9,403 research outputs found
A Riemannian low-rank method for optimization over semidefinite matrices with block-diagonal constraints
We propose a new algorithm to solve optimization problems of the form for a smooth function under the constraints that is positive
semidefinite and the diagonal blocks of are small identity matrices. Such
problems often arise as the result of relaxing a rank constraint (lifting). In
particular, many estimation tasks involving phases, rotations, orthonormal
bases or permutations fit in this framework, and so do certain relaxations of
combinatorial problems such as Max-Cut. The proposed algorithm exploits the
facts that (1) such formulations admit low-rank solutions, and (2) their
rank-restricted versions are smooth optimization problems on a Riemannian
manifold. Combining insights from both the Riemannian and the convex geometries
of the problem, we characterize when second-order critical points of the smooth
problem reveal KKT points of the semidefinite problem. We compare against state
of the art, mature software and find that, on certain interesting problem
instances, what we call the staircase method is orders of magnitude faster, is
more accurate and scales better. Code is available.Comment: 37 pages, 3 figure
Computational Geometry Column 42
A compendium of thirty previously published open problems in computational
geometry is presented.Comment: 7 pages; 72 reference
Convex Relaxations of SE(2) and SE(3) for Visual Pose Estimation
This paper proposes a new method for rigid body pose estimation based on
spectrahedral representations of the tautological orbitopes of and
. The approach can use dense point cloud data from stereo vision or an
RGB-D sensor (such as the Microsoft Kinect), as well as visual appearance data.
The method is a convex relaxation of the classical pose estimation problem, and
is based on explicit linear matrix inequality (LMI) representations for the
convex hulls of and . Given these representations, the relaxed
pose estimation problem can be framed as a robust least squares problem with
the optimization variable constrained to these convex sets. Although this
formulation is a relaxation of the original problem, numerical experiments
indicate that it is indeed exact - i.e. its solution is a member of or
- in many interesting settings. We additionally show that this method
is guaranteed to be exact for a large class of pose estimation problems.Comment: ICRA 2014 Preprin
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