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Revisiting Tropical Polynomial Division: Theory, Algorithms, and Application to Neural Networks.

Ioannis Kordonis, Petros Maragos

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    Summary
    This summary is machine-generated.

    This study introduces tropical polynomial division for simplifying neural networks. New algorithms are developed for real coefficients, showing potential for improved network analysis and learning model predictive control.

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    Area of Science:

    • Tropical geometry
    • Machine learning
    • Neural network analysis

    Background:

    • Tropical geometry is increasingly used for analyzing neural networks with piecewise linear activation functions.
    • Existing methods primarily focus on tropical polynomials with integer coefficients.

    Purpose of the Study:

    • To extend tropical polynomial division to real coefficients for neural network simplification.
    • To develop novel exact and approximate algorithms for tropical polynomial division.
    • To explore applications in machine learning and control systems.

    Main Methods:

    • Analysis of tropical polynomials with real coefficients.
    • Characterization of the quotient using convex bi-conjugates.
    • Relationship established between tropical polynomial division and convex hull computations.
    • Development of exact and approximate algorithms, including data partitioning and linear programming.
    • Special techniques for composite polynomial division.

    Main Results:

    • Existence and uniqueness of quotient-remainder pairs for tropical polynomials with real coefficients proven.
    • The quotient of tropical polynomials with integer coefficients may not have integer coefficients.
    • An exact algorithm derived from convex hull computations.
    • An approximate algorithm based on data partitioning and linear programming developed.
    • Numerical results demonstrate algorithm efficiency on benchmark datasets (MNIST, SVHN, CIFAR).

    Conclusions:

    • The proposed tropical polynomial division methods offer efficient tools for neural network simplification.
    • The algorithms show promise for applications in machine learning and learning model predictive control (LMPC).
    • Extending tropical polynomial analysis to real coefficients opens new avenues in theoretical and applied research.