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Mat. Zametki, 2018, Volume 104, Issue 6, Pages 863–871 (Mi mz12167)  

This article is cited in 2 scientific papers (total in 2 papers)

On the Recovery of an Integer Vector from Linear Measurements

S. V. Konyagin

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: Let $1\le 2l\le m<d$. A vector $x\in\mathbb Z^d$ is said to be $l$-sparse if it has at most $l$ nonzero coordinates. Let an $m\times d$ matrix $A$ be given. The problem of the recovery of an $l$-sparse vector $x\in\mathbb Z^d$ from the vector $y=A x\in\mathbb R^m$ is considered. In the case $m=2l$, we obtain necessary and sufficient conditions on the numbers $m$, $d$, and $k$ ensuring the existence of an integer matrix $A$ all of whose elements do not exceed $k$ in absolute value which makes it possible to reconstruct $l$-sparse vectors in $\mathbb Z^d$. For a fixed $m$, these conditions on $d$ differ only by a logarithmic factor depending on $k$.

Keywords: nonsingular matrix, lattices, successive minima.

Funding Agency Grant Number
Russian Science Foundation 14-50-00005
This work was supported by the Russian Science Foundation under grant 14-50-00005.


DOI: https://doi.org/10.4213/mzm12167

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English version:
Mathematical Notes, 2018, 104:6, 859–865

Bibliographic databases:

UDC: 512.643+519.21
PACS: 02.10.Yn, 02.30.Mv
Received: 29.08.2018

Citation: S. V. Konyagin, “On the Recovery of an Integer Vector from Linear Measurements”, Mat. Zametki, 104:6 (2018), 863–871; Math. Notes, 104:6 (2018), 859–865

Citation in format AMSBIB
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\paper On the Recovery of an Integer Vector from Linear Measurements
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\pages 863--871
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\pages 859--865
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. K. S. Ryutin, “Recovery of sparse integer vectors from linear measurements”, Russian Math. Surveys, 74:6 (2019), 1129–1131  mathnet  crossref  crossref  adsnasa  isi
    2. K. S. Ryutin, “Integer Sampling Matrices with Small Entries Ensuring Vector Recovery”, Math. Notes, 107:2 (2020), 363–366  mathnet  crossref  crossref  isi  elib
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