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In this paper we present an algebraic approach to the precise and globalverification and explanation of Rectifier Neural Networks, a subclass ofPiece-wise Linear Neural Networks (PLNNs), i.e., networks that semanticallyrepresent piece-wise affine functions. Key to our approach is the symbolicexecution of these networks that allows the construction of semanticallyequivalent Typed Affine Decision Structures (TADS). Due to their deterministicand sequential nature, TADS can, similarly to decision trees, be considered aswhite-box models and therefore as precise solutions to the model and outcomeexplanation problem. TADS are linear algebras which allows one to elegantlycompare Rectifier Networks for equivalence or similarity, both with precisediagnostic information in case of failure, and to characterize theirclassification potential by precisely characterizing the set of inputs that arespecifically classified or the set of inputs where two network-basedclassifiers differ. All phenomena are illustrated along a detailed discussionof a minimal, illustrative example: the continuous XOR function.

Towards Rigorous Understanding of Neural Networks via Semantics-preserving Transformations