We fully characterize a large class of feedforward neural networks in terms of truncation maps. As an application, we show how a ReLU neural network can implement a feature map which separates concentric data.

We derive explicit equations governing the cumulative biases and weights in Deep Learning with ReLU activation function, based on gradient descent for the Euclidean cost in the input layer, and under the assumption that the weights are, in a precise sense, adapted to the coordinate system distinguished by the activations. We show that gradient descent corresponds to a dynamical process in the input layer, whereby clusters of data are progressively reduced in complexity ("truncated") at an exponential rate that increases with the number of data points that have already been truncated. We provide a detailed discussion of several types of solutions to the gradient flow equations. A main motivation for this work is to shed light on the interpretability question in supervised learning.

We explicitly construct zero loss neural network classifiers. We write the weight matrices and bias vectors in terms of cumulative parameters, which determine truncation maps acting recursively on input space. The configurations for the training data considered are (i) sufficiently small, well separated clusters corresponding to each class, and (ii) equivalence classes which are sequentially linearly separable. In the best case, for $Q$ classes of data in $\mathbb{R}^M$, global minimizers can be described with $Q(M+2)$ parameters.

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.