Tropical Halfspaces
نویسنده
چکیده
As a new concept tropical halfspaces are introduced to the (linear algebraic) geometry of the tropical semiring (R,min,+). This yields exterior descriptions of the tropical polytopes that were recently studied by Develin and Sturmfels [2004] in a variety of contexts. The key tool to the understanding is a newly defined sign of the tropical determinant, which shares remarkably many properties with the ordinary sign of the determinant of a matrix. The methods are used to obtain an optimal tropical convex hull algorithm in two dimensions.
منابع مشابه
New Results in Tropical Discrete Geometry
Following the recent work of Develin and Sturmfels and others (see, e.g., [10, 16, 2, 11]), we investigate discrete geometric questions over the tropical semiring (R, min, +). Specifically, we obtain the following tropical analogues of classical theorems in convex geometry: a separation theorem for a pair of disjoint tropical polytopes by tropical halfspaces and tropical versions of Radon’s lem...
متن کاملGeneralized halfspaces in restricted-orientation convexity
We first present some basic properties of O-halfspaces and compare them with the properties of standard halfspaces. We show that O-halfspaces may be disconnected, characterize an Ohalfspace in terms of its connected components, and derive the upper bound on the number of components. We then study properties of the boundaries of O-halfspaces. Finally, we describe the complements of O-halfspaces ...
متن کاملImproved Lower Bounds for Learning Intersections of Halfspaces
We prove new lower bounds for learning intersections of halfspaces, one of the most important concept classes in computational learning theory. Our main result is that any statistical-query algorithm for learning the intersection of √ n halfspaces in n dimensions must make 2 √ n) queries. This is the first non-trivial lower bound on the statistical query dimension for this concept class (the pr...
متن کاملTesting halfspaces
This paper addresses the problem of testing whether a Boolean-valued function f is a halfspace, i.e. a function of the form f(x) = sgn(w ·x−θ).We consider halfspaces over the continuous domain R (endowed with the standard multivariate Gaussian distribution) as well as halfspaces over the Boolean cube {−1, 1} (endowed with the uniform distribution). In both cases we give an algorithm that distin...
متن کاملMinimal external representations of tropical polyhedra
Tropical polyhedra are known to be representable externally, as intersections of finitely many tropical half-spaces. However, unlike in the classical case, the extreme rays of their polar cones provide external representations containing in general superfluous half-spaces. In this paper, we prove that any tropical polyhedral cone in R (also known as “tropical polytope” in the literature) admits...
متن کامل