Network size and size of the weights in memorization with two-layers neural networks

In 1988, Eric B. Baum showed that two-layers neural networks with threshold activation function can perfectly memorize the binary labels of n points in general position in \R d using only \ulcorner n/d \urcorner neurons. We observe that with ReLU networks, using four times as many neurons one can fit arbitrary real labels. Moreover, for approximate memorization up to error \epsilon, the neural tangent kernel can also memorize with only O\left(\frac{n}{d} \cdot \log(1/\epsilon) \right) neurons (assuming that the data is well dispersed too). We show however that these constructions give rise to networks where the \emph{magnitude} of the neurons' weights are far from optimal. In contrast we propose a new training procedure for ReLU networks, based on {\em complex} (as opposed to {\em real}) recombination of the neurons, for which we show approximate memorization with both O\left(\frac{n}{d} \cdot \frac{\log(1/\epsilon)}{\epsilon}\right) neurons, as well as nearly-optimal size of the weights.