On high probability of universal approximation in random basis expansions with non-continuous weight sampling

What the math gives to ML

The paper identifies a practically relevant universality mechanism for random-feature networks: hidden weights need not come from a continuous distribution, provided their support is dense in the weight space. This suggests replacing expensive floating-point hidden-layer initialization with discrete, sparse, or multiresolution samplers while retaining approximation capability in the increasing-width limit. The most actionable transfer is a frozen random-feature layer with trainable output coefficients, followed by finite-width experiments measuring whether discrete hidden parameters preserve accuracy while reducing storage and feature-generation cost.