On exact recovery of sparse vectors from linear measurements
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Let 1 ≤ k ≤ n < N. We say that a vector x ∈ ℝ N is k-sparse if it has at most k nonzero coordinates. Let Φ be an n × N matrix. We consider the problem of recovery of a k-sparse vector x ∈ ℝ N from the vector y = Φx ∈ ℝ n . We obtain almost-sharp necessary conditions for k, n, N for this problem to be reduced to that of minimization of the ℓ1-norm of vectors z satisfying the condition y = Φz.
Keywordscompressed sensing exact recovery of a k-sparse vector restricted isometry property element of best approximation estimates of Kolmogorov widths
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