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2018 Column normalization of a random measurement matrix
Shahar Mendelson
Electron. Commun. Probab. 23: 1-8 (2018). DOI: 10.1214/17-ECP100

Abstract

In this note we answer a question of G. Lecué, by showing that column normalization of a random matrix with iid entries need not lead to good sparse recovery properties, even if the generating random variable has a reasonable moment growth. Specifically, for every $2 \leq p \leq c_1\log d$ we construct a random vector $X \in \mathbb{R} ^d$ with iid, mean-zero, variance $1$ coordinates, that satisfies $\sup _{t \in S^{d-1}} \|\bigl < {X,t} \bigr >\|_{L_q} \leq c_2\sqrt{q} $ for every $2\leq q \leq p$. We show that if $m \leq c_3\sqrt{p} d^{1/p}$ and $\tilde{\Gamma } :\mathbb{R} ^d \to \mathbb{R} ^m$ is the column-normalized matrix generated by $m$ independent copies of $X$, then with probability at least $1-2\exp (-c_4m)$, $\tilde{\Gamma } $ does not satisfy the exact reconstruction property of order $2$.

Citation

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Shahar Mendelson. "Column normalization of a random measurement matrix." Electron. Commun. Probab. 23 1 - 8, 2018. https://doi.org/10.1214/17-ECP100

Information

Received: 2 March 2017; Accepted: 13 November 2017; Published: 2018
First available in Project Euclid: 27 February 2018

zbMATH: 1390.62087
MathSciNet: MR3771771
Digital Object Identifier: 10.1214/17-ECP100

Subjects:
Primary: 62G99

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