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In this paper, we provide a geometric interpretation of the structure of DeepLearning (DL) networks, characterized by $L$ hidden layers, a ReLU rampactivation 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 traininginput size $N$ can be arbitrarily large - thus, we are considering theunderparametrized regime. We apply our recent results on shallow neuralnetworks to construct an explicit family of minimizers for the global minimumof the cost function in the case $L\geq Q$, which we show to be degenerate. Inthe context presented here, the hidden layers of the DL network "curate" thetraining inputs by recursive application of a truncation map that minimizes thenoise to signal ratio of the training inputs. Moreover, we determine a set of$2^Q-1$ distinct degenerate local minima of the cost function. Ourconstructions make no use of gradient descent algorithms at all.

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