Let $A$ be an $n^2\times K$ matrix with $K = \frac{n(n-1)}{2}$. $A$ is not orthonormal but has linearly independent columns. Let $S$ be a diagonal sampling matrix(of zeros and ones). The diagonal entries are sampled uniformly at random. The goal is to lower bound the following quantity $$ x^TSA^TASx $$ I know the structure of $A$ fairly well and can estimate coherences and such. Is there a concentration inequality that might be useful? I appreciate any suggestions.