In this paper, we study the linear separability problem for stochastic geometric objects under the well-known unipoint multipoint uncertainty models. Let @math be a given set of stochastic bichromatic points, and define @math and @math . We show that the separable-probability (SP) of @math can be computed in @math time for @math and @math time for @math , while the expected separation-margin (ESM) of @math can be computed in @math time for @math . In addition, we give an @math 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 @math and @math time, respectively. Finally, we present some applications of our algorithms to stochastic convex-hull related problems.
Recently, in @cite_3 , de studied the separability problem given a set of bichromatic imprecise points in @math in a setting that each point is drawn from an imprecision region.
@cite_3
1787197433 2964185312 2604480575 2952779762
An imprecise point is a point p with an associated imprecision region ( I _p ) indicating the set of possible locations of the point p. We study separability problems for a set R of red imprecise points and a set B of blue imprecise points in ( R ^2 ), where the imprecision regions are axis-aligned rectangles and each point p ∈ R ∪ B is drawn uniformly at random from ( I _p ). Our results include algorithms for finding certain separators (separating R from B with probability 1), possible separators (separating R from B with non-zero probability), most likely separators (separating R from B with maximal probability), and maximal separators (maximizing the expected number of correctly classified points). In this paper, we study the linear separability problem for stochastic geometric objects under the well-known unipoint multipoint uncertainty models. Let S=S_R U S_B be a given set of stochastic bichromatic points, and define n = min |S_R|, |S_B| and N = max |S_R|, |S_B| . We show that the separable-probability (SP) of S can be computed in O(nN^ d-1 ) time for d >= 3 and O(min nN log N, N^2 ) time for d=2, while the expected separation-margin (ESM) of S can be computed in O(nN^d) time for d >= 2. In addition, we give an Omega(nN^ d-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(nN^d) and O(nN^ d+1 ) time, respectively. Finally, we present some applications of our algorithms to stochastic convex-hull related problems. We describe an @math -time algorithm for computing the exact probability that two @math -dimensional probabilistic point sets are linearly separable, for any fixed @math . A probabilistic point in @math -space is a normal point, but with an associated probability of existence; the existence probabilities of all points are independent. We also show that the @math -dimensional separability problem is equivalent to a @math -dimensional convex hull membership problem, which asks for the probability that a query point lies inside the convex hull of @math probabilistic points. Using this reduction, we improve the current best bound for the convex hull membership problem by a factor of @math . In addition, our algorithms can handle input degeneracies in which more than @math points may lie on a @math -dimensional subspace, thus resolving an open problem in 2013. Finally, we prove lower bounds for the separability problem via a reduction from the @math -SUM problem, which show in particular that our @math algorithms for @math -dimensional separability and @math -dimensional convex hull membership are nearly optimal. The separability assumption (Donoho & Stodden, 2003; , 2012) turns non-negative matrix factorization (NMF) into a tractable problem. Recently, a new class of provably-correct NMF algorithms have emerged under this assumption. In this paper, we reformulate the separable NMF problem as that of finding the extreme rays of the conical hull of a finite set of vectors. From this geometric perspective, we derive new separable NMF algorithms that are highly scalable and empirically noise robust, and have several other favorable properties in relation to existing methods. A parallel implementation of our algorithm demonstrates high scalability on shared- and distributed-memory machines.