In this paper, we provide a geometric interpretation of the structure of Deep Learning (DL) networks, characterized by $L$ hidden layers, a 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 defined on spaces ${\mathbb R}^{Q}$, as well. 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.

用几何方法解释了Deep Learning（DL）网络的结构，包括L个隐藏层、坡度激活函数、L2 Schatten类（或Hilbert-Schmidt）成本函数以及相等维度的输入输出空间R^Q。隐藏层在空间R^Q上定义，并且用于通过递归应用截断映射来筛选训练输入。此外，找到了成本函数的2^Q-1个不同退化局部最小值。

深度学习网络的几何结构和全局L^2最小化器的构建