5,159 research outputs found
Large-scale Binary Quadratic Optimization Using Semidefinite Relaxation and Applications
In computer vision, many problems such as image segmentation, pixel
labelling, and scene parsing can be formulated as binary quadratic programs
(BQPs). For submodular problems, cuts based methods can be employed to
efficiently solve large-scale problems. However, general nonsubmodular problems
are significantly more challenging to solve. Finding a solution when the
problem is of large size to be of practical interest, however, typically
requires relaxation. Two standard relaxation methods are widely used for
solving general BQPs--spectral methods and semidefinite programming (SDP), each
with their own advantages and disadvantages. Spectral relaxation is simple and
easy to implement, but its bound is loose. Semidefinite relaxation has a
tighter bound, but its computational complexity is high, especially for large
scale problems. In this work, we present a new SDP formulation for BQPs, with
two desirable properties. First, it has a similar relaxation bound to
conventional SDP formulations. Second, compared with conventional SDP methods,
the new SDP formulation leads to a significantly more efficient and scalable
dual optimization approach, which has the same degree of complexity as spectral
methods. We then propose two solvers, namely, quasi-Newton and smoothing Newton
methods, for the dual problem. Both of them are significantly more efficiently
than standard interior-point methods. In practice, the smoothing Newton solver
is faster than the quasi-Newton solver for dense or medium-sized problems,
while the quasi-Newton solver is preferable for large sparse/structured
problems. Our experiments on a few computer vision applications including
clustering, image segmentation, co-segmentation and registration show the
potential of our SDP formulation for solving large-scale BQPs.Comment: Fixed some typos. 18 pages. Accepted to IEEE Transactions on Pattern
Analysis and Machine Intelligenc
Decomposition-Based Method for Sparse Semidefinite Relaxations of Polynomial Optimization Problems
We consider polynomial optimization problems pervaded by a sparsity pattern. It has been shown in [1, 2] that the optimal solution of a polynomial programming problem with structured sparsity can be computed by solving a series of semidefinite relaxations that possess the same kind of sparsity. We aim at solving the former relaxations with a decompositionbased method, which partitions the relaxations according to their sparsity pattern. The decomposition-based method that we propose is an extension to semidefinite programming of the Benders decomposition for linear programs [3] .Polynomial optimization, Semidefinite programming, Sparse SDP relaxations, Benders decomposition
Phase Retrieval via Matrix Completion
This paper develops a novel framework for phase retrieval, a problem which
arises in X-ray crystallography, diffraction imaging, astronomical imaging and
many other applications. Our approach combines multiple structured
illuminations together with ideas from convex programming to recover the phase
from intensity measurements, typically from the modulus of the diffracted wave.
We demonstrate empirically that any complex-valued object can be recovered from
the knowledge of the magnitude of just a few diffracted patterns by solving a
simple convex optimization problem inspired by the recent literature on matrix
completion. More importantly, we also demonstrate that our noise-aware
algorithms are stable in the sense that the reconstruction degrades gracefully
as the signal-to-noise ratio decreases. Finally, we introduce some theory
showing that one can design very simple structured illumination patterns such
that three diffracted figures uniquely determine the phase of the object we
wish to recover
Image labeling and grouping by minimizing linear functionals over cones
We consider energy minimization problems related to image labeling, partitioning, and grouping, which typically show up at mid-level stages of computer vision systems. A common feature of these problems is their intrinsic combinatorial complexity from an optimization pointof-view. Rather than trying to compute the global minimum - a goal we consider as elusive in these cases - we wish to design optimization approaches which exhibit two relevant properties: First, in each application a solution with guaranteed degree of suboptimality can be computed. Secondly, the computations are based on clearly defined algorithms which do not comprise any (hidden) tuning parameters. In this paper, we focus on the second property and introduce a novel and general optimization technique to the field of computer vision which amounts to compute a sub optimal solution by just solving a convex optimization problem. As representative examples, we consider two binary quadratic energy functionals related to image labeling and perceptual grouping. Both problems can be considered as instances of a general quadratic functional in binary variables, which is embedded into a higher-dimensional space such that sub optimal solutions can be computed as minima of linear functionals over cones in that space (semidefinite programs). Extensive numerical results reveal that, on the average, sub optimal solutions can be computed which yield a gap below 5% with respect to the global optimum in case where this is known
New Strings for Old Veneziano Amplitudes III. Symplectic Treatment
A d-dimensional rational polytope P is a polytope whose vertices are located
at the nodes of d-dimensional Z-lattice. Consider a number of points inside the
inflated polytope (with coefficient of inflation k, k=1,2, 3...). The Ehrhart
polynomial of P counts the number of such lattice points (nodes) inside the
inflated P and (may be) at its faces (including vertices). In Part I
(hep-th/0410242) of our four parts work we noticed that the Veneziano amplitude
is just the Laplace transform of the generating function (considered as a
partition function in the sence of statistical mechanics) for the Ehrhart
polynomial for the regular inflated simplex obtained as a deformation retract
of the Fermat (hyper) surface living in complex projective space. This
observation is sufficient for development of new symplectic (this work) and
supersymmetric (hep-th/0411241)physical models reproducing the Veneziano (and
Veneziano-like) amplitudes. General ideas (e.g.those related to the properties
of Ehrhart polynomials) are illustrated by simple practical examples (e.g. use
of mirror symmetry for explanation of available experimental data on pion-pion
scattering) worked out in some detail. Obtained final results are in formal
accord with those earlier obtained by Vergne [PNAS 93 (1996) 14238].Comment: 48 pages J.Geom.Phys.(in press, available on line
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