Tropical geometry is a relatively recent field in mathematics and computer science that combines elements of algebraic geometry and polyhedral geometry. The scalar arithmetic of its analytic part pre-existed in the form of max-plus (max, +) and its dual min-plus (min, +) semiring arithmetic used in finite automata, convex analysis, nonlinear image processing, nonlinear control, and idempotent mathematics. Thus, tropical geometry can be viewed as algebraic geometry over the tropical max-plus semiring ( { }, max, ) or its dual, and we can find tropical analogues of classical mathematics such as tropical lines, planes, hypersurfaces, and tropical varieties, represented by (max-plus or min-plus) tropical polynomials. See next Fig. for examples.

By leveraging the new ideas and initial advances we have developed in our recent work, the project TROGEMAL proposes further advancements and explorations of several new research directions in the broad intersection of tropical geometry and machine learning. Specifically, the grand vision of TROGEMAL is to greatly advance the theoretical analysis of machine learning systems and algorithms by using and improving tools from tropical geometry and max-plus algebra, as well as discover new algorithms in key areas, including (classic and deep) neural nets, graphical models and nonlinear regression, and extend all the above by advancing a generalized max-* algebra coupled with learning through novel systems evolving over nonlinear vector spaces. In short, we propose to add intuition and understanding via tropical geometry, develop optimization algorithms using max-plus algebra and lattice operators, and generalize the above via a max-* algebra on weighted lattices.