Methods, systems, and apparatus, including computer programs encoded on computer storage media, for processing inputs using a neural network that includes a circulant neural network layer. One of the methods includes receiving a layer input for the circulant layer; and processing the layer input to generate a layer output for the circulant layer, wherein processing the layer input comprises computing an activation function, wherein the activation function is dependent on the product of the circulant matrix associated with the circulant layer and the layer input, and wherein computing the activation function comprises performing a circular convolution using a Fast Fourier Transform (FFT).
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