  The conclusions provided by deep neural networks (DNNs) must be carefully
scrutinized to determine whether they are universal or architecture dependent.
The term DAG-DNN refers to a graphical representation of a DNN in which the
architecture is expressed as a direct-acyclic graph (DAG), on which arcs are
associated with functions. The level of a node denotes the maximum number of
hops between the input node and the node of interest. In the current study, we
demonstrate that DAG-DNNs can be used to derive all functions defined on
various sub-architectures of the DNN. We also demonstrate that the functions
defined in a DAG-DNN can be derived via a sequence of lower-triangular
matrices, each of which provides the transition of functions defined in
sub-graphs up to nodes at a specified level. The lifting structure associated
with lower-triangular matrices makes it possible to perform the structural
pruning of a network in a systematic manner. The fact that decomposition is
universally applicable to all DNNs means that network pruning could
theoretically be applied to any DNN, regardless of the underlying architecture.
We demonstrate that it is possible to obtain the winning ticket (sub-network
and initialization) for a weak version of the lottery ticket hypothesis, based
on the fact that the sub-network with initialization can achieve training
performance on par with that of the original network using the same number of
iterations or fewer.
