1,523 research outputs found
Generalized sequential tree-reweighted message passing
This paper addresses the problem of approximate MAP-MRF inference in general
graphical models. Following [36], we consider a family of linear programming
relaxations of the problem where each relaxation is specified by a set of
nested pairs of factors for which the marginalization constraint needs to be
enforced. We develop a generalization of the TRW-S algorithm [9] for this
problem, where we use a decomposition into junction chains, monotonic w.r.t.
some ordering on the nodes. This generalizes the monotonic chains in [9] in a
natural way. We also show how to deal with nested factors in an efficient way.
Experiments show an improvement over min-sum diffusion, MPLP and subgradient
ascent algorithms on a number of computer vision and natural language
processing problems
Parallelisation and application of AD3 as a method for solving large scale combinatorial auctions
Auctions, and combinatorial auctions (CAs), have been successfully employed to solve coordination problems in a wide range of application domains. However, the scale of CAs that can be optimally solved is small because of the complexity of the winner determination problem (WDP), namely of finding the bids that maximise the auctioneer’s revenue. A way of approximating the solution of a WDP is to solve its linear programming relaxation. The recently proposed Alternate Direction Dual Decomposition algorithm (AD3) has been shown to ef- ficiently solve large-scale LP relaxations. Hence, in this paper we show how to encode the WDP so that it can be approximated by means of AD3. Moreover, we present PAR-AD3, the first parallel implementation of AD3. PAR-AD3 shows to be up to 12.4 times faster than CPLEX in a single-thread execution, and up to 23 times faster than parallel CPLEX in an 8-core architecture. Therefore PAR- AD3 becomes the algorithm of choice to solve large-scale WDP LP relaxations for hard instances. Furthermore, PAR-AD3 has potential when considering large- scale coordination problems that must be solved as optimisation problems.Research supported by MICINN projects TIN2011-28689-C02-01, TIN2013-45732-C4-4-P and TIN2012-38876-C02-01Peer reviewe
Relax, no need to round: integrality of clustering formulations
We study exact recovery conditions for convex relaxations of point cloud
clustering problems, focusing on two of the most common optimization problems
for unsupervised clustering: -means and -median clustering. Motivations
for focusing on convex relaxations are: (a) they come with a certificate of
optimality, and (b) they are generic tools which are relatively parameter-free,
not tailored to specific assumptions over the input. More precisely, we
consider the distributional setting where there are clusters in
and data from each cluster consists of points sampled from a
symmetric distribution within a ball of unit radius. We ask: what is the
minimal separation distance between cluster centers needed for convex
relaxations to exactly recover these clusters as the optimal integral
solution? For the -median linear programming relaxation we show a tight
bound: exact recovery is obtained given arbitrarily small pairwise separation
between the balls. In other words, the pairwise center
separation is . Under the same distributional model, the
-means LP relaxation fails to recover such clusters at separation as large
as . Yet, if we enforce PSD constraints on the -means LP, we get
exact cluster recovery at center separation .
In contrast, common heuristics such as Lloyd's algorithm (a.k.a. the -means
algorithm) can fail to recover clusters in this setting; even with arbitrarily
large cluster separation, k-means++ with overseeding by any constant factor
fails with high probability at exact cluster recovery. To complement the
theoretical analysis, we provide an experimental study of the recovery
guarantees for these various methods, and discuss several open problems which
these experiments suggest.Comment: 30 pages, ITCS 201
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