  Deep Neural Networks (DNNs) have become very popular for prediction in many
areas. Their strength is in representation with a high number of parameters
that are commonly learned via gradient descent or similar optimization methods.
However, the representation is non-standardized, and the gradient calculation
methods are often performed using component-based approaches that break
parameters down into scalar units, instead of considering the parameters as
whole entities. In this work, these problems are addressed. Standard notation
is used to represent DNNs in a compact framework. Gradients of DNN loss
functions are calculated directly over the inner product space on which the
parameters are defined. This framework is general and is applied to two common
network types: the Multilayer Perceptron and the Deep Autoencoder.
