  Neural networks have been successfully used for classification tasks in a
rapidly growing number of practical applications. Despite their popularity and
widespread use, there are still many aspects of training and classification
that are not well understood. In this paper we aim to provide some new insights
into training and classification by analyzing neural networks from a
feature-space perspective. We review and explain the formation of decision
regions and study some of their combinatorial aspects. We place a particular
emphasis on the connections between the neural network weight and bias terms
and properties of decision boundaries and other regions that exhibit varying
levels of classification confidence. We show how the error backpropagates in
these regions and emphasize the important role they have in the formation of
gradients. These findings expose the connections between scaling of the weight
parameters and the density of the training samples. This sheds more light on
the vanishing gradient problem, explains the need for regularization, and
suggests an approach for subsampling training data to improve performance.
