Random Node-Asynchronous Updates on Graphs

Teke, Oguzhan; Vaidyanathan, P.P.

doi:10.1109/tsp.2019.2910485

Cited by 16 publications

(19 citation statements)

References 53 publications

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“…When the matrix A is considered as a local graph operator, i.e., the adjacency matrix, or the graph Laplacian, the scheme in (2) models the random asynchronous behavior of the nodes of a graph. 7,8 In this setting, the random asynchronous model (2) allows us to design polynomial graph filters that result in clustering algorithms for autonomous networks. 7,8 When extended to have a constant input signal, the model (2) is also useful for a node-asynchronous implementation of rational filters on graphs.…”

Section: Random Component-wise Power Methodsmentioning

confidence: 99%

“…7,8 In this setting, the random asynchronous model (2) allows us to design polynomial graph filters that result in clustering algorithms for autonomous networks. 7,8 When extended to have a constant input signal, the model (2) is also useful for a node-asynchronous implementation of rational filters on graphs. 9,20 When A is viewed as a data matrix, which is the case to be considered in this study, the component-wise updates of (2) allow distributed computation of the dominant eigenvector since the rows of A need not be accessed simultaneously.…”

Section: Random Component-wise Power Methodsmentioning

confidence: 99%

“…The values of the unselected indices remain unchanged. This randomized variant is first proposed in order to study the behavior of autonomous networks [7][8][9] in the context of graph signal processing. Here we will consider the randomized update scheme from a purely linear algebraic point of view.…”

Section: Introductionmentioning

confidence: 99%

“…When A is a normal matrix, under mild boundedness assumptions on the matrix A, it is proven that iterations indeed converge in the mean-squared sense to an eigenvector of A with eigenvalue 1. 7 However, convergence characteristics of the random updates differ significantly from that of the regular power method in two ways. The first one is the region of convergence: random component-wise updates have a larger region of convergence on the eigenvalue plane.…”

Section: Introductionmentioning

confidence: 99%

“…In addition to the enlarged convergence regions, 7 in this study we consider the effect of the randomized updates on the rate of convergence. In this regard, we first demonstrate that randomized updates may converge faster, or slower than the regular power method depending on the sign (or, phase) of the non-unit eigenvalues, and the eigenvalue gap does not have a direct effect on the convergence rate unlike the regular power method.…”

Section: Introductionmentioning

confidence: 99%

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The random component-wise power method

Teke

Vaidyanathan

2019

Wavelets and Sparsity XVIII

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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 affect the convergence rate of the randomized updates unlike the regular power method. In particular, it is shown that the rate of convergence is affected 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 different parameter settings.

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Section: Random Component-wise Power Methodsmentioning

confidence: 99%

Section: Random Component-wise Power Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The random component-wise power method

Teke

Vaidyanathan

2019

Wavelets and Sparsity XVIII

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Distributed linear-quadratic control with graph neural networks

Gama

Sojoudi

2022

Signal Processing

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Joint Vertex-Time Filtering on Graphs With Random Node-Asynchronous Updates

Teke

Vaidyanathan

2021

IEEE Access

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In graph signal processing signals are defined over a graph, and filters are designed to manipulate the variation of signals over the graph. On the other hand, time domain signal processing treats signals as time series, and digital filters are designed to manipulate the variation of signals in time. This study focuses on the notion of vertex-time filters, which manipulates the variation of a time-dependent graph signal both in the time domain and graph domain simultaneously. The key aspects of the proposed filtering operations are due to the random and asynchronous behavior of the nodes, in which they follow a collectcompute-broadcast scheme. For the analysis of the randomized vertex-time filtering operations, this study first considers the random asynchronous variant of linear discrete-time state-space models, in which each state variable gets updated randomly and independently (and asynchronously) in every iteration. Unlike previous studies that analyzed similar models under certain assumptions on the input signal, this study considers the model in the most general setting with arbitrary time-dependent input signals, which lay the foundations for the vertex-time graph filtering operations. This analysis shows that exponentials continue to be eigenfunctions in a statistical sense in spite of the random asynchronous nature of the model. This study also presents the necessary and sufficient condition for the mean-squared stability and shows that stability of the underlying state transition matrix is neither necessary nor sufficient for the mean-squared stability of the randomized asynchronous recursions. Then, the proposed filtering operations are proven to be mean-square stable if and only if the filter, the graph operator and the update probabilities satisfy a certain condition. The results show that some unstable vertex-time graph filters (in the synchronous case) can be implemented in a stable manner in the presence of randomized asynchronicity, which is also demonstrated by numerical examples.INDEX TERMS Autonomous networks, asynchronous updates, randomized iterations, vertex-time filters.

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Random Node-Asynchronous Updates on Graphs

Cited by 16 publications

References 53 publications

The random component-wise power method

The random component-wise power method

Distributed linear-quadratic control with graph neural networks

Joint Vertex-Time Filtering on Graphs With Random Node-Asynchronous Updates

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