6. Approximation by finite-width networks🔗

Monte Carlo approximation turns a finite coefficient measure into a finite-width network. The Banach-valued Rademacher lemma precedes the general bound; a two-coordinate comparison then gives the dimension-free Lipschitz rate. We prove spectral moments and qualitative discretization before compact-open universality. The final subsection gives the Hilbert variance and vector rates. Appendix C records additional concentration and input-truncation consequences.

1. 6.1. Rademacher complexity and the general approximation bound
2. 6.2. Explicit Lipschitz approximation rate
3. 6.3. Finite variation and moments from a spectral density
4. 6.4. Constructive universal approximation
5. 6.5. Vector-valued approximation