A spline theory of deep networks

Richard G. Baraniuk, Randall Balestriero

Research output: Chapter in Book/Report/Conference proceedingConference contribution

4 Scopus citations

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 languageEnglish (US)
Title of host publication35th International Conference on Machine Learning, ICML 2018
EditorsAndreas Krause, Jennifer Dy
PublisherInternational Machine Learning Society (IMLS)
Pages646-660
Number of pages15
Volume1
ISBN (Electronic)9781510867963
StatePublished - Jan 1 2018
Event35th International Conference on Machine Learning, ICML 2018 - Stockholm, Sweden
Duration: Jul 10 2018Jul 15 2018

Other

Other35th International Conference on Machine Learning, ICML 2018
CountrySweden
CityStockholm
Period7/10/187/15/18

ASJC Scopus subject areas

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

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