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
15 - Learning Data-Driven Optimized Kernels
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
Standard kernels with scalar shape parameters lack flexibility for capturing anisotropic data structures in high dimensions. We present two-layered kernels built on a matrix transformation used to learn optimal feature directions. For Matérn kernels, we prove convergence rates depend on the effective dimension of this matrix, with faster rates when null space aligns with invariant subspaces of the target function. The 2L-KOGA algorithm combines kernel optimization via stochastic gradient descent with greedy node selection through VKOGA. Numerical experiments on regression benchmarks show 2L-KOGA outperforms standard kernels when significant dimensions can be reduced, with performance correlating to the concentration of singular values.
Information
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- Kernel-Based MethodsMilestones and New Trends, pp. 140 - 148Publisher: Cambridge University PressPrint publication year: 2026
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