TROGEMAL: Tropical Geometry and Machine Learning
About the Project
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.
- WP1: Tropical Regression
- T1.1: Algorithms for Multivariate Tropical Regression with Convex PWL Models
- T1.2: Slope transforms and Tropical Regression with General Convex Models
- T1.3: Extensions of Tropical Regression to Non-convex PWL Models
- WP2. Tropical Geometry of Neural Networks
- T2.1: Tropical Geometric Analysis of NNs with PWL Activations
- T2.2: Simplification – Minimization of Neural Networks
- WP3. Tropical Geometry of Graphical Models & Inference Algorithms
- T3.1: Tropical Modeling and Spectral Characterization of WFST Algorithms
- T3.2: Tropical Geometry of Statistical Models
- WP4 Generalized Tropical Geometry & Learning on Weighted Lattices
- T4.1: Extensions of Tropical Geometry and Regression using Max-* Algebra
- T4.2: Max-* generalization of inference algorithms
- WP5. Project Management and Dissemination
- T5.1: Management and Administration
- T5.2: Dissemination of results
– Dr. George Retsinas (Post-Doctoral Researcher at the IRAL & CVSP Group at NTUA )
– Georgios Smyrnis (PhD student in the Electrical & Computer Engineering Dept. at the University of Texas at Austin
– Emmanouil Theodosis (PhD student in Computer Science, School of Engineering & Applied Sciences atHarvard University)
– Despina Kassianidi (Technician at the IRAL & CVSP Group at NTUA )
– I. Kordonis, E. Theodosis, G. Retsinas, and P. Maragos, “Matrix Factorization in Tropical and Mixed Tropical-Linear Algebras”, Proc. 2024 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2024), Seoul, Korea, April 2024.
Presentation of achieved results at conferences and workshops
- June 2023: TROGEMAL results presented in ICASSP 2023 in Rhodes, Greece.
- November 2023: TROGEMAL results presented in BMVC 2023 in Aberdeen, UK.
- April 2024: TROGEMAL results presented in ICASSP 2024 in Seoul, South Korea.