28,533 research outputs found
On the convex hull of a space curve
The boundary of the convex hull of a compact algebraic curve in real 3-space
defines a real algebraic surface. For general curves, that boundary surface is
reducible, consisting of tritangent planes and a scroll of stationary
bisecants. We express the degree of this surface in terms of the degree, genus
and singularities of the curve. We present algorithms for computing their
defining polynomials, and we exhibit a wide range of examples.Comment: 19 pages, 4 figures, minor change
Convex hulls of curves of genus one
Let C be a real nonsingular affine curve of genus one, embedded in affine
n-space, whose set of real points is compact. For any polynomial f which is
nonnegative on C(R), we prove that there exist polynomials f_i with f \equiv
\sum_i f_i^2 (modulo I_C) and such that the degrees deg(f_i) are bounded in
terms of deg(f) only. Using Lasserre's relaxation method, we deduce an explicit
representation of the convex hull of C(R) in R^n by a lifted linear matrix
inequality. This is the first instance in the literature where such a
representation is given for the convex hull of a nonrational variety. The same
works for convex hulls of (singular) curves whose normalization is C. We then
make a detailed study of the associated degree bounds. These bounds are
directly related to size and dimension of the projected matrix pencils. In
particular, we prove that these bounds tend to infinity when the curve C
degenerates suitably into a singular curve, and we provide explicit lower
bounds as well.Comment: 1 figur
Convex Hulls in Polygonal Domains
We study generalizations of convex hulls to polygonal domains with holes. Convexity in Euclidean space is based on the notion of shortest paths, which are straight-line segments. In a polygonal domain, shortest paths are polygonal paths called geodesics. One possible generalization of convex hulls is based on the "rubber band" conception of the convex hull boundary as a shortest curve that encloses a given set of sites. However, it is NP-hard to compute such a curve in a general polygonal domain. Hence, we focus on a different, more direct generalization of convexity, where a set X is geodesically convex if it contains all geodesics between every pair of points x,y in X. The corresponding geodesic convex hull presents a few surprises, and turns out to behave quite differently compared to the classic Euclidean setting or to the geodesic hull inside a simple polygon. We describe a class of geometric objects that suffice to represent geodesic convex hulls of sets of sites, and characterize which such domains are geodesically convex. Using such a representation we present an algorithm to construct the geodesic convex hull of a set of O(n) sites in a polygonal domain with a total of n vertices and h holes in O(n^3h^{3+epsilon}) time, for any constant epsilon > 0
Convex Hull-Based Multi-objective Genetic Programming for Maximizing ROC Performance
ROC is usually used to analyze the performance of classifiers in data mining.
ROC convex hull (ROCCH) is the least convex major-ant (LCM) of the empirical
ROC curve, and covers potential optima for the given set of classifiers.
Generally, ROC performance maximization could be considered to maximize the
ROCCH, which also means to maximize the true positive rate (tpr) and minimize
the false positive rate (fpr) for each classifier in the ROC space. However,
tpr and fpr are conflicting with each other in the ROCCH optimization process.
Though ROCCH maximization problem seems like a multi-objective optimization
problem (MOP), the special characters make it different from traditional MOP.
In this work, we will discuss the difference between them and propose convex
hull-based multi-objective genetic programming (CH-MOGP) to solve ROCCH
maximization problems. Convex hull-based sort is an indicator based selection
scheme that aims to maximize the area under convex hull, which serves as a
unary indicator for the performance of a set of points. A selection procedure
is described that can be efficiently implemented and follows similar design
principles than classical hyper-volume based optimization algorithms. It is
hypothesized that by using a tailored indicator-based selection scheme CH-MOGP
gets more efficient for ROC convex hull approximation than algorithms which
compute all Pareto optimal points. To test our hypothesis we compare the new
CH-MOGP to MOGP with classical selection schemes, including NSGA-II, MOEA/D)
and SMS-EMOA. Meanwhile, CH-MOGP is also compared with traditional machine
learning algorithms such as C4.5, Naive Bayes and Prie. Experimental results
based on 22 well-known UCI data sets show that CH-MOGP outperforms
significantly traditional EMOAs
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