The geometry of deep networks: Power diagram subdivision

Randall Balestriero, Romain Cosentino, Behnaam Aazhang, Richard G. Baraniuk

Research output: Contribution to journalConference articlepeer-review

21 Scopus citations


We study the geometry of deep (neural) networks (DNs) with piecewise affine and convex nonlinearities. The layers of such DNs have been shown to be max-affine spline operators (MASOs) that partition their input space and apply a region-dependent affine mapping to their input to produce their output. We demonstrate that each MASO layer's input space partition corresponds to a power diagram (an extension of the classical Voronoi tiling) with a number of regions that grows exponentially with respect to the number of units (neurons). We further show that a composition of MASO layers (e.g., the entire DN) produces a progressively subdivided power diagram and provide its analytical form. The subdivision process constrains the affine maps on the potentially exponentially many power diagram regions with respect to the number of neurons to greatly reduce their complexity. For classification problems, we obtain a formula for the DN's decision boundary in the input space plus a measure of its curvature that depends on the DN's architecture, nonlinearities, and weights. Numerous numerical experiments support and extend our theoretical results.

Original languageEnglish (US)
JournalAdvances in Neural Information Processing Systems
StatePublished - 2019
Event33rd Annual Conference on Neural Information Processing Systems, NeurIPS 2019 - Vancouver, Canada
Duration: Dec 8 2019Dec 14 2019

ASJC Scopus subject areas

  • Computer Networks and Communications
  • Information Systems
  • Signal Processing


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