Abstract
In this paper, we study the linear separability problem for stochastic geometric objects under the well-known unipoint and multipoint uncertainty models. Let S=SR∪SB be a given set of stochastic bichromatic points, and define n=min{|SR|,|SB|} and N=max{|SR|,|SB|}. We show that the separable-probability (SP) of S can be computed in O(nNd−1) time for d≥3 and O(min{nNlogN,N2}) time for d=2, while the expected separation-margin (ESM) of S can be computed in O(nNd) time for d≥2. In addition, we give an Ω(nNd−1)witness-based lower bound for computing SP, which implies the optimality of our algorithm among all those in this category. Also, a hardness result for computing ESM is given to show the difficulty of further improving our algorithm. As an extension, we generalize the same problems from points to general geometric objects, i.e., polytopes and/or balls, and extend our algorithms to solve the generalized SP and ESM problems in O(nNd) and O(nNd+1) time, respectively. Finally, we present some applications of our algorithms to stochastic convex hull-related problems.
Highlights
Linear separability describes the property that a set of d-dimensional bichromatic points can be separated by a hyperplane such that all the red points lie on one side of the hyperplane and all the blue points lie on the other side
Our focus is to compute the separable-probability (SP) and the expected separation-margin (ESM) for a given set of bichromatic stochastic points in Rd for d ≥ 2, where the former is the probability that the existent points are linearly separable, and the latter is the expectation of the separation-margin of the existent points. (See Section 3.1 for a detailed and formal definition of the latter.)
Problem when given a set of bichromatic stochastic points in Rd for d ≥ 3. (The runtime is O(min{N 2, nN log N }) for d = 2.) We show an Ω(nN d−1) lower bound for all witness-based algorithms, which implies the optimality of our algorithm among all witness-based methods for d ≥ 3. (See Section 2.) 2
Summary
Linear separability describes the property that a set of d-dimensional bichromatic (red and blue) points can be separated by a hyperplane such that all the red points lie on one side of the hyperplane and all the blue points lie on the other side This problem has been well studied for years in computational geometry, and is widely used in machine learning and data mining for data classification. We study the linear separability problem under two different uncertainty models, i.e., the unipoint and multipoint models [4] In the former, each stochastic data point has a fixed and known location, and has a positive probability to exist at that location; whereas in the latter, each stochastic data point occurs in one of discretely-many possible locations with known probabilities, and the existence probabilities of each point sum up to at most 1 to allow for its absence. Most proofs and some details are omitted here, but can be found in the full version [18]
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