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

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.