Tropical Geometry and Machine Learning

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