During the last years, Convolutional Neural Networks (CNNs) have achieved
state-of-the-art performance in image classification. Their architectures have
largely drawn inspiration by models of the primate visual system. However,
while recent research results of neuroscience prove the existence of non-linear
operations in the response of complex visual cells, little effort has been
devoted to extend the convolution technique to non-linear forms. Typical
convolutional layers are linear systems, hence their expressiveness is limited.
To overcome this, various non-linearities have been used as activation
functions inside CNNs, while also many pooling strategies have been applied. We
address the issue of developing a convolution method in the context of a
computational model of the visual cortex, exploring quadratic forms through the
Volterra kernels. Such forms, constituting a more rich function space, are used
as approximations of the response profile of visual cells. Our proposed
second-order convolution is tested on CIFAR-10 and CIFAR-100. We show that a
network which combines linear and non-linear filters in its convolutional
layers, can outperform networks that use standard linear filters with the same
architecture, yielding results competitive with the state-of-the-art on these
datasets.

研究表明，在计算模型的视觉皮层中开发基于 Volterra 核的二阶卷积方法，可以增加卷积层的表达能力，在 CIFAR-10 和 CIFAR-100 数据集上，使用线性和非线性滤波器相结合的卷积网络可以超越使用相同体系结构的标准线性滤波器的网络的性能，取得与最先进水平相当的结果。