Title: Revisiting Tropical Polynomial Division: Theory, Algorithms and Application to Neural Networks Show Chinese title

Title: 重新审视热带多项式除法：理论、算法及其在神经网络中的应用

Abstract: Tropical geometry has recently found several applications in the analysis of neural networks with piecewise linear activation functions. This paper presents a new look at the problem of tropical polynomial division and its application to the simplification of neural networks. We analyze tropical polynomials with real coefficients, extending earlier ideas and methods developed for polynomials with integer coefficients. We first prove the existence of a unique quotient-remainder pair and characterize the quotient in terms of the convex bi-conjugate of a related function. Interestingly, the quotient of tropical polynomials with integer coefficients does not necessarily have integer coefficients. Furthermore, we develop a relationship of tropical polynomial division with the computation of the convex hull of unions of convex polyhedra and use it to derive an exact algorithm for tropical polynomial division. An approximate algorithm is also presented, based on an alternation between data partition and linear programming. We also develop special techniques to divide composite polynomials, described as sums or maxima of simpler ones. Finally, we present some numerical results to illustrate the efficiency of the algorithms proposed, using the MNIST handwritten digit and CIFAR-10 datasets. Show Chinese abstract

Abstract: 热带几何最近在分析具有分段线性激活函数的神经网络中找到了几个应用。 本文从一个新的角度审视了热带多项式除法的问题及其在神经网络简化中的应用。 我们分析了具有实系数的热带多项式，扩展了之前为整数系数多项式开发的想法和方法。 我们首先证明了商-余数对的存在性，并通过相关函数的凸双共轭来描述商。 有趣的是，具有整数系数的热带多项式的商不一定具有整数系数。 此外，我们建立了热带多项式除法与凸包计算之间的关系，以及凸多面体并集的凸包计算，并利用它推导出一种精确的热带多项式除法算法。 还提出了一种近似算法，该算法基于数据分区和线性规划的交替进行。 我们还开发了专门的技术来划分复合多项式，这些多项式被描述为更简单多项式的和或最大值。 最后，我们展示了一些数值结果，以说明所提出的算法的效率，使用了MNIST手写数字和CIFAR-10数据集。