Sharp embeddings between quasi-Banach Besov spaces and shallow ReLU variation spaces
arXiv:2609.00680
2026
Regularization
2 ideas extracted · analyzed Sep 2, 2026
What the math gives to ML
The paper establishes a sharp correspondence between shallow ReLU^k ridge networks with bounded coefficient variation and isotropic Besov smoothness. Its transferable asset is a mathematically grounded complexity measure: total variation of the representing measure controls both approximation structure and frequency regularity, rather than merely penalizing Euclidean parameter magnitudes. This supports two practical mechanisms: multiscale Besov regularization of network outputs and active-set neuron growth or pruning under an explicit variation budget. The strongest initial validation should use low-dimensional function regression, where Fourier spectra, active width, and approximation error can be measured directly.
Ideas from this paper
Unverified
2026
Add a multiscale Besov penalty to the output of a shallow ReLU^k network, targeting the smoothness threshold that the paper proves is sufficient for finite ridge-variation representation. This suppresses pathological high-frequency output while preserving low-frequency approximation, providing a principled alternative to ordinary parameter weight decay.
Useful6/10
Difficulty4/10
Novelty6/10
Unverified
2026
Treat a finite shallow network as a discrete signed measure over ridge atoms and grow or prune neurons according to their contribution to total variation. This turns width selection into an atomic approximation procedure: add neurons correlated with the current residual and remove coefficients that consume budget without contributing materially.
Useful5/10
Difficulty5/10
Novelty5/10