3,381 research outputs found

    On the Separability of Stochastic Geometric Objects, with Applications

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    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

    Largest separable balls around the maximally mixed bipartite quantum state

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    For finite-dimensional bipartite quantum systems, we find the exact size of the largest balls, in spectral lpl_p norms for 1p1 \le p \le \infty, of separable (unentangled) matrices around the identity matrix. This implies a simple and intutively meaningful geometrical sufficient condition for separability of bipartite density matrices: that their purity \tr \rho^2 not be too large. Theoretical and experimental applications of these results include algorithmic problems such as computing whether or not a state is entangled, and practical ones such as obtaining information about the existence or nature of entanglement in states reached by NMR quantum computation implementations or other experimental situations.Comment: 7 pages, LaTeX. Motivation and verbal description of results and their implications expanded and improved; one more proof included. This version differs from the PRA version by the omission of some erroneous sentences outside the theorems and proofs, which will be noted in an erratum notice in PRA (and by minor notational differences

    Fast Conical Hull Algorithms for Near-separable Non-negative Matrix Factorization

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    The separability assumption (Donoho & Stodden, 2003; Arora et al., 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.Comment: 15 pages, 6 figure

    On the expected diameter, width, and complexity of a stochastic convex-hull

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    We investigate several computational problems related to the stochastic convex hull (SCH). Given a stochastic dataset consisting of nn points in Rd\mathbb{R}^d each of which has an existence probability, a SCH refers to the convex hull of a realization of the dataset, i.e., a random sample including each point with its existence probability. We are interested in computing certain expected statistics of a SCH, including diameter, width, and combinatorial complexity. For diameter, we establish the first deterministic 1.633-approximation algorithm with a time complexity polynomial in both nn and dd. For width, two approximation algorithms are provided: a deterministic O(1)O(1)-approximation running in O(nd+1logn)O(n^{d+1} \log n) time, and a fully polynomial-time randomized approximation scheme (FPRAS). For combinatorial complexity, we propose an exact O(nd)O(n^d)-time algorithm. Our solutions exploit many geometric insights in Euclidean space, some of which might be of independent interest
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