Book contents
- Frontmatter
- Dedication
- Reviews
- Contents
- Figures
- Tables
- Preface
- Acknowledgments
- 1 Introduction
- 2 Positive Definite Kernels and Functions
- 3 RBF Interpolation and Polynomial Precision
- 4 Error Estimates and Examples of Kernels
- 5 On Uncertainty Principles and Shape Parameter Selection
- 6 Rational Kernels
- 7 Partition of Unity
- 8 Efficient Reduced-Basis Approximation
- 9 Approximating via Variably Scaled Discontinuous Kernels
- 10 Application of VSDKs to Magnetic Particle Imaging
- 11 Applications to Astronomical Imaging
- 12 Kernels in Machine Learning
- 13 Support Vector Machines and other Methods
- 14 Local-to-global SVMs
- 15 Learning Data-Driven Optimized Kernels
- 16 Choosing and Learning the Scaling Function for the VSKs
- 17 Kernels for Persistent Homology
- 18 Software
- 19 Conclusions
- References
- Index
5 - On Uncertainty Principles and Shape Parameter Selection
Published online by Cambridge University Press: 08 August 2026
Book contents
- Frontmatter
- Dedication
- Reviews
- Contents
- Figures
- Tables
- Preface
- Acknowledgments
- 1 Introduction
- 2 Positive Definite Kernels and Functions
- 3 RBF Interpolation and Polynomial Precision
- 4 Error Estimates and Examples of Kernels
- 5 On Uncertainty Principles and Shape Parameter Selection
- 6 Rational Kernels
- 7 Partition of Unity
- 8 Efficient Reduced-Basis Approximation
- 9 Approximating via Variably Scaled Discontinuous Kernels
- 10 Application of VSDKs to Magnetic Particle Imaging
- 11 Applications to Astronomical Imaging
- 12 Kernels in Machine Learning
- 13 Support Vector Machines and other Methods
- 14 Local-to-global SVMs
- 15 Learning Data-Driven Optimized Kernels
- 16 Choosing and Learning the Scaling Function for the VSKs
- 17 Kernels for Persistent Homology
- 18 Software
- 19 Conclusions
- References
- Index
Summary
The shape parameter in kernel functions affects both accuracy and numerical stability, leading to fundamental uncertainty principles. We recall three main trade-offs: accuracy versus stability, accuracy and stability versus problem size, and accuracy versus efficiency. We present empirical strategies for shape parameter tuning, including power function minimization and cross-validation techniques such as the Extended Rippa's Algorithm (ERA). These methods enable practical, data-driven parameter selection without requiring knowledge of the underlying function.
Information
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- Kernel-Based MethodsMilestones and New Trends, pp. 33 - 38Publisher: Cambridge University PressPrint publication year: 2026
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