122 research outputs found
Large-Scale Sensor Network Localization via Rigid Subnetwork Registration
In this paper, we describe an algorithm for sensor network localization (SNL)
that proceeds by dividing the whole network into smaller subnetworks, then
localizes them in parallel using some fast and accurate algorithm, and finally
registers the localized subnetworks in a global coordinate system. We
demonstrate that this divide-and-conquer algorithm can be used to leverage
existing high-precision SNL algorithms to large-scale networks, which could
otherwise only be applied to small-to-medium sized networks. The main
contribution of this paper concerns the final registration phase. In
particular, we consider a least-squares formulation of the registration problem
(both with and without anchor constraints) and demonstrate how this otherwise
non-convex problem can be relaxed into a tractable convex program. We provide
some preliminary simulation results for large-scale SNL demonstrating that the
proposed registration algorithm (together with an accurate localization scheme)
offers a good tradeoff between run time and accuracy.Comment: 5 pages, 8 figures, 1 table. To appear in Proc. IEEE International
Conference on Acoustics, Speech, and Signal Processing, April 19-24, 201
Approximating the Little Grothendieck Problem over the Orthogonal and Unitary Groups
The little Grothendieck problem consists of maximizing
over binary variables , where C is a
positive semidefinite matrix. In this paper we focus on a natural
generalization of this problem, the little Grothendieck problem over the
orthogonal group. Given C a dn x dn positive semidefinite matrix, the objective
is to maximize restricting to take
values in the group of orthogonal matrices, where denotes the (ij)-th
d x d block of C. We propose an approximation algorithm, which we refer to as
Orthogonal-Cut, to solve this problem and show a constant approximation ratio.
Our method is based on semidefinite programming. For a given , we show
a constant approximation ratio of , where is
the expected average singular value of a d x d matrix with random Gaussian
i.i.d. entries. For d=1 we recover the known
approximation guarantee for the classical little Grothendieck problem. Our
algorithm and analysis naturally extends to the complex valued case also
providing a constant approximation ratio for the analogous problem over the
Unitary Group.
Orthogonal-Cut also serves as an approximation algorithm for several
applications, including the Procrustes problem where it improves over the best
previously known approximation ratio of~. The little
Grothendieck problem falls under the class of problems approximated by a recent
algorithm proposed in the context of the non-commutative Grothendieck
inequality. Nonetheless, our approach is simpler and it provides a more
efficient algorithm with better approximation ratios and matching integrality
gaps.
Finally, we also provide an improved approximation algorithm for the more
general little Grothendieck problem over the orthogonal (or unitary) group with
rank constraints.Comment: Updates in version 2: extension to the complex valued (unitary group)
case, sharper lower bounds on the approximation ratios, matching integrality
gap, and a generalized rank constrained version of the problem. Updates in
version 3: Improvement on the expositio
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
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