A Randomized Distributed Kaczmarz Algorithm and Anomaly Detection

The Kaczmarz algorithm is an iterative method for solving systems of linear equations. We introduce a randomized Kaczmarz algorithm for solving systems of linear equations in a distributed environment, i.e., the equations within the system are distributed over multiple nodes within a network. The mo...

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Vydáno v:Axioms Ročník 11; číslo 3; s. 106
Hlavní autoři: Keinert, Fritz, Weber, Eric S.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Basel MDPI AG 01.03.2022
Témata:
Algorithms
Anomalies
Control theory
Convergence
distributed optimization
Eigenvalues
Estimates
Gossip
Kaczmarz method
Linear equations
Mathematical analysis
Nodes
random control
sparse noise
stochastic gradient descent
ISSN:2075-1680, 2075-1680
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Shrnutí:The Kaczmarz algorithm is an iterative method for solving systems of linear equations. We introduce a randomized Kaczmarz algorithm for solving systems of linear equations in a distributed environment, i.e., the equations within the system are distributed over multiple nodes within a network. The modification we introduce is designed for a network with a tree structure that allows for passage of solution estimates between the nodes in the network. We demonstrate that the algorithm converges to the solution, or the solution of minimal norm, when the system is consistent. We also prove convergence rates of the randomized algorithm that depend on the spectral data of the coefficient matrix and the random control probability distribution. In addition, we demonstrate that the randomized algorithm can be used to identify anomalies in the system of equations when the measurements are perturbed by large, sparse noise.
Bibliografie:ObjectType-Article-1
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ISSN:2075-1680
2075-1680
DOI:10.3390/axioms11030106