Towards Rigorous Understanding of Neural Networks via Semantics-preserving Transformations

TL;DR

This paper introduces an algebraic symbolic execution method to verify and explain Rectifier Neural Networks by transforming them into transparent decision structures, enabling precise comparison and understanding of their classification behavior.

Contribution

It presents a novel algebraic approach to verify and explain piece-wise linear neural networks through symbolic execution and the construction of Typed Affine Decision Structures (TADS).

Findings

TADS provide exact explanations of network decisions.

The method enables precise comparison of neural networks for equivalence.

Illustrated with a detailed example of the XOR function.

Abstract

In this paper we present an algebraic approach to the precise and global verification and explanation of Rectifier Neural Networks, a subclass of Piece-wise Linear Neural Networks (PLNNs), i.e., networks that semantically represent piece-wise affine functions. Key to our approach is the symbolic execution of these networks that allows the construction of semantically equivalent Typed Affine Decision Structures (TADS). Due to their deterministic and sequential nature, TADS can, similarly to decision trees, be considered as white-box models and therefore as precise solutions to the model and outcome explanation problem. TADS are linear algebras which allows one to elegantly compare Rectifier Networks for equivalence or similarity, both with precise diagnostic information in case of failure, and to characterize their classification potential by precisely characterizing the set of inputs that are specifically classified or the set of inputs where two network-based classifiers differ. All phenomena are illustrated along a detailed discussion of a minimal, illustrative example: the continuous XOR function.