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Controlling the parameters' norm often yields good generalisation whentraining neural networks. Beyond simple intuitions, the relation betweenregularising parameters' norm and obtained estimators remains theoreticallymisunderstood. For one hidden ReLU layer networks with unidimensional data,this work shows the parameters' norm required to represent a function is givenby 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 termsis not regularised. The presence of this additional weighting factor is ofutmost significance as it is shown to enforce the uniqueness and sparsity (inthe number of kinks) of the minimal norm interpolator. Conversely, omitting thebias' norm allows for non-sparse solutions. Penalising the bias terms in theregularisation, either explicitly or implicitly, thus leads to sparseestimators.

Penalising the biases in norm regularisation enforces sparsity