Universal admissibility for scattering transforms

What the math gives to ML

This paper removes the usual wavelet-specific frequency-covering and analyticity assumptions from scattering transforms: a Parseval filter bank with a genuinely nonzero low-pass response at zero already yields a norm-preserving infinite-depth representation. The transferable asset is a principled way to build stable, geometry-agnostic convolution-modulus front ends from arbitrary learned or handcrafted filter banks. Its finite-depth guarantee is only polynomial, with residual energy bounded by O(N^{-min{s,1}/d}) for H^s inputs, and the counterexample rules out assuming exponential depth convergence. A practical neural-network use is a certified finite-depth scattering stem whose filters are constrained to Parseval form and whose depth is selected from an empirical residual-energy estimate rather than an unjustified exponential-rate heuristic.