  Controlling the parameters' norm often yields good generalisation when
training neural networks. Beyond simple intuitions, the relation between
regularising parameters' norm and obtained estimators remains theoretically
misunderstood. For one hidden ReLU layer networks with unidimensional data,
this work shows the parameters' norm required to represent a function is given
by the total variation of its second derivative, weighted by a $\sqrt{1+x^2}$
factor. Notably, this weighting factor disappears when the norm of bias terms
is not regularised. The presence of this additional weighting factor is of
utmost significance as it is shown to enforce the uniqueness and sparsity (in
the number of kinks) of the minimal norm interpolator. Conversely, omitting the
bias' norm allows for non-sparse solutions. Penalising the bias terms in the
regularisation, either explicitly or implicitly, thus leads to sparse
estimators.
