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    On the Decision Boundaries of Deep Neural Networks: A Tropical Geometry Perspective

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    Type
    Preprint
    Authors
    Alfarra, Motasem
    Bibi, Adel cc
    Hammoud, Hasan
    Gaafar, Mohamed
    Ghanem, Bernard cc
    KAUST Department
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Electrical Engineering
    Electrical Engineering Program
    VCC Analytics Research Group
    Date
    2020-02-20
    Permanent link to this record
    http://hdl.handle.net/10754/661768
    
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    Abstract
    This work tackles the problem of characterizing and understanding the decision boundaries of neural networks with piecewise linear non-linearity activations. We use tropical geometry, a new development in the area of algebraic geometry, to characterize the decision boundaries of a simple neural network of the form (Affine, ReLU, Affine). Our main finding is that the decision boundaries are a subset of a tropical hypersurface, which is intimately related to a polytope formed by the convex hull of two zonotopes. The generators of these zonotopes are functions of the neural network parameters. This geometric characterization provides new perspective to three tasks. Specifically, we propose a new tropical perspective to the lottery ticket hypothesis, where we see the effect of different initializations on the tropical geometric representation of a network's decision boundaries. Moreover, we use this characterization to propose a new set of tropical regularizers, which directly deal with the decision boundaries of a network. We investigate the use of these regularizers in neural network pruning (by removing network parameters that do not contribute to the tropical geometric representation of the decision boundaries) and in generating adversarial input attacks (by producing input perturbations that explicitly perturb the decision boundaries' geometry and ultimately change the network's prediction).
    Sponsors
    This work was supported by the King Abdullah University of Science and Technology (KAUST) Office of Sponsored Research.
    Publisher
    arXiv
    arXiv
    2002.08838
    Additional Links
    https://arxiv.org/pdf/2002.08838
    Collections
    Preprints; Electrical Engineering Program; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

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