  It has been repeatedly observed that convolutional architectures when applied
to image understanding tasks learn oriented bandpass filters. A standard
explanation of this result is that these filters reflect the structure of the
images that they have been exposed to during training: Natural images typically
are locally composed of oriented contours at various scales and oriented
bandpass filters are matched to such structure. We offer an alternative
explanation based not on the structure of images, but rather on the structure
of convolutional architectures. In particular, complex exponentials are the
eigenfunctions of convolution. These eigenfunctions are defined globally;
however, convolutional architectures operate locally. To enforce locality, one
can apply a windowing function to the eigenfunctions, which leads to oriented
bandpass filters as the natural operators to be learned with convolutional
architectures. From a representational point of view, these filters allow for a
local systematic way to characterize and operate on an image or other signal.
We offer empirical support for the hypothesis that convolutional networks learn
such filters at all of their convolutional layers. While previous research has
shown evidence of filters having oriented bandpass characteristics at early
layers, ours appears to be the first study to document the predominance of such
filter characteristics at all layers. Previous studies have missed this
observation because they have concentrated on the cumulative compositional
effects of filtering across layers, while we examine the filter characteristics
that are present at each layer.
