Resolution-Consistent Greedy Neural Approximation on Infinite-Dimensional Spaces
arXiv:2608.20812
2026
Architecture
2 ideas extracted · analyzed Sep 1, 2026
What the math gives to ML
The paper's central transferable asset is a resolution-independent approximation framework for functions of infinite-dimensional inputs observed through finitely many coordinates. Representing each retained-resolution input as a uniformly bounded element of a common Hilbert space, together with normalized neuron parameters, prevents statistical complexity from growing with the number of coordinates. This suggests neural operators and sensor models whose weights are invariant to discretization resolution, with performance decomposed into coordinate-truncation, finite-width, and sampling errors. Fully-corrective greedy neuron addition is a second practical transfer, providing a constructive low-width training method and an explicit width-versus-error diagnostic.
Ideas from this paper
✓✓ Beats tuned baseline
2026
Build a shallow neural model whose input at every discretization level is embedded into a common Hilbert space with uniformly bounded norm, and constrain every neuron parameter in the corresponding dual norm. The statistical complexity then depends on the Hilbert norm bound rather than the number of retained coordinates, allowing one model design to operate across increasingly fine measurements.
Useful8/10
Difficulty4/10
Novelty7/10
✓✓ Beats tuned baseline
2026
Train a low-width network by repeatedly selecting a normalized neuron that is maximally correlated with the current residual, then refit all output coefficients jointly. This gives a constructive alternative to random initialization of all hidden units and exposes an empirical width-versus-error curve that can guide early stopping or architecture selection.
Useful7/10
Difficulty5/10
Novelty5/10