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In the last decade, we have witnessed the introduction of several novel deepneural network (DNN) architectures exhibiting ever-increasing performanceacross diverse tasks. Explaining the upward trend of their performance,however, remains difficult as different DNN architectures of comparable depthand width -- common factors associated with their expressive power -- mayexhibit a drastically different performance even when trained on the samedataset. In this paper, we introduce the concept of the non-linearity signatureof DNN, the first theoretically sound solution for approximately measuring thenon-linearity of deep neural networks. Built upon a score derived fromclosed-form optimal transport mappings, this signature provides a betterunderstanding of the inner workings of a wide range of DNN architectures andlearning paradigms, with a particular emphasis on the computer vision task. Weprovide extensive experimental results that highlight the practical usefulnessof the proposed non-linearity signature and its potential for long-reachingimplications. The code for our work is available athttps://github.com/qbouniot/AffScoreDeep

From Alexnet to Transformers: Measuring the Non-linearity of Deep Neural Networks with Affine Optimal Transport