1.3. Related work🔗

Ridgelet analysis, developed by Murata, Candès and subsequent authors, relates ridge functions to Fourier, Radon and wavelet analysis; its distributional extension admits unbounded activations, and weighted Sobolev activation spaces provide continuous synthesis pairings. Earlier Hilbert-space constructions integrate over a finite-dimensional subspace.

Integration over an infinite-dimensional space with respect to a Gaussian measure is classical: Fourier–Wiener transforms, Gaussian Radon transforms, and a ridgelet transform for Wiener functionals whose atoms are generated by a functional on the Cameron–Martin space and whose basic identity relates two Gaussian spaces of different variance rather than reconstructing the target. Here the atoms are neurons, the direction measure is homogeneous, and the Plancherel identity holds on a single space.

Universal approximation with infinite-dimensional inputs goes back to the approximation of functionals; compact-open universality is known on Fréchet spaces with operator activations, in L^p of a Radon measure, for scalar ridge neurons on locally convex spaces with every continuous non-polynomial activation, and globally on weighted spaces. These are existence theorems; here the approximants are obtained from an explicit coefficient, one for each admissible analysis filter.

Finite-width bounds by sampling a parameter distribution follow Barron and the variation-norm viewpoint, with Hilbert-input, Banach-valued and encoder–decoder versions. Here the sampling measure is the homogeneous measure \nu_\alpha, the coefficient is characterized by an analysis operator with a Plancherel identity and a closed-range projection, and the uniform rate covers Hilbert-valued outputs. Lower bounds exclude such rates for all Lipschitz operators; the rate here concerns a Barron-type class. The manuscript's Appendix F expands these comparisons.