7,677 research outputs found
Rotation Averaging and Strong Duality
In this paper we explore the role of duality principles within the problem of
rotation averaging, a fundamental task in a wide range of computer vision
applications. In its conventional form, rotation averaging is stated as a
minimization over multiple rotation constraints. As these constraints are
non-convex, this problem is generally considered challenging to solve globally.
We show how to circumvent this difficulty through the use of Lagrangian
duality. While such an approach is well-known it is normally not guaranteed to
provide a tight relaxation. Based on spectral graph theory, we analytically
prove that in many cases there is no duality gap unless the noise levels are
severe. This allows us to obtain certifiably global solutions to a class of
important non-convex problems in polynomial time.
We also propose an efficient, scalable algorithm that out-performs general
purpose numerical solvers and is able to handle the large problem instances
commonly occurring in structure from motion settings. The potential of this
proposed method is demonstrated on a number of different problems, consisting
of both synthetic and real-world data
Rotation Coordinate Descent for Fast Globally Optimal Rotation Averaging
Under mild conditions on the noise level of the measurements, rotation
averaging satisfies strong duality, which enables global solutions to be
obtained via semidefinite programming (SDP) relaxation. However, generic
solvers for SDP are rather slow in practice, even on rotation averaging
instances of moderate size, thus developing specialised algorithms is vital. In
this paper, we present a fast algorithm that achieves global optimality called
rotation coordinate descent (RCD). Unlike block coordinate descent (BCD) which
solves SDP by updating the semidefinite matrix in a row-by-row fashion, RCD
directly maintains and updates all valid rotations throughout the iterations.
This obviates the need to store a large dense semidefinite matrix. We
mathematically prove the convergence of our algorithm and empirically show its
superior efficiency over state-of-the-art global methods on a variety of
problem configurations. Maintaining valid rotations also facilitates
incorporating local optimisation routines for further speed-ups. Moreover, our
algorithm is simple to implement; see supplementary material for a
demonstration program.Comment: Accepted to CVPR 2021 as an oral presentatio
Strings as Solitons & Black Holes as Strings
Supersymmetric closed string theories contain an infinite tower of
BPS-saturated, oscillating, macroscopic strings in the perturbative spectrum.
When these theories have dual formulations, this tower of states must exist
nonperturbatively as solitons in the dual theories. We present a general class
of exact solutions of low-energy supergravity that corresponds to all these
states. After dimensional reduction they can be interpreted as supersymmetric
black holes with a degeneracy related to the degeneracy of the string states.
{}For example, in four dimensions we obtain a point-like solution which is
asymptotic to a stationary, rotating, electrically-charged black hole with
Regge-bounded angular momentum and with the usual ring-singularity replaced by
a string source. This further supports the idea that the entropy of
supersymmetric black holes can be understood in terms of counting of string
states. We also discuss some applications of these solutions to string duality.Comment: 52 pages, harvmac (b
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