Before revision:  We give an \Omega(} 1 ^n\rightarrow\mathbb{R one-sided \sigma-smooth , then the approximation factor only depends on \sigma. We apply the new algorithm to a broad class of quadratic supermodular functions arising in diversity maximization. This captures metric diversity maximization ( \sigma =2) and negative-type diversity (\sigma=4) .
After revision:  We give an \Omega(} 1 ^n\rightarrow\mathbb{R one-sided \sigma-smooth F with an additional property: non-positive third order partial derivatives. Using the multilinear framework and new matroid rounding techniques for quadratic objectives, we give an \Omega(1/ \sigma =2) and negative-type diversity (\sigma=4) .