TROGEMAL: Tropical Geometry and Machine Learning
The TROGEMAL (Tropical Geometry and Machine Learning) project, led by Professor Petros Maragos at the National Technical University of Athens, aims to advance the theoretical analysis and development of machine learning systems and algorithms using tools from tropical geometry and max-plus algebra. This three-year research project explores innovative approaches in key areas such as neural networks, graphical models, and nonlinear regression.
The research focuses on four main objectives:
- Developing New Tropical Regression Techniques: Creating methods for data fitting using tropical polynomials and piecewise-linear (PWL) functions, with applications in multivariate nonlinear regression.
- Analyzing and Simplifying Neural Networks: Leveraging tropical geometry to understand and reduce the complexity of neural networks, focusing on networks with piecewise-linear activations.
- Applying Tropical Methods to Graphical Models and Inference Algorithms: Enhancing algorithms like the Viterbi algorithm and probabilistic graphical models by viewing them through the lens of tropical geometry.
- Extending Tropical Geometry to More General Algebraic Structures: Developing a generalized max-* algebra to create new frameworks for machine learning problems over weighted lattices.
By leveraging tropical geometry, the project aims to provide new insights, develop optimization algorithms, and generalize existing approaches in machine learning. The outcomes are expected to have a significant impact on both the theoretical foundations and practical applications of machine learning and artificial intelligence.
- 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
– Prof. Petros Maragos (PI, Director of IRAL and Head of CVSP Group at NTUA)
– Dr. George Retsinas (Post-Doctoral Researcher at the IRAL & CVSP Group at NTUA )
– Dr. George Retsinas (Post-Doctoral Researcher at the IRAL & CVSP Group at NTUA )
– Dr. Ioannis Kordonis (Post-Doctoral Researcher at the IRAL & CVSP Group at NTUA)
– Vasileios Charisopoulos (PhD student in the Dept. of Operations Research & Information Engineering at Cornell University)
– 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)
– 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)
– Nikolaos Tsilivis (PhD student at New York University, Center for Data Science )
– Despina Kassianidi (Technician at the IRAL & CVSP Group at NTUA )
Publications
– I.Kordonis and P.Maragos, “Revisiting Tropical Polynomial Division: Theory, Algorithms and Application to Neural Networks“, arXiv:2306.15157, June 2023.
– G. Retsinas, G. Sfikas, P.P. Filntisis, and P. Maragos, “Newton-Based Trainable Learning Rate”, Proc. 48th IEEE Int’l Conf. on Acoustics, Speech and Signal Processing (ICASSP 2023), Rhodes, Greece, June 2023.
– A. Glentis-Georgoulakis, G. Retsinas and P. Maragos, “Feather: An Elegant Solution to Effective DNN Sparsification”, Proc. 34th British Machine Vision Conference (BMVC), Aberdeen, UK, Nov. 2023.
– 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.
– 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.
Talks
P. Maragos, “Tropical Geometry for Machine Learning”, Plenary Talk, CaLISTA Workshop on Geometry-Informed Machine Learning, Paris, 02 Sep. 2024. [Slides]
Project Details
The research project is supported by the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the “2nd Call for H.F.R.I. Research Projects to support Faculty Members & Researchers” (Project Number:2656).