StateSpace-SSL: Integrated Sequence & Vision Learning

• StateSpace-SSL is a family of methods that integrate linear state-space models with self-supervised learning to capture both local and global dependencies.
• It employs efficient techniques like FFT-based convolutions and HiPPO-inspired parameterizations to enhance memory and sequence modeling in deep learning architectures.
• The approach extends to robust time series analysis and vision tasks, achieving competitive accuracy through structured regression and prototype-driven SSL methods.

StateSpace-SSL denotes a distinct family of approaches integrating state-space models (SSMs) with self-supervised learning (SSL) or structured statistical learning for time series, sequence, and vision applications. Recent developments under the StateSpace-SSL nomenclature include linear state-space layers for deep sequence modeling, high-dimensional regularized SSM regression for time series, and prototype-driven SSL with Vision Mamba architectures for plant disease imagery. Variants differ in architectural details, but all leverage the ability of state-space recurrences to model long dependencies, combine local and global structure, and enable efficient inference and training.

1. Linear State-Space Foundations

StateSpace-SSL methods fundamentally build on linear state-space systems, typically modeled as continuous- or discrete-time dynamical systems:

$$\dot{x}(t) = A x(t) + B u(t), \qquad y(t) = C x(t) + D u(t),$$

where $x(t)$ is the state, $u(t)$ the input, $y(t)$ the output, and $(A,B,C,D)$ are system matrices. Discretization yields recurrent equations:

$$x_t = \overline A x_{t-1} + \overline B u_t, \qquad y_t = C x_t + D u_t,$$

and the model can be expressed equivalently as a temporal convolution, with computationally efficient FFT-based training ($O(L \log L)$ per sequence of length $L$), or as an RNN with $O(N)$ per-timestep cost (Gu et al., 2021).

To address the challenge of modeling long-range dependencies, StateSpace-SSL restricts the state matrix $A$ to parameterizations motivated by the HiPPO framework, leading to low-recurrence-width or quasiseparable structures that provide both expressivity and computational tractability. For instance, the HiPPO-LegS initialization for $A$ ensures the model can efficiently memorize and transmit sequence information across time (Gu et al., 2021).

2. StateSpace-SSL for Deep Sequence and Time Series Modeling

In deep learning, StateSpace-SSL (specifically the Linear State-Space Layer, LSSL) is instantiated as a building block for sequence models, generalizing and subsuming RNNs, CNNs, and neural ODEs:

• Convolutional perspective: The layer computes convolutions with theoretically infinite receptive field using kernels derived from system dynamics, enabling efficient parallel training.
• Recurrent perspective: The state update is carried forward through sequence steps, providing memory and time-adaptivity.
• Continuous-time interpretation: This enables flexibility for irregular sampling and non-uniform time-series intervals.

Architecturally, LSSL/StateSpace-SSL networks alternate LSSL blocks with GeLU nonlinearities, feed-forward mixing, LayerNorm, and residual connections, with model sizes ranging from $N=128, H=128$ to larger setups (up to $2$M parameters). This structure achieves state-of-the-art accuracy on long-range sequential tasks, such as raw audio classification (95.87% on 16,000-step Speech Commands vs prior top-1 scores of 71.66%), dense pixel-wise image classification, and complex physiological regression with massively improved convergence speed and memory efficiency relative to transformer and RNN baselines (Gu et al., 2021).

3. State Space Learning (SSL) for Regularized Time Series Analysis

State Space Learning (SSL) reinterprets additive linear state-space models for time series as a high-dimensional regression problem (Ramos et al., 17 Aug 2024). By unrolling standard structural SSM recursions (level, trend, seasonality, exogenous factors), the SSM is rewritten as

$Y = X \Theta + \varepsilon,$

where $Y$ is the time series, $X$ is a design matrix encoding step/ramp/seasonal functions and external regressors, and $\Theta$ contains the structural parameters, shocks, and optional exogenous and outlier coefficients.

SSL solves the parameter estimation by convex penalized regression (elastic net or group-lasso variants):

$$\min_{\Theta} \Big\{ \frac{1}{T}\|Y - X\Theta\|_2^2 + \lambda \big[ (1-\alpha)/2 \cdot \|\Theta_{\mathrm{innov}}\|_2^2 + \alpha \cdot \|\Theta_{\mathrm{innov}}\|_1 \big] \Big\},$$

optionally applying adaptive weighting akin to the Adaptive Lasso. The approach supports exogenous variable selection, robust outlier detection via $L_1$-penalty dummy columns, and deterministic closed-form extrapolation for forecasting. Empirical benchmarks on the M4 competition (48,000 monthly series) demonstrate superior accuracy (OWA = 0.890, sMAPE = 12.98%, MASE = 0.936) and subset selection (exact recovery rate >84%) compared to traditional Kalman filtering and other sparse learning approaches, at substantially reduced computation cost (Ramos et al., 17 Aug 2024).

4. StateSpace-SSL in Vision: Self-supervised Plant Disease Detection

StateSpace-SSL has also been adapted for image-based tasks, most prominently in plant health diagnosis under SSL (Mamun et al., 10 Dec 2025). This approach employs a Vision Mamba (VM) encoder—a variant of structured SSMs—configured to process high-resolution leaf imagery by raster-ordered or serpentine scanning of image patches, modeling lesion continuity as a unidimensional state sequence:

$h_k = g_k \odot (W_s h_{k-1} + W_x x_k)$

for patch index $k$, with learnable gating $g_k$, recurrence $W_s$, and input projection $W_x$.

A prototype-driven teacher–student SSL objective is employed: Each encoder (teacher and student) projects input views to K-dimensional prototype scores via a two-layer MLP, aligning student outputs across multiple crops to a stable, exponentially averaged teacher via cross-entropy of softmax distributions. Cropping strategy ensures generalization from global to local levels.

StateSpace-SSL achieves state-of-the-art downstream accuracy on three datasets (e.g., PlantVillage 94.61%, PlantDoc 91.24%, Citrus 89.83%), offers linear scaling in image token count, requires less VRAM and compute than transformer or CNN SSL variants, and produces sharper, lesion-centered Grad-CAM activations. The linear SSM-based recurrence captures elongated lesion shapes along leaf veins, a class of patterns poorly handled by local CNNs or quadratic-cost transformers (Mamun et al., 10 Dec 2025).

5. Statistical Inference and Generalizations

StateSpace-SSL variants extend to stochastic and nonlinear settings by augmenting SSM transitions with expressive latent functions, including LSTM-based transitions in the context of State Space LSTM (SSL) with Particle MCMC inference (Zheng et al., 2017). Here, the latent state evolves according to an LSTM, yielding posterior distributions that cannot be factorized over time. Sequential Monte Carlo (SMC) with Particle Gibbs sampling enables unbiased joint inference over the state path, improving accuracy and stability in tracking, language modeling, and user-click prediction compared to factorized or EM inference. This formulation combines the interpretability of SSMs (latent trajectory modeling) with the representational power of deep sequence models (Zheng et al., 2017).

6. Computational Complexity and Implementation

StateSpace-SSL approaches achieve attractive computational characteristics. Training and inference in the linear structured cases can be performed in $O(L \log L)$ per sequence or $O(N)$ per timestep for inference, leveraging Krylov methods and quasiseparable state-matrix algebra. In time series regression, polynomial-time coordinate descent (GLMNet) yields global-optimal solutions in milliseconds per series, even with high-dimensional regressors. In vision, linear scaling with patch count enables practical SSL on large, high-resolution datasets, reducing overall training time (StateSpace-SSL: 9h/8.7GB vs ViT-based MAE: 30h/22.6GB) (Mamun et al., 10 Dec 2025, Ramos et al., 17 Aug 2024, Gu et al., 2021).

The Julia library StateSpaceLearning.jl implements core regression-based SSL workflows, and deep learning instantiations are available in a variety of frameworks for structured LSSLs and Vision Mamba modules (Ramos et al., 17 Aug 2024).

7. Empirical Findings and Limitations

StateSpace-SSL models show superior or highly competitive empirical performance across time series, long-sequence, and vision tasks. They specifically excel in settings requiring:

• Modeling of long-range dependencies and non-local structures.
• Efficient handling of high-dimensional inputs or outputs.
• Accurate forecasting, smoothing, and latent trajectory estimation.
• Robust subset selection and outlier accommodation.

Nevertheless, limitations include possible under-representation of small or low-contrast features in vision contexts, and reliance on accurate structural or recurrence assumptions for interpretability and convergence. Future extensions may include multi-scale or hierarchical SSMs and domain-specific inductive biases (Mamun et al., 10 Dec 2025).

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Summary table: Selected StateSpace-SSL Approaches and Domains

Variant	Domain	Key Features
Linear State-Space Layer (LSSL)	Deep sequence	FFT convolutions, HiPPO memory, SOTA accuracy
State Space Learning (SSL)	Time series	Convex regularized regression, outliers, fast
StateSpace-SSL (Vision Mamba)	Vision/SSL (plant)	Linear-time SSM, prototype SSL, lesion-focused
State Space LSTM + PG Inference	Stochastic SSM	LSTM transitions, SMC+PG, full joint posterior

These variants collectively demonstrate StateSpace-SSL’s unification of classical dynamical systems theory, efficient statistical learning, and modern self-supervised paradigms across structured data modalities (Mamun et al., 10 Dec 2025, Ramos et al., 17 Aug 2024, Gu et al., 2021, Zheng et al., 2017).