Abstract
Let 1 ≤ 2l ≤ m < d. A vector x ∈ ℤd is said to be l-sparse if it has at most l nonzero coordinates. Let an m × d matrix A be given. The problem of the recovery of an l-sparse vector x ∈ Zd from the vector y = Ax ∈ Rm is considered. In the case m = 2l, we obtain necessary conditions 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 ℤd. For a fixed m, these conditions on d differ only by a logarithmic factor depending on k.
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Original Russian Text © S. V. Konyagin, 2018, published in Matematicheskie Zametki, 2018, Vol. 104, No. 6, pp. 863–871.
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Konyagin, S.V. On the Recovery of an Integer Vector from Linear Measurements. Math Notes 104, 859–865 (2018). https://doi.org/10.1134/S0001434618110305
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DOI: https://doi.org/10.1134/S0001434618110305