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During the last years, Convolutional Neural Networks (CNNs) have achievedstate-of-the-art performance in image classification. Their architectures havelargely drawn inspiration by models of the primate visual system. However,while recent research results of neuroscience prove the existence of non-linearoperations in the response of complex visual cells, little effort has beendevoted to extend the convolution technique to non-linear forms. Typicalconvolutional layers are linear systems, hence their expressiveness is limited.To overcome this, various non-linearities have been used as activationfunctions inside CNNs, while also many pooling strategies have been applied. Weaddress the issue of developing a convolution method in the context of acomputational model of the visual cortex, exploring quadratic forms through theVolterra kernels. Such forms, constituting a more rich function space, are usedas approximations of the response profile of visual cells. Our proposedsecond-order convolution is tested on CIFAR-10 and CIFAR-100. We show that anetwork which combines linear and non-linear filters in its convolutionallayers, can outperform networks that use standard linear filters with the samearchitecture, yielding results competitive with the state-of-the-art on thesedatasets.

Non-linear Convolution Filters for CNN-based Learning