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Technologies on Effectiveness and Efficiency: A Survey of State Spaces Models

Xingtai Lv, Youbang Sun, Kaiyan Zhang, Shang Qu, Xuekai Zhu, Yuchen Fan, Yi Wu, Ermo Hua, Xinwei Long, Ning Ding, Bowen Zhou

TL;DR

This survey examines State Space Models (SSMs) as efficient alternatives to Transformer-based architectures for long-range sequence modeling. It traces the evolution from original continuous-to-discrete SSMs to Structured SSMs (notably S4) and Selective SSMs (notably Mamba), detailing the mathematical formulations, structural constraints, and optimization techniques that enhance effectiveness and efficiency. The work highlights key methods such as HiPPO-based memory constraints, Diagonal Plus Low-Rank decompositions, and input-dependent selectivity, as well as practical integrations with LTCs and Transformers. Across applications in video, speech, molecular biology, 3D signals, time series, and structured data, SSMs demonstrate strong long-range modeling with favorable computational profiles, suggesting growing relevance in real-world AI systems.

Abstract

State Space Models (SSMs) have emerged as a promising alternative to the popular transformer-based models and have been increasingly gaining attention. Compared to transformers, SSMs excel at tasks with sequential data or longer contexts, demonstrating comparable performances with significant efficiency gains. In this survey, we provide a coherent and systematic overview for SSMs, including their theoretical motivations, mathematical formulations, comparison with existing model classes, and various applications. We divide the SSM series into three main sections, providing a detailed introduction to the original SSM, the structured SSM represented by S4, and the selective SSM typified by Mamba. We put an emphasis on technicality, and highlight the various key techniques introduced to address the effectiveness and efficiency of SSMs. We hope this manuscript serves as an introduction for researchers to explore the theoretical foundations of SSMs.

Technologies on Effectiveness and Efficiency: A Survey of State Spaces Models

TL;DR

This survey examines State Space Models (SSMs) as efficient alternatives to Transformer-based architectures for long-range sequence modeling. It traces the evolution from original continuous-to-discrete SSMs to Structured SSMs (notably S4) and Selective SSMs (notably Mamba), detailing the mathematical formulations, structural constraints, and optimization techniques that enhance effectiveness and efficiency. The work highlights key methods such as HiPPO-based memory constraints, Diagonal Plus Low-Rank decompositions, and input-dependent selectivity, as well as practical integrations with LTCs and Transformers. Across applications in video, speech, molecular biology, 3D signals, time series, and structured data, SSMs demonstrate strong long-range modeling with favorable computational profiles, suggesting growing relevance in real-world AI systems.

Abstract

State Space Models (SSMs) have emerged as a promising alternative to the popular transformer-based models and have been increasingly gaining attention. Compared to transformers, SSMs excel at tasks with sequential data or longer contexts, demonstrating comparable performances with significant efficiency gains. In this survey, we provide a coherent and systematic overview for SSMs, including their theoretical motivations, mathematical formulations, comparison with existing model classes, and various applications. We divide the SSM series into three main sections, providing a detailed introduction to the original SSM, the structured SSM represented by S4, and the selective SSM typified by Mamba. We put an emphasis on technicality, and highlight the various key techniques introduced to address the effectiveness and efficiency of SSMs. We hope this manuscript serves as an introduction for researchers to explore the theoretical foundations of SSMs.

Paper Structure

This paper contains 30 sections, 21 equations, 4 figures.

Table of Contents

  1. Introduction
  2. State Space Models: From Continuous To Discrete
  3. Continuous-Time State Space Models
  4. Discretization of Continuous-Time SSMs
  5. Fundamental Properties of SSMs
  6. Models with Original SSMs
  7. Structured State Space Models
  8. Overview of Structured State Space Models
  9. Effectiveness: High-Order Polynomial Projection Operators (HiPPO)
  10. Efficiency: Diagonal Plus Low-Rank
  11. Additional Structural Constraints for Structured SSMs
  12. Integration of Structured SSMs with Other Models
  13. Selective State Space Models
  14. Overview of Selective State Space Models
  15. Effectiveness: Selectivity
  16. ...and 15 more sections

Figures (4)

  • Figure 1: Technologies of SSM Series and their corresponding relationships. Each technical description card includes the technique’s name, mathematical formulation, and objective.
  • Figure 2: Typology of SSM Series. The “Architecture Variants” presents the common SSM architectures, while the “Application” shows the practical implementations of SSMs in real-world scenarios.
  • Figure 3: The illustration of a RC oscillator circuit, where $L$, $C$, and $R$ denote the inductance, capacitance and resistance, respectively.
  • Figure 4: Applications of State Space Models Across Diverse Data Types