4,827 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
Gossip Algorithms for Distributed Signal Processing
Gossip algorithms are attractive for in-network processing in sensor networks
because they do not require any specialized routing, there is no bottleneck or
single point of failure, and they are robust to unreliable wireless network
conditions. Recently, there has been a surge of activity in the computer
science, control, signal processing, and information theory communities,
developing faster and more robust gossip algorithms and deriving theoretical
performance guarantees. This article presents an overview of recent work in the
area. We describe convergence rate results, which are related to the number of
transmitted messages and thus the amount of energy consumed in the network for
gossiping. We discuss issues related to gossiping over wireless links,
including the effects of quantization and noise, and we illustrate the use of
gossip algorithms for canonical signal processing tasks including distributed
estimation, source localization, and compression.Comment: Submitted to Proceedings of the IEEE, 29 page
A distributed optimization framework for localization and formation control: applications to vision-based measurements
Multiagent systems have been a major area of research for the last 15 years. This interest has been motivated by tasks that can be executed more rapidly in a collaborative manner or that are nearly impossible to carry out otherwise. To be effective, the agents need to have the notion of a common goal shared by the entire network (for instance, a desired formation) and individual control laws to realize the goal. The common goal is typically centralized, in the sense that it involves the state of all the agents at the same time. On the other hand, it is often desirable to have individual control laws that are distributed, in the sense that the desired action of an agent depends only on the measurements and states available at the node and at a small number of neighbors. This is an attractive quality because it implies an overall system that is modular and intrinsically more robust to communication delays and node failures
Fast ADMM Algorithm for Distributed Optimization with Adaptive Penalty
We propose new methods to speed up convergence of the Alternating Direction
Method of Multipliers (ADMM), a common optimization tool in the context of
large scale and distributed learning. The proposed method accelerates the speed
of convergence by automatically deciding the constraint penalty needed for
parameter consensus in each iteration. In addition, we also propose an
extension of the method that adaptively determines the maximum number of
iterations to update the penalty. We show that this approach effectively leads
to an adaptive, dynamic network topology underlying the distributed
optimization. The utility of the new penalty update schemes is demonstrated on
both synthetic and real data, including a computer vision application of
distributed structure from motion.Comment: 8 pages manuscript, 2 pages appendix, 5 figure
A Convex Approach to Consensus on SO(n)
This paper introduces several new algorithms for consensus over the special
orthogonal group. By relying on a convex relaxation of the space of rotation
matrices, consensus over rotation elements is reduced to solving a convex
problem with a unique global solution. The consensus protocol is then
implemented as a distributed optimization using (i) dual decomposition, and
(ii) both semi and fully distributed variants of the alternating direction
method of multipliers technique -- all with strong convergence guarantees. The
convex relaxation is shown to be exact at all iterations of the dual
decomposition based method, and exact once consensus is reached in the case of
the alternating direction method of multipliers. Further, analytic and/or
efficient solutions are provided for each iteration of these distributed
computation schemes, allowing consensus to be reached without any online
optimization. Examples in satellite attitude alignment with up to 100 agents,
an estimation problem from computer vision, and a rotation averaging problem on
validate the approach.Comment: Accepted to 52nd Annual Allerton Conference on Communication,
Control, and Computin
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