Optimal and algorithmic norm regularization of random matrices
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- by Vishesh Jain, Ashwin Sah and Mehtaab Sawhney PDF
- Proc. Amer. Math. Soc. 150 (2022), 4503-4518 Request permission
Abstract:
Let $A$ be an $n\times n$ random matrix whose entries are i.i.d. with mean $0$ and variance $1$. We present a deterministic polynomial time algorithm which, with probability at least $1-2\exp (-\Omega (\epsilon n))$ in the choice of $A$, finds an $\epsilon n \times \epsilon n$ sub-matrix such that zeroing it out results in $\widetilde {A}$ with \[ \lVert \widetilde {A}\rVert = O(\sqrt {n/\epsilon }). \] Our result is optimal up to a constant factor and improves previous results of Rebrova and Vershynin, and Rebrova. We also prove an analogous result for $A$ a symmetric $n\times n$ random matrix whose upper-diagonal entries are i.i.d. with mean $0$ and variance $1$.References
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Additional Information
- Vishesh Jain
- Affiliation: Department of Statistics, Stanford University, Stanford, California 94305
- MR Author ID: 1101424
- Email: visheshj@stanford.edu
- Ashwin Sah
- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
- MR Author ID: 1279710
- ORCID: 0000-0003-3438-5175
- Email: asah@mit.edu
- Mehtaab Sawhney
- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
- MR Author ID: 1204694
- Email: msawhney@mit.edu
- Received by editor(s): December 2, 2020
- Received by editor(s) in revised form: December 19, 2021
- Published electronically: April 29, 2022
- Communicated by: Deane Yang
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 4503-4518
- MSC (2020): Primary 60B20
- DOI: https://doi.org/10.1090/proc/15964
- MathSciNet review: 4470191