9 for Mamba2 (Terzić et al., 18 May 2026).

5. Domain-specific extensions and applications

SSMs have been adapted aggressively to modalities in which irregular sampling, long horizons, or non-Euclidean structure are intrinsic. In event-camera object detection, a continuous-time linear SSM with learnable timescales can be re-discretized at new inference frequencies without retraining. In that setting, training uses 50 ms windows at 20 Hz, evaluation is performed at 20, 40, 80, 100, and 200 Hz, SSM-based models train 33% faster than recurrent Transformer approaches, and the average performance drop from 20 Hz to higher frequencies is 3.76 mAP for the SSM models, versus 21.25 mAP for RVT and 24.53 mAP for GET (Zubić et al., 2024).

For fully asynchronous event streams, a hardware-aware SSM can be written as

$$p_\theta(x_t \mid y_{1:t-1}) = \int p_\theta(x_t \mid x_{t-1}) p_\theta(x_{t-1} \mid y_{1:t-1}) dx_{t-1},$$0

and simplified to a real-valued shared-decay form $$p_\theta(x_t \mid y_{1:t-1}) = \int p_\theta(x_t \mid x_{t-1}) p_\theta(x_{t-1} \mid y_{1:t-1}) dx_{t-1},$$1. This permits direct mapping of hidden-state decay onto short-term-memory memristors and static linear maps onto RRAM crossbars. The resulting compute-in-memory system reports 95.7% on SHD, 84.7% on SSC, 97.3% on DVS128 Gesture, and 63.5% on DVS128 Lips, with fixed layer-wise $$p_\theta(x_t \mid y_{1:t-1}) = \int p_\theta(x_t \mid x_{t-1}) p_\theta(x_{t-1} \mid y_{1:t-1}) dx_{t-1},$$2 often improving both stability and accuracy (Zhang et al., 17 Nov 2025).

Graph-structured extensions change the state space itself. One approach replaces fixed scan order with a dynamically generated minimum spanning tree over token features, then propagates state along tree paths: $$p_\theta(x_t \mid y_{1:t-1}) = \int p_\theta(x_t \mid x_{t-1}) p_\theta(x_{t-1} \mid y_{1:t-1}) dx_{t-1},$$3 This yields GG-SSM, which reaches 84.9% top-1 on ImageNet and outperforms VMamba across several visual and time-series settings. A different line formulates the latent state itself as a graph $$p_\theta(x_t \mid y_{1:t-1}) = \int p_\theta(x_t \mid x_{t-1}) p_\theta(x_{t-1} \mid y_{1:t-1}) dx_{t-1},$$4, with stochastic latent topology

$$p_\theta(x_t \mid y_{1:t-1}) = \int p_\theta(x_t \mid x_{t-1}) p_\theta(x_{t-1} \mid y_{1:t-1}) dx_{t-1},$$5

and message-passing state updates; on controlled GPVAR forecasting it reaches MAE 0.328 versus 0.323 for a model with known ground-truth graph and far outperforms non-graph RNN baselines (Zubić et al., 2024, Zambon et al., 2023).

Biosignal and code applications expose a complementary pattern. EEG-SSM combines a temporal Mamba-style branch with a spectral branch based on wavelet bands and Welch PSD, reaching 91.0% accuracy on 2 s EEG segments for HC/FTD/AD classification and remaining at 77.8% at 256 s, whereas the temporal-only branch drops from 89.1% to 46.8% over the same length range (Tran et al., 2024). In code modeling, SSM-based encoders outperform comparable Transformers on AST sibling prediction, AST distance prediction, and DFG edge prediction during pretraining, but frequency-domain analysis shows a spectral shift toward short-range dependencies during type-inference fine-tuning; guided modifications such as an explicit high-frequency path or 8 kernels per layer improve NLCodeSearch from 25.39 to 30.89 MRR, SQA from 76.08 to 79.57 MRR, and Type Inference from 59.70 to 60.98 F1 (Wu et al., 6 Feb 2026).

6. Limitations, pathologies, and current directions

Despite their breadth, SSMs have recurring failure modes. A cautionary result from ecology shows that even the univariate linear Gaussian model

$$p_\theta(x_t \mid y_{1:t-1}) = \int p_\theta(x_t \mid x_{t-1}) p_\theta(x_{t-1} \mid y_{1:t-1}) dx_{t-1},$$6

can have serious parameter- and state-estimation problems when measurement error dominates process stochasticity. In the main simulation study, 29.6% of fits had state RMSE more than 50% larger when using estimated rather than true parameters; fixing $$p_\theta(x_t \mid y_{1:t-1}) = \int p_\theta(x_t \mid x_{t-1}) p_\theta(x_{t-1} \mid y_{1:t-1}) dx_{t-1},$$7 to its true value reduced this to 5.0%. The same paper emphasizes flat or multimodal likelihoods, boundary estimates such as $$p_\theta(x_t \mid y_{1:t-1}) = \int p_\theta(x_t \mid x_{t-1}) p_\theta(x_{t-1} \mid y_{1:t-1}) dx_{t-1},$$8, and the fact that Bayesian posteriors can appear well behaved while still being biased (Auger-Méthé et al., 2015).

Other limitations are structural. The continuous-time approximate-likelihood framework for irregularly sampled nonlinear and non-Gaussian SSMs is especially attractive for one-dimensional latent processes with known transition densities, but the authors explicitly note a curse of dimensionality for multivariate state processes (Mews et al., 2020). The data-dependent generalization theory for modern sequence SSMs is proved only for single-layer linear SSMs, while practical multi-layer implementations treat features as independent when computing the proposed complexity term (Liu et al., 2024). Hardware-oriented event SSMs gain efficiency from static, non-input-dependent transitions, but this makes them less expressive than selective SSMs such as Mamba (Zhang et al., 17 Nov 2025).

Current directions therefore concentrate on three fronts. One is richer propagation topology, as in graph-generated scanning and latent graph-state models. A second is the expressivity-efficiency trade-off in transition structure, exemplified by Flash PD-SSM’s attempt to retain FSA expressiveness while matching structured SSM throughput (Terzić et al., 18 May 2026). A third is theory: recent work formulates SSM training itself as an ensemble optimal control problem, with shared parameters viewed as a control $$p_\theta(x_t \mid y_{1:t-1}) = \int p_\theta(x_t \mid x_{t-1}) p_\theta(x_{t-1} \mid y_{1:t-1}) dx_{t-1},$$9 acting on a population of input-dependent systems

$$p_\theta(x_t \mid y_{1:t}) \propto p_\theta(y_t \mid x_t) p_\theta(x_t \mid y_{1:t-1}).$$0

derives a Pontryagin maximum principle for this setting, and proposes a method of successive approximations with subsequence convergence to a PMP-stationary limit and sufficient conditions for global optimality (Feng et al., 13 Mar 2026). This suggests that the future development of SSMs will continue to be shaped simultaneously by statistical inference, sequence-model architecture, numerical analysis, and control theory.