Recovery of sparsest signals via ℓq-minimization

In this paper, it is proved that every s-sparse vector x ∈ R can be exactly recovered from the measurement vector z = Ax ∈ R via some `-minimization with 0 < q ≤ 1, as soon as each s-sparse vector x ∈ R is uniquely determined by the measurement z. Moreover it is shown that the exponent q in the `-minimization can be so chosen to be about 0.6796× (1− δ2s(A)), where δ2s(A) is the restricted isometry constant of order 2s for the measurement matrix A.