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It is an elementary fact in the scientific literature that the Lipschitz normof the realization function of a feedforward fully-connected rectified linearunit (ReLU) artificial neural network (ANN) can, up to a multiplicativeconstant, be bounded from above by sums of powers of the norm of the ANNparameter vector. Roughly speaking, in this work we reveal in the case ofshallow ANNs that the converse inequality is also true. More formally, we provethat the norm of the equivalence class of ANN parameter vectors with the samerealization function is, up to a multiplicative constant, bounded from above bythe sum of powers of the Lipschitz norm of the ANN realization function (withthe exponents $ 1/2 $ and $ 1 $). Moreover, we prove that this upper bound onlyholds when employing the Lipschitz norm but does neither hold for H\"oldernorms nor for Sobolev-Slobodeckij norms. Furthermore, we prove that this upperbound only holds for sums of powers of the Lipschitz norm with the exponents $1/2 $ and $ 1 $ but does not hold for the Lipschitz norm alone.

On bounds for norms of reparameterized ReLU artificial neural network  parameters: sums of fractional powers of the Lipschitz norm control the  network parameter vector