Tropical Fermat-Weber Polytropes

In this talk we focus on the geometry of tropical Fermat-Weber points in terms of the symmetric tropical metric over the tropical projective torus. It is well-known that a tropical Fermat-Weber point of a given sample is not unique and, in this talk, we show that the set of all possible Fermat-Weber points forms a polytrope, which is a classical polytope and a tropical polytope. Then, we introduce the tropical Fermat-Weber gradient and, using them, we show that the tropical Fermat-Weber polytrope is a bounded cell of a tropical hyperplane arrangement given by both min- and max-tropical hyperplanes with apices which are observations in the input data. If time permits, we discuss its application to robust neural networks against adversarial attacks on image data. This is a joint with J. Sabol, D. Barnhill, and K. Miura.