An Analytical Approach to Compute the Exact Preimage of Feed-Forward Neural Networks

TL;DR

This paper introduces an analytical method to compute the exact preimage of feed-forward neural networks with linear or piecewise linear activations, enhancing understanding of network outputs by providing complete preimages.

Contribution

It presents a novel analytical approach to determine the entire preimage of such neural networks, unlike previous numerical or approximate methods.

Findings

Computes exact preimages for networks with linear or piecewise linear activations.

Provides the entire set of inputs corresponding to a given output.

Offers a non-unique, analytical solution for preimages.

Abstract

Neural networks are a convenient way to automatically fit functions that are too complex to be described by hand. The downside of this approach is that it leads to build a black-box without understanding what happened inside. Finding the preimage would help to better understand how and why such neural networks had given such outputs. Because most of the neural networks are noninjective function, it is often impossible to compute it entirely only by a numerical way. The point of this study is to give a method to compute the exact preimage of any Feed-Forward Neural Network with linear or piecewise linear activation functions for hidden layers. In contrast to other methods, this one is not returning a unique solution for a unique output but returns analytically the entire and exact preimage.