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Randomized Kaczmarz Converges Along Small Singular Vectors

Randomized Kaczmarz is a simple iterative method for finding solutions of linear systems $Ax = b$. We point out that the arising sequence $(x_k)_{k=1}^{\infty}$ tends to converge to the solution $x$ in an interesting way: generically, as $k \rightarrow \infty$, $x_k - x$ tends to the singular vector of $A$ corresponding to the smallest singular value. This has interesting consequences: in particular, the error analysis of Strohmer and Vershynin is optimal. It also quantifies the “preconvergence” phenomenon where the method initially seems to converge faster. This fact also allows for a fast computation of vectors $x$ for which the Rayleigh quotient $\|Ax\|/\|x\|$ is small: solve $Ax = 0$ via randomized Kaczmarz.

Ketersediaan

Informasi Detail

Judul Seri

SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS; Vol.42 No.2 April-June 2021

No. Panggil

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Penerbit

: .,

Deskripsi Fisik

p. 608 - 615

Bahasa

English

ISBN/ISSN

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Klasifikasi

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Tipe Isi

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Tipe Media

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Tipe Pembawa

-

Edisi

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Subjek

ART
SINGULAR VECTOR
ALGEBRAIC RECONSTRUCTION TECHNIQUE
KACZMARZ METHOD
PROJECTION ONTO CONVEX SETS
POCS
RANDOMIZED KACZMARZ METHOD

Info Detail Spesifik

https://doi.org/10.1137/20M1350947

Pernyataan Tanggungjawab

Stefan Steinerberger

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