## Abstract

We build a rigorous bridge between deep networks (DNs) and approximation theory via spline functions and operators. Our key result is that a large class of DNs can be written as a composition of max-affine spline operators (MASOs), which provide a powerful portal through which to view and analyze their inner workings. For instance, conditioned on the input signal, the output of a MASO DN can be written as a simple affine transformation of the input. This implies that a DN constructs a set of signal-dependent, class-specific templates against which the signal is compared via a simple inner product; we explore the links to the classical theory of optimal classification via matched filters and the effects of data memorization. Going further, we propose a simple penalty term that can be added to the cost function of any DN learning algorithm to force the templates to be orthogonal with each other; this leads to significantly improved classification performance and reduccd ovcrfitting with no change to the DN architecture. The spline partition of the input signal space opens up a new geometric avenue to study how DNs organize signals in a hierarchical fashion. As an application, we develop and validate a new distance metric for signals that quantifies the difference between their partition encodings.

Original language | English (US) |
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Title of host publication | 35th International Conference on Machine Learning, ICML 2018 |

Editors | Andreas Krause, Jennifer Dy |

Publisher | International Machine Learning Society (IMLS) |

Pages | 646-660 |

Number of pages | 15 |

Volume | 1 |

ISBN (Electronic) | 9781510867963 |

State | Published - Jan 1 2018 |

Event | 35th International Conference on Machine Learning, ICML 2018 - Stockholm, Sweden Duration: Jul 10 2018 → Jul 15 2018 |

### Other

Other | 35th International Conference on Machine Learning, ICML 2018 |
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Country/Territory | Sweden |

City | Stockholm |

Period | 7/10/18 → 7/15/18 |

## ASJC Scopus subject areas

- Computational Theory and Mathematics
- Human-Computer Interaction
- Software