Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Notation
- Part One Machine Learning
- Part Two Optimal Recovery
- Part Three Compressive Sensing
- Executive Summary
- 14 Sparse Recovery from Linear Observations
- 15 The Complexity of Sparse Recovery
- 16 Low-Rank Recovery from Linear Observations
- 17 Sparse Recovery from One-Bit Observations
- 18 Group Testing
- Part Four Optimization
- Part Five Neural Networks
- Appendices
- References
- Index
14 - Sparse Recovery from Linear Observations
from Part Three - Compressive Sensing
Published online by Cambridge University Press: 21 April 2022
- Frontmatter
- Dedication
- Contents
- Preface
- Notation
- Part One Machine Learning
- Part Two Optimal Recovery
- Part Three Compressive Sensing
- Executive Summary
- 14 Sparse Recovery from Linear Observations
- 15 The Complexity of Sparse Recovery
- 16 Low-Rank Recovery from Linear Observations
- 17 Sparse Recovery from One-Bit Observations
- 18 Group Testing
- Part Four Optimization
- Part Five Neural Networks
- Appendices
- References
- Index
Summary
This chapter introduces the standard compressive sensing problem, where one tries to recover sparse vectors from few linear observations. The problem is proved to be solvable using ?1-minimization as a recovery map if and only if the observation matrix satisfies the so-called null space property. This property is then shown to be a consequence of an atypical restricted isometry property from ?2 to ?1, which holds with high probability for Gaussian matrices.
- Type
- Chapter
- Information
- Mathematical Pictures at a Data Science Exhibition , pp. 116 - 122Publisher: Cambridge University PressPrint publication year: 2022