Infinite-dimensional operator ridgelet transform
This Blueprint follows the manuscript in reading order, from the motivation and the Gaussian-weighted construction to reconstruction, finite-width approximation and examples. Sections 1–9 contain the main exposition, numerical experiments and discussion. Prerequisites and detailed proofs appear in the main exposition before their use. Appendices A–E give optional kernel and projection formulas, additional approximation bounds, finite-dimensional comparison and the dilation obstruction, and numerical methods. The manuscript's Appendix F is an extended overview of related work with comparison tables; it contains no mathematical statements and is not reproduced here.
Numbered mathematical statements have the manuscript's identifiers and numbers. Their Lean panels show the precise formal statements and proof status. Descriptive auxiliary nodes provide definitions and formalization infrastructure without changing manuscript numbering. The dependency graph links these mathematical ingredients; prose, remarks and numerical experiments are not marked as machine-checked theorems.
Contents
- 1. Introduction
- 2. Neural networks with Hilbert-space inputs
- 3. The Gaussian-weighted ridgelet transform
- 4. Representation and reconstruction
- 5. Reconstruction formulas and activation functions
- 6. Approximation by finite-width networks
- 7. Examples on infinite-dimensional spaces
- 8. Numerical experiments
- 9. Discussion
- Appendix A. A kernel interpretation of the frame operator
- Appendix B. Coefficient projections and minimum-norm solutions
- Appendix C. Additional approximation bounds
- Appendix D. The finite-dimensional case and the dilation obstruction
- Appendix E. Numerical methods and additional results
- Formalization infrastructure
- Dependency Graph
- Blueprint Summary