  Mutual information is widely applied to learn latent representations of
observations, whilst its implication in classification neural networks remain
to be better explained. We show that optimising the parameters of
classification neural networks with softmax cross-entropy is equivalent to
maximising the mutual information between inputs and labels under the balanced
data assumption. Through experiments on synthetic and real datasets, we show
that softmax cross-entropy can estimate mutual information approximately. When
applied to image classification, this relation helps approximate the point-wise
mutual information between an input image and a label without modifying the
network structure. To this end, we propose infoCAM, informative class
activation map, which highlights regions of the input image that are the most
relevant to a given label based on differences in information. The activation
map helps localise the target object in an input image. Through experiments on
the semi-supervised object localisation task with two real-world datasets, we
evaluate the effectiveness of our information-theoretic approach.
