A Formal Proof of the Expressiveness of Deep Learning

  • Alexander BentkampEmail author
  • Jasmin Christian Blanchette
  • Dietrich Klakow
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10499)


Deep learning has had a profound impact on computer science in recent years, with applications to image recognition, language processing, bioinformatics, and more. Recently, Cohen et al. provided theoretical evidence for the superiority of deep learning over shallow learning. We formalized their mathematical proof using Isabelle/HOL. The Isabelle development simplifies and generalizes the original proof, while working around the limitations of the HOL type system. To support the formalization, we developed reusable libraries of formalized mathematics, including results about the matrix rank, the Borel measure, and multivariate polynomials as well as a library for tensor analysis.



We thank Lukas Bentkamp, Robert Lewis, Anders Schlichtkrull, Mark Summerfield, and the anonymous reviewers for suggesting many textual improvements. The work has received funding from the European Research Council under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 713999, Matryoshka).


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Alexander Bentkamp
    • 1
    • 2
    Email author
  • Jasmin Christian Blanchette
    • 1
    • 3
  • Dietrich Klakow
    • 2
  1. 1.Vrije Universiteit AmsterdamAmsterdamThe Netherlands
  2. 2.Universität des SaarlandesSaarbrückenGermany
  3. 3.Max-Planck-Institut für InformatikSaarbrückenGermany

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