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In a previous work, we proposed a geometric framework to study a deep neuralnetwork, seen as sequence of maps between manifolds, employing singularRiemannian geometry. In this paper, we present an application of thisframework, proposing a way to build the class of equivalence of an input point:such class is defined as the set of the points on the input manifold mapped tothe same output by the neural network. In other words, we build the preimage ofa point in the output manifold in the input space. In particular. we focus forsimplicity on the case of neural networks maps from n-dimensional real spacesto (n - 1)-dimensional real spaces, we propose an algorithm allowing to buildthe set of points lying on the same class of equivalence. This approach leadsto two main applications: the generation of new synthetic data and it mayprovides some insights on how a classifier can be confused by smallperturbation on the input data (e.g. a penguin image classified as an imagecontaining a chihuahua). In addition, for neural networks from 2D to 1D realspaces, we also discuss how to find the preimages of closed intervals of thereal line. We also present some numerical experiments with several neuralnetworks trained to perform non-linear regression tasks, including the case ofa binary classifier.

A singular Riemannian geometry approach to Deep Neural Networks II. Reconstruction of 1-D equivalence classes