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We study the structural and statistical properties of $\mathcal{R}$-normminimizing interpolants of datasets labeled by specific target functions. The$\mathcal{R}$-norm is the basis of an inductive bias for two-layer neuralnetworks, recently introduced to capture the functional effect of controllingthe size of network weights, independently of the network width. We find thatthese interpolants are intrinsically multivariate functions, even when thereare ridge functions that fit the data, and also that the $\mathcal{R}$-norminductive bias is not sufficient for achieving statistically optimalgeneralization for certain learning problems. Altogether, these results shednew light on an inductive bias that is connected to practical neural networktraining.

Intrinsic dimensionality and generalization properties of the $\mathcal{R}$-norm inductive bias