5,061 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
An Effective Feature Selection Method Based on Pair-Wise Feature Proximity for High Dimensional Low Sample Size Data
Feature selection has been studied widely in the literature. However, the
efficacy of the selection criteria for low sample size applications is
neglected in most cases. Most of the existing feature selection criteria are
based on the sample similarity. However, the distance measures become
insignificant for high dimensional low sample size (HDLSS) data. Moreover, the
variance of a feature with a few samples is pointless unless it represents the
data distribution efficiently. Instead of looking at the samples in groups, we
evaluate their efficiency based on pairwise fashion. In our investigation, we
noticed that considering a pair of samples at a time and selecting the features
that bring them closer or put them far away is a better choice for feature
selection. Experimental results on benchmark data sets demonstrate the
effectiveness of the proposed method with low sample size, which outperforms
many other state-of-the-art feature selection methods.Comment: European Signal Processing Conference 201
Modulation-mode assignment for SVD-assisted and iteratively detected downlink multiuser MIMO transmission schemes
In this contribution we jointly optimize the number of multiple-input multiple-output (MIMO) layers and the number of bits per symbol within an iteratively-detected multiuser MIMO downlink (DL) transmission scheme under the constraint of a given fixed data throughput and integrity. Instead of treating all the users jointly as in zero-forcing (ZF) multiuser transmission techniques, the investigated singular value decomposition (SVD) assisted DL multiuser MIMO system takes the individual user's channel characteristics into account. In analogy to bit-interleaved coded irregular modulation, we introduce a MIMO-BICM scheme, where different user-specific signal constellations and mapping arrangement were used within a single codeword. Extrinsic information transfer (EXIT) charts are used for analyzing and optimizing the convergence behaviour of the iterative demapping and decoding. Our results show that in order to achieve the best bit-error rate, not necessarily all user-specific MIMO layers have to be activate
Belief-Propagation for Weighted b-Matchings on Arbitrary Graphs and its Relation to Linear Programs with Integer Solutions
We consider the general problem of finding the minimum weight \bm-matching
on arbitrary graphs. We prove that, whenever the linear programming (LP)
relaxation of the problem has no fractional solutions, then the belief
propagation (BP) algorithm converges to the correct solution. We also show that
when the LP relaxation has a fractional solution then the BP algorithm can be
used to solve the LP relaxation. Our proof is based on the notion of graph
covers and extends the analysis of (Bayati-Shah-Sharma 2005 and Huang-Jebara
2007}.
These results are notable in the following regards: (1) It is one of a very
small number of proofs showing correctness of BP without any constraint on the
graph structure. (2) Variants of the proof work for both synchronous and
asynchronous BP; it is the first proof of convergence and correctness of an
asynchronous BP algorithm for a combinatorial optimization problem.Comment: 28 pages, 2 figures. Submitted to SIAM journal on Discrete
Mathematics on March 19, 2009; accepted for publication (in revised form)
August 30, 2010; published electronically July 1, 201
Approximation Algorithms for Stochastic Boolean Function Evaluation and Stochastic Submodular Set Cover
Stochastic Boolean Function Evaluation is the problem of determining the
value of a given Boolean function f on an unknown input x, when each bit of x_i
of x can only be determined by paying an associated cost c_i. The assumption is
that x is drawn from a given product distribution, and the goal is to minimize
the expected cost. This problem has been studied in Operations Research, where
it is known as "sequential testing" of Boolean functions. It has also been
studied in learning theory in the context of learning with attribute costs. We
consider the general problem of developing approximation algorithms for
Stochastic Boolean Function Evaluation. We give a 3-approximation algorithm for
evaluating Boolean linear threshold formulas. We also present an approximation
algorithm for evaluating CDNF formulas (and decision trees) achieving a factor
of O(log kd), where k is the number of terms in the DNF formula, and d is the
number of clauses in the CNF formula. In addition, we present approximation
algorithms for simultaneous evaluation of linear threshold functions, and for
ranking of linear functions.
Our function evaluation algorithms are based on reductions to the Stochastic
Submodular Set Cover (SSSC) problem. This problem was introduced by Golovin and
Krause. They presented an approximation algorithm for the problem, called
Adaptive Greedy. Our main technical contribution is a new approximation
algorithm for the SSSC problem, which we call Adaptive Dual Greedy. It is an
extension of the Dual Greedy algorithm for Submodular Set Cover due to Fujito,
which is a generalization of Hochbaum's algorithm for the classical Set Cover
Problem. We also give a new bound on the approximation achieved by the Adaptive
Greedy algorithm of Golovin and Krause
On the Usefulness of Predicates
Motivated by the pervasiveness of strong inapproximability results for
Max-CSPs, we introduce a relaxed notion of an approximate solution of a
Max-CSP. In this relaxed version, loosely speaking, the algorithm is allowed to
replace the constraints of an instance by some other (possibly real-valued)
constraints, and then only needs to satisfy as many of the new constraints as
possible.
To be more precise, we introduce the following notion of a predicate
being \emph{useful} for a (real-valued) objective : given an almost
satisfiable Max- instance, there is an algorithm that beats a random
assignment on the corresponding Max- instance applied to the same sets of
literals. The standard notion of a nontrivial approximation algorithm for a
Max-CSP with predicate is exactly the same as saying that is useful for
itself.
We say that is useless if it is not useful for any . This turns out to
be equivalent to the following pseudo-randomness property: given an almost
satisfiable instance of Max- it is hard to find an assignment such that the
induced distribution on -bit strings defined by the instance is not
essentially uniform.
Under the Unique Games Conjecture, we give a complete and simple
characterization of useful Max-CSPs defined by a predicate: such a Max-CSP is
useless if and only if there is a pairwise independent distribution supported
on the satisfying assignments of the predicate. It is natural to also consider
the case when no negations are allowed in the CSP instance, and we derive a
similar complete characterization (under the UGC) there as well.
Finally, we also include some results and examples shedding additional light
on the approximability of certain Max-CSPs
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