Geometric structure of Deep Learning networks and construction of global ${\mathcal L}^2$ minimizers

In this paper, we provide a geometric interpretation of the structure of Deep Learning (DL) networks, characterized by $L$ hidden layers, a ReLU ramp activation function, an $\mathcal{L}^2$ Schatten class (or Hilbert-Schmidt) cost function, and input and output spaces $\mathbb{R}^Q$ with equal dimension $Q\geq1$. The hidden layers are also defined on $\mathbb{R}^{Q}$; the training input size $N$ can be arbitrarily large - thus, we are considering the underparametrized regime. We apply our recent results on shallow neural networks to construct an explicit family of minimizers for the global minimum of the cost function in the case $L\geq Q$, which we show to be degenerate. In the context presented here, the hidden layers of the DL network "curate" the training inputs by recursive application of a truncation map that minimizes the noise to signal ratio of the training inputs. Moreover, we determine a set of $2^Q-1$ distinct degenerate local minima of the cost function. Our constructions make no use of gradient descent algorithms at all.