  Convolution utilizes a shift-equivalent prior of images, thus leading to
great success in image processing tasks. However, commonly used poolings in
convolutional neural networks (CNNs), such as max-pooling, average-pooling, and
strided-convolution, are not shift-equivalent. Thus, the shift-equivalence of
CNNs is destroyed when convolutions and poolings are stacked. Moreover,
anti-aliasing is another essential property of poolings from the perspective of
signal processing. However, recent poolings are neither shift-equivalent nor
anti-aliasing. To address this issue, we propose a new pooling method that is
shift-equivalent and anti-aliasing, named frequency pooling. Frequency pooling
first transforms the features into the frequency domain, and then removes the
frequency components beyond the Nyquist frequency. Finally, it transforms the
features back to the spatial domain. We prove that frequency pooling is
shift-equivalent and anti-aliasing based on the property of Fourier transform
and Nyquist frequency. Experiments on image classification show that frequency
pooling improves accuracy and robustness with respect to the shifts of CNNs.
