State-space models have gained popularity in sequence modelling due to their simple and efficient network structures. However, the absence of nonlinear activation along the temporal direction limits the model's capacity.

Large-scale traffic forecasting relies on fixed sensor networks that often exhibit blackouts: contiguous intervals of missing measurements caused by detector or communication failures. These outages are typically handled under a Missing At Random (MAR) assumption, even though blackout events may correlate with unobserved traffic conditions (e.g., congestion or anomalous flow), motivating a Missing Not At Random (MNAR) treatment. We propose a latent state-space framework that jointly models (i) traffic dynamics via a linear dynamical system and (ii) sensor dropout via a Bernoulli observation channel whose probability depends on the latent traffic state. Inference uses an Extended Kalman Filter with Rauch-Tung-Striebel smoothing, and parameters are learned via an approximate EM procedure with a dedicated update for detector-specific missingness parameters. On the Seattle inductive loop detector data, introducing latent dynamics yields large gains over naive baselines, reducing blackout imputation RMSE from 7.02 (LOCF) and 5.02 (linear interpolation + seasonal naive) to 4.23 (MAR LDS), corresponding to about a 64% reduction in MSE relative to LOCF. Explicit MNAR modeling provides a consistent but smaller additional improvement on real data (imputation RMSE 4.20; 0.8% RMSE reduction relative to MAR), with similar modest gains for short-horizon post-blackout forecasts (evaluated at 1, 3, and 6 steps). In controlled synthetic experiments, the MNAR advantage increases as the true missingness dependence on latent state strengthens. Overall, temporal dynamics dominate performance, while MNAR modeling offers a principled refinement that becomes most valuable when missingness is genuinely informative.

Structured state-space models (SSMs) are gaining popularity as effective foundational architectures for sequential data, demonstrating outstanding performance across a diverse set of domains alongside desirable scalability properties. Recent developments show that if the linear recurrence powering SSMs allows for a selectivity mechanism leveraging multiplicative interactions between inputs and hidden states (e.g. Mamba, GLA, Hawk/Griffin, HGRN2), then the resulting architecture can surpass attention-powered foundation models trained on text in both accuracy and efficiency, at scales of billion parameters. In this paper, we give theoretical grounding to the selectivity mechanism, often linked to in-context learning, using tools from Rough Path Theory. We provide a framework for the theoretical analysis of generalized selective SSMs, fully characterizing their expressive power and identifying the gating mechanism as the crucial architectural choice. Our analysis provides a closed-form description of the expressive powers of modern SSMs, such as Mamba, quantifying theoretically the drastic improvement in performance from the previous generation of models, such as S4. Our theory not only motivates the success of modern selective state-space models, but also provides a solid framework to understand the expressive power of future SSM variants. In particular, it suggests cross-channel interactions could play a vital role in future improvements.

Probabilistic inference in high-dimensional state-space models is computationally challenging. For many spatiotemporal systems, however, prior knowledge about the dependency structure of state variables is available. We leverage this structure to develop a computationally efficient approach to state estimation and learning in graph-structured state-space models with (partially) unknown dynamics and limited historical data. Building on recent methods that combine ideas from deep learning with principled inference in Gaussian Markov random fields (GMRF), we reformulate graph-structured state-space models as Deep GMRFs defined by simple spatial and temporal graph layers. This results in a flexible spatiotemporal prior that can be learned efficiently from a single time sequence via variational inference. Under linear Gaussian assumptions, we retain a closed-form posterior, which can be sampled efficiently using the conjugate gradient method, scaling favourably compared to classical Kalman filter based approaches.

State-space models have gained popularity in sequence modelling due to their simple and efficient network structures. However, the absence of nonlinear activation along the temporal direction limits the model's capacity. In this paper, we prove that stacking state-space models with layer-wise nonlinear activation is sufficient to approximate any continuous sequence-to-sequence relationship. Our findings demonstrate that the addition of layer-wise nonlinear activation enhances the model's capacity to learn complex sequence patterns. Meanwhile, it can be seen both theoretically and empirically that the state-space models do not fundamentally resolve the issue of exponential decaying memory. Theoretical results are justified by numerical verifications.

Inferring effective connectivity between spatially segregated brain regions is important for understanding human brain dynamics in health and disease. Non-invasive neuroimaging modalities, such as electroencephalography (EEG) and functional magnetic resonance imaging (fMRI), are often used to make measurements and infer connectivity. However most studies do not consider integrating the two modalities even though each is an indirect measure of the latent neural dynamics and each has its own spatial and/or temporal limitations. In this study, we develop a linear state-space model to infer the effective connectivity in a distributed brain network based on simultaneously recorded EEG and fMRI data. Our method first identifies task-dependent and subject-dependent regions of interest (ROI) based on the analysis of fMRI data.

Models of disease progression are instrumental for predicting patient outcomes and understanding disease dynamics. Existing models provide the patient with pragmatic (supervised) predictions of risk, but do not provide the clinician with intelligible (unsupervised) representations of disease pathophysiology. In this paper, we develop the attentive state-space model, a deep probabilistic model that learns accurate and interpretable structured representations for disease trajectories. Unlike Markovian state-space models, in which the dynamics are memoryless, our model uses an attention mechanism to create memoryful dynamics, whereby attention weights determine the dependence of future disease states on past medical history. To learn the model parameters from medical records, we develop an infer ence algorithm that simultaneously learns a compiled inference network and the model parameters, leveraging the attentive state-space representation to construct a Rao-Blackwellized variational approximation of the posterior state distribution. Experiments on data from the UK Cystic Fibrosis registry show that our model demonstrates superior predictive accuracy and provides insights into the progression of chronic disease.

Recent advancements in foundation models for tabular data, such as TabPFN, demonstrated that pretrained Transformer architectures can approximate Bayesian inference with high predictive performance. However, Transformers suffer from quadratic complexity with respect to sequence length, motivating the exploration of more efficient sequence models. In this work, we investigate the potential of using Hydra, a bidirectional linear-time structured state space model (SSM), as an alternative to Transformers in TabPFN. A key challenge lies in SSM's inherent sensitivity to the order of input tokens - an undesirable property for tabular datasets where the row order is semantically meaningless. We investigate to what extent a bidirectional approach can preserve efficiency and enable symmetric context aggregation. Our experiments show that this approach reduces the order-dependence, achieving predictive performance competitive to the original TabPFN model.

Human activity recognition (HAR) from inertial sensors is essential for ubiquitous computing, mobile health, and ambient intelligence. Conventional deep models such as Convolutional Neural Networks (CNNs), Recurrent Neural Networks (RNNs), and transformers have advanced HAR but remain limited by vanishing or exloding gradients, high computational cost, and difficulty in capturing long-range dependencies. Structured state-space models (SSMs) like Mamba address these challenges with linear complexity and effective temporal modeling, yet they are restricted to first-order dynamics without stable longterm memory mechanisms. We introduce Momentum Mamba, a momentum-augmented SSM that incorporates second-order dynamics to improve stability of information flow across time steps, robustness, and long-sequence modeling. Two extensions further expand its capacity: Complex Momentum Mamba for frequency-selective memory scaling. Experiments on multiple HAR benchmarks demonstrate consistent gains over vanilla Mamba and Transformer baselines in accuracy, robustness, and convergence speed. With only moderate increases in training cost, momentum-augmented SSMs offer a favorable accuracy-efficiency balance, establishing them as a scalable paradigm for HAR and a promising principal framework for broader sequence modeling applications.