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Matroid Prophet Inequalities
, 2012
"... Consider a gambler who observes a sequence of independent, nonnegative random numbers and is allowed to stop the sequence at any time, claiming a reward equal to the most recent observation. The famous prophet inequality of Krengel, Sucheston, and Garling asserts that a gambler who knows the distri ..."
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Cited by 19 (4 self)
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selection, subject to a matroid constraint. We show that the gambler can still achieve at least half as much reward as the prophet; this result is the best possible, since it is known that the ratio cannot be improved even in the original prophet inequality, which corresponds to the special case of rank
Capacity of Fading Channels with Channel Side Information
, 1997
"... We obtain the Shannon capacity of a fading channel with channel side information at the transmitter and receiver, and at the receiver alone. The optimal power adaptation in the former case is "waterpouring" in time, analogous to waterpouring in frequency for timeinvariant frequencysele ..."
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Cited by 579 (23 self)
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We obtain the Shannon capacity of a fading channel with channel side information at the transmitter and receiver, and at the receiver alone. The optimal power adaptation in the former case is "waterpouring" in time, analogous to waterpouring in frequency for timeinvariant frequencyselective fading channels. Inverting the channel results in a large capacity penalty in severe fading.
Graphical models, exponential families, and variational inference
, 2008
"... The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building largescale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fiel ..."
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Cited by 800 (26 self)
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The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building largescale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fields, including bioinformatics, communication theory, statistical physics, combinatorial optimization, signal and image processing, information retrieval and statistical machine learning. Many problems that arise in specific instances — including the key problems of computing marginals and modes of probability distributions — are best studied in the general setting. Working with exponential family representations, and exploiting the conjugate duality between the cumulant function and the entropy for exponential families, we develop general variational representations of the problems of computing likelihoods, marginal probabilities and most probable configurations. We describe how a wide varietyof algorithms — among them sumproduct, cluster variational methods, expectationpropagation, mean field methods, maxproduct and linear programming relaxation, as well as conic programming relaxations — can all be understood in terms of exact or approximate forms of these variational representations. The variational approach provides a complementary alternative to Markov chain Monte Carlo as a general source of approximation methods for inference in largescale statistical models.
Planning Algorithms
, 2004
"... This book presents a unified treatment of many different kinds of planning algorithms. The subject lies at the crossroads between robotics, control theory, artificial intelligence, algorithms, and computer graphics. The particular subjects covered include motion planning, discrete planning, planning ..."
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Cited by 1108 (51 self)
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This book presents a unified treatment of many different kinds of planning algorithms. The subject lies at the crossroads between robotics, control theory, artificial intelligence, algorithms, and computer graphics. The particular subjects covered include motion planning, discrete planning, planning under uncertainty, sensorbased planning, visibility, decisiontheoretic planning, game theory, information spaces, reinforcement learning, nonlinear systems, trajectory planning, nonholonomic planning, and kinodynamic planning.
From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images
, 2007
"... A fullrank matrix A ∈ IR n×m with n < m generates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries: can it ever be unique? If so, when? As optimization of sparsity is combin ..."
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Cited by 423 (37 self)
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A fullrank matrix A ∈ IR n×m with n < m generates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries: can it ever be unique? If so, when? As optimization of sparsity is combinatorial in nature, are there efficient methods for finding the sparsest solution? These questions have been answered positively and constructively in recent years, exposing a wide variety of surprising phenomena; in particular, the existence of easilyverifiable conditions under which optimallysparse solutions can be found by concrete, effective computational methods. Such theoretical results inspire a bold perspective on some important practical problems in signal and image processing. Several wellknown signal and image processing problems can be cast as demanding solutions of undetermined systems of equations. Such problems have previously seemed, to many, intractable. There is considerable evidence that these problems often have sparse solutions. Hence, advances in finding sparse solutions to underdetermined systems energizes research on such signal and image processing problems – to striking effect. In this paper we review the theoretical results on sparse solutions of linear systems, empirical
Prophet inequalities with limited information
 IN: SODA
, 2014
"... In the classical prophet inequality, a gambler observes a sequence of stochastic rewards V1,..., Vn and must decide, for each reward Vi, whether to keep it and stop the game or to forfeit the reward forever and reveal the next value Vi. The gambler’s goal is to obtain a constant fraction of the expe ..."
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Cited by 5 (1 self)
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of the expected reward that the optimal offline algorithm would get. Recently, prophet inequalities have been generalized to settings where the gambler can choose k items, and, more generally, where he can choose any independent set in a matroid. However, all the existing algorithms require the gambler to know
FUTURE PATHS FOR INTEGER PROGRAMMING AND LINKS TO Artificial Intelligence
, 1986
"... Scope and PurposeA summary is provided of some of the recent (and a few notsorecent) developments that otTer promise for enhancing our ability to solve combinatorial optimization problems. These developments may be usefully viewed as a synthesis of the perspectives of operations research and arti ..."
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Cited by 356 (8 self)
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Scope and PurposeA summary is provided of some of the recent (and a few notsorecent) developments that otTer promise for enhancing our ability to solve combinatorial optimization problems. These developments may be usefully viewed as a synthesis of the perspectives of operations research and artificial intelligence. Although compatible with the use of algorithmic subroutines, the frameworks examined are primarily heuristic, based on the supposition that etTective solution of complex combinatorial structures in some cases may require a level of flexibility beyond that attainable by methods with formally demonstrable convergence properties. AbstractInteger programming has benefited from many innovations in models and methods. Some of the promising directions for elaborating these innovations in the future may be viewed from a framework that links the perspectives of artificial intelligence and operations research. To demonstrate this, four key areas are examined: (1) controlled randomization, (2) learning strategies, (3) induced decomposition and (4) tabu search. Each of these is shown to have characteristics that appear usefully relevant to developments on the horizon.
Results 1  10
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