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

12/17/2021
by   Alessandro Benfenati, et al.
0

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.

READ FULL TEXT

page 21

page 22

page 23

research
12/17/2021

A singular Riemannian geometry approach to Deep Neural Networks I. Theoretical foundations

Deep Neural Networks are widely used for solving complex problems in sev...
research
06/17/2022

Sheaf Neural Networks with Connection Laplacians

A Sheaf Neural Network (SNN) is a type of Graph Neural Network (GNN) tha...
research
07/10/2020

Transformations between deep neural networks

We propose to test, and when possible establish, an equivalence between ...
research
12/10/2020

On the emergence of tetrahedral symmetry in the final and penultimate layers of neural network classifiers

A recent numerical study observed that neural network classifiers enjoy ...
research
09/11/2018

ManifoldNet: A Deep Network Framework for Manifold-valued Data

Deep neural networks have become the main work horse for many tasks invo...
research
11/16/2017

LDMNet: Low Dimensional Manifold Regularized Neural Networks

Deep neural networks have proved very successful on archetypal tasks for...
research
05/29/2018

Statistical Recurrent Models on Manifold valued Data

In a number of disciplines, the data (e.g., graphs, manifolds) to be ana...

Please sign up or login with your details

Forgot password? Click here to reset