  This paper introduces two straightforward, effective indices to evaluate the
input data and the data flowing through layers of a feedforward deep neural
network. For classification problems, the separation rate of target labels in
the space of dataflow is explained as a key factor indicating the performance
of designed layers in improving the generalization of the network. According to
the explained concept, a shapeless distance-based evaluation index is proposed.
Similarly, for regression problems, the smoothness rate of target outputs in
the space of dataflow is explained as a key factor indicating the performance
of designed layers in improving the generalization of the network. According to
the explained smoothness concept, a shapeless distance-based smoothness index
is proposed for regression problems. To consider more strictly concepts of
separation and smoothness, their extended versions are introduced, and by
interpreting a regression problem as a classification problem, it is shown that
the separation and smoothness indices are related together. Through four case
studies, the profits of using the introduced indices are shown. In the first
case study, for classification and regression problems , the challenging of
some known input datasets are compared respectively by the proposed separation
and smoothness indices. In the second case study, the quality of dataflow is
evaluated through layers of two pre-trained VGG 16 networks in classification
of Cifar10 and Cifar100. In the third case study, it is shown that the correct
classification rate and the separation index are almost equivalent through
layers particularly while the serration index is increased. In the last case
study, two multi-layer neural networks, which are designed for the prediction
of Boston Housing price, are compared layer by layer by using the proposed
smoothness index.
