  Among the several paradigms of artificial intelligence (AI) or machine
learning (ML), a remarkably successful paradigm is deep learning. Deep
learning's phenomenal success has been hoped to be interpreted via fundamental
research on the theory of deep learning. Accordingly, applied research on deep
learning has spurred the theory of deep learning-oriented depth and breadth of
developments. Inspired by such developments, we pose these fundamental
questions: can we accurately approximate an arbitrary matrix-vector product
using deep rectified linear unit (ReLU) feedforward neural networks (FNNs)? If
so, can we bound the resulting approximation error? In light of these
questions, we derive error bounds in Lebesgue and Sobolev norms that comprise
our developed deep approximation theory. Guided by this theory, we have
successfully trained deep ReLU FNNs whose test results justify our developed
theory. The developed theory is also applicable for guiding and easing the
training of teacher deep ReLU FNNs in view of the emerging teacher-student AI
or ML paradigms that are essential for solving several AI or ML problems in
wireless communications and signal processing; network science and graph signal
processing; and network neuroscience and brain physics.
