On the Subsample Size of Quantile-Based Randomized Kaczmarz

[2507.15185] On the Subsample Size of Quantile-Based Randomized Kaczmarz Skip to main content Search Submit Donate Log in Mathematics > Numerical Analysis arXiv:2507.15185 (math) [Submitted on 21 Jul 2025 ( v1 ), last revised 28 Aug 2026 (this version, v2)] Title: On the Subsample Size of Quantile-Based Randomized Kaczmarz Authors: Jian-Feng Cai , Junren Chen , Anna Ma , Tong Wu View PDF Abstract: Quantile-based randomized Kaczmarz (QRK) was recently introduced to efficiently solve sparsely corrupted linear systems $\mathbf{A} \mathbf{x}^*+\mathbf{\epsilon} = \mathbf{b}$ [SIAM J. Matrix Anal. Appl., 43(2), 605-637], where $\mathbf{A}\in \mathbb{R}^{m\times n}$ and $\mathbf{\epsilon}$ is an arbitrary $(\beta m)$-sparse corruption. However, all existing theoretical guarantees for QRK require quantiles to be computed using all $m$ samples (or a subsample of the same order), thus negating the computational advantage of Kaczmarz-type methods. This paper overcomes the bottleneck. We analyze a subsampling QRK, which computes quantiles from $D$ uniformly chosen samples at each iteration. Under some standard scaling assumptions on the coefficient matrix, we show that QRK with subsample size $D\ge\frac{C\log (T)}{\log(1/\beta)}$ linearly converges over the first $T$ iterations with high probability, where $C$ is some absolute constant. This subsample size is a substantial reduction from $O(m)$ in prior results. For instance, it translates into $O(\log(n))$ even if an approximation error of $\exp(-n^2)$ is desired. Intriguingly, our subsample size is also tight up to a multiplicative constant: if $D\le \frac{c\log(T)}{\log(1/\beta)}$ for some constant $c$, the error of the $T$-th iterate could be arbitrarily large with high probability. Numerical results are provided to corroborate our theory. Comments: main tex: 21 pages Subjects: Numerical Analysis (math.NA) MSC classes: 65F10, 68W20, 60B20 Cite as: arXiv:2507.15185 [math.NA] (or arXiv:2507.15185v2 [math.NA] for this version) https://doi.org/10.48550/arXiv.2507.15185 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Tong Wu [ view email ] [v1] Mon, 21 Jul 2025 02:06:39 UTC (2,121 KB) [v2] Fri, 28 Aug 2026 06:37:48 UTC (2,163 KB) Full-text links: Access Paper: View PDF TeX Source view license Current browse context: math.NA < prev | next > new | recent | 2025-07 Change to browse by: cs cs.NA math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

Related documents

• On the Subsample Size of Quantile-Based Randomized Kaczmarz#682470
• Adaptive Row Selection Meets Asynchrony in Randomized Kaczmarz#665841
• Adaptive Row Selection Meets Asynchrony in Randomized Kaczmarz#664341
• Adaptive Row Selection Meets Asynchrony in Randomized Kaczmarz#353029
• Smooth Partial Lotteries for Stable Randomized Selection#204785