30,287 research outputs found
Convergence of multivariate belief propagation, with applications to cuckoo hashing and load balancing.
International audienceThis paper is motivated by two applications, namely i) generalizations of cuckoo hashing, a computationally simple approach to assigning keys to objects, and ii) load balancing in content distribution networks, where one is interested in determining the impact of content replication on performance. These two problems admit a common abstraction: in both scenarios, performance is characterized by the maximum weight of a generalization of a matching in a bipartite graph, featuring node and edge capacities. Our main result is a law of large numbers characterizing the asymptotic maximum weight matching in the limit of large bipartite random graphs, when the graphs admit a local weak limit that is a tree. This result specializes to the two application scenarios, yielding new results in both contexts. In contrast with previous results, the key novelty is the ability to handle edge capacities with arbitrary integer values. An analysis of belief propagation algorithms (BP) with multivariate belief vectors underlies the proof. In particular, we show convergence of the corresponding BP by exploiting monotonicity of the belief vectors with respect to the so-called upshifted likelihood ratio stochastic order. This auxiliary result can be of independent interest, providing a new set of structural conditions which ensure convergence of BP
Weight Space Structure and Internal Representations: a Direct Approach to Learning and Generalization in Multilayer Neural Network
We analytically derive the geometrical structure of the weight space in
multilayer neural networks (MLN), in terms of the volumes of couplings
associated to the internal representations of the training set. Focusing on the
parity and committee machines, we deduce their learning and generalization
capabilities both reinterpreting some known properties and finding new exact
results. The relationship between our approach and information theory as well
as the Mitchison--Durbin calculation is established. Our results are exact in
the limit of a large number of hidden units, showing that MLN are a class of
exactly solvable models with a simple interpretation of replica symmetry
breaking.Comment: 12 pages, 1 compressed ps figure (uufile), RevTeX fil
Improving Christofides' Algorithm for the s-t Path TSP
We present a deterministic (1+sqrt(5))/2-approximation algorithm for the s-t
path TSP for an arbitrary metric. Given a symmetric metric cost on n vertices
including two prespecified endpoints, the problem is to find a shortest
Hamiltonian path between the two endpoints; Hoogeveen showed that the natural
variant of Christofides' algorithm is a 5/3-approximation algorithm for this
problem, and this asymptotically tight bound in fact has been the best
approximation ratio known until now. We modify this algorithm so that it
chooses the initial spanning tree based on an optimal solution to the Held-Karp
relaxation rather than a minimum spanning tree; we prove this simple but
crucial modification leads to an improved approximation ratio, surpassing the
20-year-old barrier set by the natural Christofides' algorithm variant. Our
algorithm also proves an upper bound of (1+sqrt(5))/2 on the integrality gap of
the path-variant Held-Karp relaxation. The techniques devised in this paper can
be applied to other optimization problems as well: these applications include
improved approximation algorithms and improved LP integrality gap upper bounds
for the prize-collecting s-t path problem and the unit-weight graphical metric
s-t path TSP.Comment: 31 pages, 5 figure
Decomposition approaches to integration without a measure
Extending the idea of Even and Lehrer [3], we discuss a general approach to
integration based on a given decomposition system equipped with a weighting
function, and a decomposition of the integrated function. We distinguish two
type of decompositions: sub-decomposition based integrals (in economics linked
with optimization problems to maximize the possible profit) and
super-decomposition based integrals (linked with costs minimization). We
provide several examples (both theoretical and realistic) to stress that our
approach generalizes that of Even and Lehrer [3] and also covers problems of
linear programming and combinatorial optimization. Finally, we introduce some
new types of integrals related to optimization tasks.Comment: 15 page
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