arXiv:2608.24884 2026 Architecture 1 ideas extracted · analyzed Aug 29, 2026
The paper builds a polynomial-like function space from iterated Stieltjes integrals, while preserving interpolation, Newton-coordinate, Hermite, and best-approximation properties. Its transferable asset is an adaptive one-dimensional geometry: the pseudometric d_g(x,y)=|g(x)-g(y)| measures distance in a transformed coordinate, allowing basis resolution to follow the data distribution rather than the raw coordinate. A practical neural-network adaptation is a compact Stieltjes-Newton feature head using a fixed or learned monotone g, with incremental basis expansion when additional capacity is needed.
Replace raw polynomial or Fourier features for a scalar coordinate x with iterated Stieltjes-integral features generated by a monotone data-adaptive coordinate g. Use a Newton-form basis so the model can add higher-order terms or new interpolation knots without recomputing all previous coefficients.