577 research outputs found
A path following algorithm for the graph matching problem
We propose a convex-concave programming approach for the labeled weighted
graph matching problem. The convex-concave programming formulation is obtained
by rewriting the weighted graph matching problem as a least-square problem on
the set of permutation matrices and relaxing it to two different optimization
problems: a quadratic convex and a quadratic concave optimization problem on
the set of doubly stochastic matrices. The concave relaxation has the same
global minimum as the initial graph matching problem, but the search for its
global minimum is also a hard combinatorial problem. We therefore construct an
approximation of the concave problem solution by following a solution path of a
convex-concave problem obtained by linear interpolation of the convex and
concave formulations, starting from the convex relaxation. This method allows
to easily integrate the information on graph label similarities into the
optimization problem, and therefore to perform labeled weighted graph matching.
The algorithm is compared with some of the best performing graph matching
methods on four datasets: simulated graphs, QAPLib, retina vessel images and
handwritten chinese characters. In all cases, the results are competitive with
the state-of-the-art.Comment: 23 pages, 13 figures,typo correction, new results in sections 4,5,
On recovery guarantees for angular synchronization
The angular synchronization problem of estimating a set of unknown angles
from their known noisy pairwise differences arises in various applications. It
can be reformulated as a optimization problem on graphs involving the graph
Laplacian matrix. We consider a general, weighted version of this problem,
where the impact of the noise differs between different pairs of entries and
some of the differences are erased completely; this version arises for example
in ptychography. We study two common approaches for solving this problem,
namely eigenvector relaxation and semidefinite convex relaxation. Although some
recovery guarantees are available for both methods, their performance is either
unsatisfying or restricted to the unweighted graphs. We close this gap,
deriving recovery guarantees for the weighted problem that are completely
analogous to the unweighted version.Comment: 20 pages, 5 figure
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