Jiazhou Wang

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me:

Can you give me a brief literature review on deep learning with state space model?

OpenAI:

Below is a brief literature review that outlines how deep learning techniques have been integrated with state space models (SSMs) to better capture and model temporal dynamics and complex sequences:

Overview

Classical state space models—such as Kalman filters and hidden Markov models—have long provided a mathematical framework for modeling dynamical systems. Recent research has focused on enhancing these models using deep neural networks to learn nonlinear representations and latent dynamics from data. Early approaches incorporated variational inference techniques (e.g., variational autoencoders) to estimate the latent states and transitions, thereby offering improved robustness and flexibility in modeling complex sequences.

Deep Learning-Based Approaches for SSMs

A recent selective review summarizes how deep learning methods have been employed to extend both discrete-time and continuous-time state space models. This work discusses neural network formulations for latent variable inference, neural ordinary differential equations, and stochastic differential equations, all integrated within the SSM framework to handle irregular time series and multi-frequency data efficiently citeturn0search0. Such deep SSMs have been successfully applied in various domains including time series forecasting, speech processing, and even neuroimaging.

New Architectures and Theoretical Advances

Building on these advances, researchers have proposed novel architectures that use state space layers as a core component. For example, approaches like S4 and its extensions (e.g., Mamba) reinterpret the deep network’s layers as evolving states in a dynamical system. This perspective not only provides computational efficiency—offering alternatives to attention-based models like Transformers—but also improves the model’s ability to capture long-range dependencies citeturn0search2. Complementary surveys such as the “State Space Model for New-Generation Network Alternative to Transformers: A Survey” and “State Space Models as Foundation Models: A Control Theoretic Overview” further compare these architectures and discuss their control theoretic foundations citeturn0academia8 citeturn0academia9.

On the theoretical side, recent work is beginning to analyze the learning dynamics of deep state space models. For instance, “Towards a theory of learning dynamics in deep state space models” investigates how network depth, covariance structures, and initialization affect parameter evolution and the ability to capture long-term dependencies citeturn0academia11.

Conclusion

Overall, integrating deep learning with state space modeling has led to architectures that are more expressive and computationally efficient. These models not only enhance our ability to model sequential data across diverse applications but also open new theoretical avenues for understanding dynamic neural systems.

Below is the list of the key papers and reviews mentioned in the literature review along with their corresponding links:

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