Lower Bound Proof U = U (1) ; U (2) ; …, U (log (cn)) ; 0s –U (k) is an £ ( ε -2 ) x c/2 submatrix with entries in {-10 k, 10 k } –U (k) i, j = 10 k if matched entry of x is 0, else U (k) i, j = -10 k Bobs index i corresponds to U (k*) i*, j* U is such that A = U+U = U (1) ; U (2) ; …, U (k*) ; 0s –U is determined from x i+1, …, x s A T B is i * -th row of U (k*) ||A|| ¼ ||U (k *) || since the entries of A grow geometrically ε 2 ||A|| 2 ¢ ||B|| 2, the squared error, is small, so most entries of the approximation to A T B have the correct sign
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