State Space Models

A guide to structured memory and complex dynamics in deep learning

Open Edition · 2026

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Changelog

• [Jun 2026] First public preview. Foundations and Structured State Spaces & S4 chapters available.

Authors

Cosmo Santoni Imperial College London

Cite this book

@book{santoni2026ssm,
  author = {Santoni, Cosmo},
  title  = {State Space Models: A guide to structured memory and complex dynamics in deep learning},
  year   = {2026},
  note   = {Open Edition},
  doi    = {10.5281/zenodo.20736327},
  url    = {https://ssm.guide}
}

Table of contents

• Preface
• Reader’s Guide
• Notation & Preliminaries
• Roadmap & Contributing

• 1 Introduction
• 2 State as Memory
• 3 Continuous-Time State Space Models
• 4 Discretisation
• 5 From Recurrence to Convolution
• 6 Transfer Functions
• 7 Polynomial Projection
• 8 HiPPO
• 9 Structured State Matrices
• 10 The S4 Kernel Algorithm
• 11 Diagonal State Space Models

• References