Rank Diminishing in Deep Neural Networks

It is an instance of a key structural condition that applies across broad domains of machine learning. In particular, the assumption of low-rank feature representations led to algorithmic developments in many architectures. For neural networks, however, the intrinsic mechanism that yields low-rank structures remains vague and unclear. To fill this gap, we perform a rigorous study on the behavior of network rank, focusing particularly on the notion of rank deficiency. We theoretically establish a universal monotone decreasing property of network ranks from the basic rules of differential and algebraic composition, and uncover rank deficiency of network blocks and deep function coupling.