Abstract
The convex hull of a set of points, $C$, serves to expose extremal properties of $C$ and can help identify elements in $C$ of high interest. For many problems, particularly in the presence of noise, the true vertex set (and facets) may be difficult to determine. One solution is to expand the list of high interest candidates to points lying near the boundary of the convex hull. We propose a quadratic program for the purpose of stratifying points in a data cloud based on proximity to the boundary of the convex hull. For each data point, a quadratic program is solved to determine an associated weight vector. We show that the weight vector encodes geometric information concerning the point's relationship to the boundary of the convex hull. The computation of the weight vectors can be carried out in parallel, and for a fixed number of points and fixed neighborhood size, the overall computational complexity of the algorithm grows linearly with dimension. As a consequence, meaningful computations can be complete...
Highlights
The convex hull of a set of points in Euclidean space is defined as the smallest convex set containing the points
We propose a convex optimization problem and algorithm for coarsely ranking and stratifying data based on proximity to the boundary of the convex hull which we call the convex hull stratification algorithm (CHSA)
We have proposed an optimization problem that can be used to identify vertices of the convex hull of a data set in high dimensions and to stratify points in the data set based on proximity to the convex hull, which we call the convex hull stratification algorithm (CHSA)
Summary
The convex hull of a set of points in Euclidean space is defined as the smallest convex set containing the points. Convex hull computations have applications in a diverse range of areas, including number theory, combinatorics, algebraic geometry, pattern recognition, endmember detection, data visualization, path planning, and geographical information systems [3, 8, 13, 14, 16]. Vertices and boundary points of the convex hull may carry crucial information, or clues, as to the intrinsic nature of the data. These points may capture pure or unmixed distinguishing features, or they may be potential optima for linear. We propose a convex optimization problem and algorithm for coarsely ranking and stratifying data based on proximity to the boundary of the convex hull which we call the convex hull stratification algorithm (CHSA)
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