The notion of graph shift, introduced recently in graph signal processing, extends many classical signal processing techniques to graphs. Its practical importance follows from its localization: a single graph shift requires nodes to communicate only with their neighbors. However, communications should happen simultaneously, which requires a synchronization over the graph. In order to overcome this restriction, recent studies consider a random asynchronous variant of the graph shift, which is also suitable for autonomous networks. A graph signal under this randomized scheme is shown to converge (under mild conditions) to an eigenvector of the eigenvalue 1 of the operator even if the operator has other eigenvalues with magnitudes larger than unity. If the eigenvalue 1 does not exist, the operator can be easily normalized in theory. However, in practice, the normalization requires one to know the (dominant) eigenvalues, which may not be possible to obtain in large autonomous networks. To eliminate this limitation, this study considers the use of a nonlinearity in the updates making the scheme similar in spirit to the Hopfield neural network model. Our simulation results show that a graph signal still approaches the eigenvector of the dominant eigenvalue although the convergence is not exact. Nevertheless, approximation ismore »
The random component-wise power method
This paper considers a random component-wise variant of the unnormalized power method, which is similar to the regular power iteration except that only a random subset of indices is updated in each iteration. For the case of normal matrices, it was previously shown that random component-wise updates converge in the mean-squared sense to an eigenvector of eigenvalue 1 of the underlying matrix even in the case of the matrix having spectral radius larger than unity. In addition to the enlarged convergence regions, this study shows that the eigenvalue gap does not directly aect the convergence rate of the randomized updates unlike the regular power method. In particular, it is shown that the rate of convergence is aected by the phase of the eigenvalues in the case of random component-wise updates, and the randomized updates favor negative eigenvalues over positive ones. As an application, this study considers a reformulation of the component-wise updates revealing a randomized algorithm that is proven to converge to the dominant left and right singular vectors of a normalized data matrix. The algorithm is also extended to handle large-scale distributed data when computing an arbitrary rank approximation of an arbitrary data matrix. Numerical simulations verify the convergence of the proposed algorithms under dierent parameter settings.
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- Proc. SPIE 11138, Wavelets and Sparsity XVIII
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- National Science Foundation
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