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In this note, we study how neural networks with a single hidden layer andReLU activation interpolate data drawn from a radially symmetric distributionwith target labels 1 at the origin and 0 outside the unit ball, if no labelsare known inside the unit ball. With weight decay regularization and in theinfinite neuron, infinite data limit, we prove that a unique radially symmetricminimizer exists, whose weight decay regularizer and Lipschitz constant grow as$d$ and $\sqrt{d}$ respectively.  We furthermore show that the weight decay regularizer grows exponentially in$d$ if the label $1$ is imposed on a ball of radius $\varepsilon$ rather thanjust at the origin. By comparison, a neural networks with two hidden layers canapproximate the target function without encountering the curse ofdimensionality.

Optimal bump functions for shallow ReLU networks: Weight decay, depth separation and the curse of dimensionality