Tropical geometry is a relatively recent field in mathematics and computer science, combining elements of algebraic geometry and polyhedral geometry. The scalar arithmetic of its analytic part preexisted in the form of max-plus and min-plus semiring arithmetic used in finite automata, nonlinear image processing, convex analysis, nonlinear control, optimization, and idempotent mathematics. Tropical geometry recently emerged in the analysis and extension of several classes of problems and systems in both classical machine learning and deep learning. Three such areas include: 1) deep neural networks with piecewise linear (PWL) activation functions; 2) probabilistic graphical models; and 3) nonlinear regression with PWL functions. In this article, we first summarize introductory ideas and objects of tropical geometry, providing a theoretical framework for both the max-plus algebra that underlies tropical geometry and its extensions to general max algebras. This unifies scalar and vector/signal operations over a class of nonlinear spaces, called weighted lattices, and allows us to provide optimal solutions for algebraic equations used in tropical geometry and generalize tropical geometric objects. Then, we survey the state of the art and recent progress in the aforementioned areas. First, we illustrate a purely geometric approach for studying the representation power of neural networks with PWL activations. Then, we review the tropical geometric analysis of parametric statistical models, such as HMMs; later, we focus on the Viterbi algorithm and related methods for weighted finite-state transducers and provide compact and elegant representations via their formal tropical modeling. Finally, we provide optimal solutions and an efficient algorithm for the convex regression problem, using concepts and tools from tropical geometry and max-plus algebra. Throughout this article, we also outline problems and future directions in machine learning that can benefit from the tropical-geometric point of view.

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热带几何与机器学习

热带几何是数学和计算机科学中相对较新的领域，结合了代数几何和多面体几何元素。其解析部分的标量算法以存在于有限自动机，非线性图像处理，凸分析，非线性控制，优化和幂等数学中的max-plus和min-plus半环算法的形式预先存在。热带几何最近出现在分析和扩展经典机器学习和深度学习中的几类问题和系统中。三个方面包括：1）具有分段线性（PWL）激活功能的深度神经网络；2）概率图形模型；和3）具有PWL函数的非线性回归。在本文中，我们首先总结了热带几何的入门思想和对象，为构成热带几何基础的max-plus代数及其对一般max代数的扩展提供了理论框架。这将一类非线性空间（称为加权格）上的标量和矢量/信号运算统一起来，并使我们能够为热带几何中使用的代数方程式提供最佳解，并推广热带几何对象。然后，我们调查了上述领域的最新技术和最新进展。首先，我们说明了一种纯几何方法，用于研究具有PWL激活的神经网络的表示能力。然后，我们回顾了参数统计模型（例如HMM）的热带几何分析；之后，我们专注于加权有限状态传感器的Viterbi算法和相关方法，并通过其正式的热带建模提供紧凑而优雅的表示形式。最后，我们使用热带几何和max-plus代数的概念和工具，为凸回归问题提供了最佳解决方案和有效算法。在整篇文章中，我们还概述了可以从热带几何角度受益的机器学习中的问题和未来方向。