and are less prone to overfitting [@Srivastava2014; @Kingma; @Poole2014].

Even though the regularisation conferred by Gaussian noise injections (GNIs) can be observed empirically, and the benefits of noising data are well understood theoretically [@Bishop1995; @Cohen2019; @Webb1994], there have been few studies on understanding the benefits of methods that inject noise *throughout* a network. Here we study the *explicit* regularisation of such injections, which is the added positive term we obtain to the loss function when we marginalise out the noise we have injected. In particular, we show that the regularisers we derive are a form of Tikhonov Regularisation which penalises some function space norm of a neural network [@Girosi1990; @Burger2003; @Bishop1995]. Here we show that noise injections induce a penalisation of a Sobolev space norm and in particular we show that this corresponds to a prior that penalises functions with high-frequency components in the Fourier domain.

Here we show analytically that such regularisation promotes larger classification margins which make models robust to noise, explaining the empirical results of @Rahaman2019 which established that neural networks with lower-frequency spectra are more robust to input perturbations. Finally we show empirically that the explicit regulariser also improves model calibration.

Concretely our contributions are:

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