  Convolutional neural networks have had a great success in numerous tasks,
including image classification, object detection, sequence modelling, and many
more. It is generally assumed that such neural networks are translation
invariant, meaning that they can detect a given feature independent of its
location in the input image. While this is true for simple cases, where
networks are composed of a restricted number of layer classes and where images
are fairly simple, complex images with common state-of-the-art networks do not
usually enjoy this property as one might hope. This paper shows that most of
the existing convolutional architectures define, at initialisation, a specific
feature importance landscape that conditions their capacity to attend to
different locations of the images later during training or even at test time.
We demonstrate how this phenomenon occurs under specific conditions and how it
can be adjusted under some assumptions. We derive the P-objective, or PILCRO
for Pixel-wise Importance Landscape Curvature Regularised Objective, a simple
regularisation technique that favours weight configurations that produce
smooth, low-curvature importance landscapes that are conditioned on the data
and not on the chosen architecture. Through extensive experiments, we further
show that P-regularised versions of popular computer vision networks have a
flat importance landscape, train faster, result in a better accuracy and are
more robust to noise at test time, when compared to their original counterparts
in common computer-vision classification settings.
