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

Abstract: In a previous work, we proposed a geometric framework to study a deep neural network, seen as sequence of maps between manifolds, employing singular Riemannian geometry. In this paper, we present an application of this framework, 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 to the same output by the neural network. In other words, we build the preimage of a point in the output manifold in the input space. In particular. we focus for simplicity on the case of neural networks maps from n-dimensional real spaces to (n - 1)-dimensional real spaces, we propose an algorithm allowing to build the set of points lying on the same class of equivalence. This approach leads to two main applications: the generation of new synthetic data and it may provides some insights on how a classifier can be confused by small perturbation on the input data (e.g. a penguin image classified as an image containing a chihuahua). In addition, for neural networks from 2D to 1D real spaces, we also discuss how to find the preimages of closed intervals of the real line. We also present some numerical experiments with several neural networks trained to perform non-linear regression tasks, including the case of a binary classifier.