33,014 research outputs found
R\'enyi generalizations of quantum information measures
Quantum information measures such as the entropy and the mutual information
find applications in physics, e.g., as correlation measures. Generalizing such
measures based on the R\'enyi entropies is expected to enhance their scope in
applications. We prescribe R\'enyi generalizations for any quantum information
measure which consists of a linear combination of von Neumann entropies with
coefficients chosen from the set {-1,0,1}. As examples, we describe R\'enyi
generalizations of the conditional quantum mutual information, some quantum
multipartite information measures, and the topological entanglement entropy.
Among these, we discuss the various properties of the R\'enyi conditional
quantum mutual information and sketch some potential applications. We
conjecture that the proposed R\'enyi conditional quantum mutual informations
are monotone increasing in the R\'enyi parameter, and we have proofs of this
conjecture for some special cases.Comment: 9 pages, related to and extends the results from arXiv:1403.610
Small Extended Formulation for Knapsack Cover Inequalities from Monotone Circuits
Initially developed for the min-knapsack problem, the knapsack cover
inequalities are used in the current best relaxations for numerous
combinatorial optimization problems of covering type. In spite of their
widespread use, these inequalities yield linear programming (LP) relaxations of
exponential size, over which it is not known how to optimize exactly in
polynomial time. In this paper we address this issue and obtain LP relaxations
of quasi-polynomial size that are at least as strong as that given by the
knapsack cover inequalities.
For the min-knapsack cover problem, our main result can be stated formally as
follows: for any , there is a -size LP relaxation with an integrality gap of at most ,
where is the number of items. Prior to this work, there was no known
relaxation of subexponential size with a constant upper bound on the
integrality gap.
Our construction is inspired by a connection between extended formulations
and monotone circuit complexity via Karchmer-Wigderson games. In particular,
our LP is based on -depth monotone circuits with fan-in~ for
evaluating weighted threshold functions with inputs, as constructed by
Beimel and Weinreb. We believe that a further understanding of this connection
may lead to more positive results complementing the numerous lower bounds
recently proved for extended formulations.Comment: 21 page
Message-Passing Algorithms for Quadratic Minimization
Gaussian belief propagation (GaBP) is an iterative algorithm for computing
the mean of a multivariate Gaussian distribution, or equivalently, the minimum
of a multivariate positive definite quadratic function. Sufficient conditions,
such as walk-summability, that guarantee the convergence and correctness of
GaBP are known, but GaBP may fail to converge to the correct solution given an
arbitrary positive definite quadratic function. As was observed in previous
work, the GaBP algorithm fails to converge if the computation trees produced by
the algorithm are not positive definite. In this work, we will show that the
failure modes of the GaBP algorithm can be understood via graph covers, and we
prove that a parameterized generalization of the min-sum algorithm can be used
to ensure that the computation trees remain positive definite whenever the
input matrix is positive definite. We demonstrate that the resulting algorithm
is closely related to other iterative schemes for quadratic minimization such
as the Gauss-Seidel and Jacobi algorithms. Finally, we observe, empirically,
that there always exists a choice of parameters such that the above
generalization of the GaBP algorithm converges
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