  This paper introduces a new loss function induced by the Fourier-based
Metric. This metric is equivalent to the Wasserstein distance but is computed
very efficiently using the Fast Fourier Transform algorithm. We prove that the
Fourier loss function is twice differentiable, and we provide the explicit
formula for both its gradient and its Hessian matrix. More importantly, we show
that minimising the Fourier loss function is equivalent to maximising the
likelihood of the data under a Gaussian noise in the space of frequencies. We
apply our loss function to a multi-class classification task using MNIST,
Fashion-MNIST, and CIFAR10 datasets. The computational results show that, while
its accuracy is competitive with other state-of-the-art loss functions, the
Fourier loss function is significantly more robust to noisy data.
