Construct a reusable ReLU trunk that emits approximate univariate powers or Legendre-polynomial features for each input coordinate, then combine them with a linear or low-rank polynomial head. This gives a compact explicit basis for smooth functions and can replace a large generic MLP in low-dimensional scientific regression or serve as a frozen or partially trainable front-end.
For smooth coordinate-based regression, replace a width-heavy MLP with a deliberately narrow but deeper ReLU network and choose depth and width using the paper's analytic-function approximation law. The hypothesis is that, at fixed parameter count, increasing depth gives a larger reduction in approximation error than increasing width when the target is close to analytic.
Replace a noisy or expensive per-layer spectral-norm estimate with a sharp upper bound obtained by maximizing the largest squared singular value subject to several layer spectral moments. The bound uses the paper's few-distinct-values structure, so the optimization scales with the number of moments rather than the width of the layer.