
Distributed methods for synchronization of orthogonal matrices over graphs
This paper addresses the problem of synchronizing orthogonal matrices ov...
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Learning Transformation Synchronization
Reconstructing the 3D model of a physical object typically requires us t...
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Graph Consistency as a Graduated Property: ConsistencySustaining and Improving Graph Transformations
Where graphs are used for modelling and specifying systems, consistency ...
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True Parallel Graph Transformations: an Algebraic Approach Based on Weak Spans
We address the problem of defining graph transformations by the simultan...
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Simultaneous hollowisation, joint numerical range, and stabilization by noise
We consider orthogonal transformations of arbitrary square matrices to a...
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Extending the DavisKahan theorem for comparing eigenvectors of two symmetric matrices II: Computation and Applications
The extended DavisKahan theorem makes use of polynomial matrix transfor...
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Nearestneighbour Markov point processes on graphs with Euclidean edges
We define nearestneighbour point processes on graphs with Euclidean edg...
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On Transitive Consistency for Linear Invertible Transformations between Euclidean Coordinate Systems
Transitive consistency is an intrinsic property for collections of linear invertible transformations between Euclidean coordinate frames. In practice, when the transformations are estimated from data, this property is lacking. This work addresses the problem of synchronizing transformations that are not transitively consistent. Once the transformations have been synchronized, they satisfy the transitive consistency condition  a transformation from frame A to frame C is equal to the composite transformation of first transforming A to B and then transforming B to C. The coordinate frames correspond to nodes in a graph and the transformations correspond to edges in the same graph. Two direct or centralized synchronization methods are presented for different graph topologies; the first one for quasistrongly connected graphs, and the second one for connected graphs. As an extension of the second method, an iterative GaussNewton method is presented, which is later adapted to the case of affine and Euclidean transformations. Two distributed synchronization methods are also presented for orthogonal matrices, which can be seen as distributed versions of the two direct or centralized methods; they are similar in nature to standard consensus protocols used for distributed averaging. When the transformations are orthogonal matrices, a bound on the optimality gap can be computed. Simulations show that the gap is almost right, even for noise large in magnitude. This work also contributes on a theoretical level by providing linear algebraic relationships for transitively consistent transformations. One of the benefits of the proposed methods is their simplicity  basic linear algebraic methods are used, e.g., the Singular Value Decomposition (SVD). For a wide range of parameter settings, the methods are numerically validated.
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