Column normalization of a random measurement matrix
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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 ≤ p ≤ c_1 d we construct a random vector X ∈ R^d with iid, mean-zero, variance 1 coordinates, that satisfies _t ∈ S^d-1<X,t>_L_q≤ c_2√(q) for every 2≤ q ≤ p. We show that if m ≤ c_3√(p)d^1/p and Γ̃:R^d → R^m is the column-normalized matrix generated by m independent copies of X, then with probability at least 1-2(-c_4m), Γ̃ does not satisfy the exact reconstruction property of order 2.
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