6,879 research outputs found
Belief propagation : an asymptotically optimal algorithm for the random assignment problem
The random assignment problem asks for the minimum-cost perfect matching in
the complete bipartite graph \Knn with i.i.d. edge weights, say
uniform on . In a remarkable work by Aldous (2001), the optimal cost was
shown to converge to as , as conjectured by M\'ezard and
Parisi (1987) through the so-called cavity method. The latter also suggested a
non-rigorous decentralized strategy for finding the optimum, which turned out
to be an instance of the Belief Propagation (BP) heuristic discussed by Pearl
(1987). In this paper we use the objective method to analyze the performance of
BP as the size of the underlying graph becomes large. Specifically, we
establish that the dynamic of BP on \Knn converges in distribution as
to an appropriately defined dynamic on the Poisson Weighted
Infinite Tree, and we then prove correlation decay for this limiting dynamic.
As a consequence, we obtain that BP finds an asymptotically correct assignment
in time only. This contrasts with both the worst-case upper bound for
convergence of BP derived by Bayati, Shah and Sharma (2005) and the best-known
computational cost of achieved by Edmonds and Karp's algorithm
(1972).Comment: Mathematics of Operations Research (2009
A survey of max-type recursive distributional equations
In certain problems in a variety of applied probability settings (from
probabilistic analysis of algorithms to statistical physics), the central
requirement is to solve a recursive distributional equation of the form X =^d
g((\xi_i,X_i),i\geq 1). Here (\xi_i) and g(\cdot) are given and the X_i are
independent copies of the unknown distribution X. We survey this area,
emphasizing examples where the function g(\cdot) is essentially a ``maximum''
or ``minimum'' function. We draw attention to the theoretical question of
endogeny: in the associated recursive tree process X_i, are the X_i measurable
functions of the innovations process (\xi_i)?Comment: Published at http://dx.doi.org/10.1214/105051605000000142 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
An Optimal Medium Access Control with Partial Observations for Sensor Networks
We consider medium access control (MAC) in multihop sensor networks, where only partial information about the shared medium is available to the transmitter. We model our setting as a queuing problem in which the service rate of a queue is a function of a partially observed Markov chain representing the available bandwidth, and in which the arrivals are controlled based on the partial observations so as to keep the system in a desirable mildly unstable regime. The optimal controller for this problem satisfies a separation property: we first compute a probability measure on the state space of the chain, namely the information state, then use this measure as the new state on which the control decisions are based. We give a formal description of the system considered and of its dynamics, we formalize and solve an optimal control problem, and we show numerical simulations to illustrate with concrete examples properties of the optimal control law. We show how the ergodic behavior of our queuing model is characterized by an invariant measure over all possible information states, and we construct that measure. Our results can be specifically applied for designing efficient and stable algorithms for medium access control in multiple-accessed systems, in particular for sensor networks
Fast Estimation of the Median Covariation Matrix with Application to Online Robust Principal Components Analysis
The geometric median covariation matrix is a robust multivariate indicator of
dispersion which can be extended without any difficulty to functional data. We
define estimators, based on recursive algorithms, that can be simply updated at
each new observation and are able to deal rapidly with large samples of high
dimensional data without being obliged to store all the data in memory.
Asymptotic convergence properties of the recursive algorithms are studied under
weak conditions. The computation of the principal components can also be
performed online and this approach can be useful for online outlier detection.
A simulation study clearly shows that this robust indicator is a competitive
alternative to minimum covariance determinant when the dimension of the data is
small and robust principal components analysis based on projection pursuit and
spherical projections for high dimension data. An illustration on a large
sample and high dimensional dataset consisting of individual TV audiences
measured at a minute scale over a period of 24 hours confirms the interest of
considering the robust principal components analysis based on the median
covariation matrix. All studied algorithms are available in the R package
Gmedian on CRAN
An adaptive fixed-mesh ALE method for free surface flows
In this work we present a Fixed-Mesh ALE method for the numerical simulation of free surface flows capable of using an adaptive finite element mesh covering a background domain. This mesh is successively refined and unrefined at each time step in order to focus the computational effort on the spatial regions where it is required. Some of the main ingredients of the formulation are the use of an Arbitrary-LagrangianâEulerian formulation for computing temporal derivatives, the use of stabilization terms for stabilizing convection, stabilizing the lack of compatibility between velocity and pressure interpolation spaces, and stabilizing the ill-conditioning introduced by the cuts on the background finite element mesh, and the coupling of the algorithm with an adaptive mesh refinement procedure suitable for running on distributed memory environments. Algorithmic steps for the projection between meshes are presented together with the algebraic fractional step approach used for improving the condition number of the linear systems to be solved. The method is tested in several numerical examples. The expected convergence rates both in space and time are observed. Smooth solution fields for both velocity and pressure are obtained (as a result of the contribution of the stabilization terms). Finally, a good agreement between the numerical results and the reference experimental data is obtained.Postprint (published version
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