Sharp Mixed Spectral Barron Regularity of Coulombic Many-Electron Wave Functions

What the math gives to ML

The paper identifies a mixed Fourier-L1 regularity structure in which global isotropic frequency control is supplemented by products of coordinate-wise frequency weights. This is useful for neural models of high-dimensional functions whose singularities or interactions are localized to subsets of coordinates, because an isotropic spectral penalty can miss smoothness that factorizes across particles or variables. The most direct transfer is a mixed spectral regularizer or Fourier-feature architecture for particle and molecular networks, with coordinate-product weights explicitly encouraging low-complexity dependence on selected input blocks.