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Appl."],"published-print":{"date-parts":[[2021,1]]},"abstract":"<jats:p>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 \u201cpreconvergence\u201d 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.<\/jats:p>","DOI":"10.1137\/20m1350947","type":"journal-article","created":{"date-parts":[[2021,4,8]],"date-time":"2021-04-08T14:01:16Z","timestamp":1617890476000},"page":"608-615","source":"Crossref","is-referenced-by-count":28,"title":["Randomized Kaczmarz Converges Along Small Singular Vectors"],"prefix":"10.1137","volume":"42","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7745-4217","authenticated-orcid":true,"given":"Stefan","family":"Steinerberger","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,4,8]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2018.05.009"},{"key":"atypb2","first-page":"299","author":"Cenker C.","year":"1992","journal-title":"Proceedings of SPIE: Visual Communications and Image Processing"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1007\/s11075-011-9451-z"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1016\/0022-5193(70)90109-8"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1137\/15M1025487"},{"key":"atypb6","unstructured":"G. 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