NDLI: Randomized Kaczmarz algorithm for inconsistent linear systems: An exact MSE analysis

Content Provider	IEEE Xplore Digital Library
Author	Chuang Wang Agaskar, A. Lu, Y.M.
Copyright Year	2015
Description	Author affiliation: Harvard Univ., Cambridge, MA, USA (Chuang Wang; Agaskar, A.; Lu, Y.M.)
Abstract	We provide a complete characterization of the randomized Kaczmarz algorithm (RKA) for inconsistent linear systems. The Kaczmarz algorithm, known in some fields as the algebraic reconstruction technique, is a classical method for solving large-scale overdetermined linear systems through a sequence of projection operators; the randomized Kaczmarz algorithm is a recent proposal by Strohmer and Vershynin to randomize the sequence of projections in order to guarantee exponential convergence (in mean square) to the solutions. A flurry of work followed this development, with renewed interest in the algorithm, its extensions, and various bounds on their performance. Earlier, we studied the special case of consistent linear systems and provided an exact formula for the mean squared error (MSE) in the value reconstructed by RKA, as well as a simple way to compute the exact decay rate of the error. In this work, we consider the case of inconsistent linear systems, which is a more relevant scenario for most applications. First, by using a “lifting trick”, we derive an exact formula for the MSE given a fixed noise vector added to the measurements. Then we show how to average over the noise when it is drawn from a distribution with known first and second-order statistics. Finally, we demonstrate the accuracy of our exact MSE formulas through numerical simulations, which also illustrate that previous upper bounds in the literature may be several orders of magnitude too high.
Starting Page	498
Ending Page	502
File Size	211192
Page Count	5
File Format	PDF
ISBN	9781467373531
DOI	10.1109/SAMPTA.2015.7148941
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2015-05-25
Publisher Place	USA
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Algorithm design and analysis Linear systems Upper bound Limiting Noise Overdetermined linear systems Signal processing algorithms Kaczmarz Algorithm Randomized Kaczmarz algorithm Noise measurement
Content Type	Text
Resource Type	Article

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