  Marcus, Spielman, and Srivastava recently solved the Kadison-Singer problem
by showing that if u_1, ..., u_m are column vectors in C^d such that \sum
u_iu_i^* = I, then a set of indices S \subseteq {1, ..., m} can be chosen so
that \sum_{i \in S} u_iu_i^* is approximately (1/2)I, with the approximation
good in operator norm to order \epsilon^{1/2} where \epsilon = \max \|u_i\|^2.
We extend their result to show that every linear combination of the matrices
u_iu_i^* with coefficients in [0,1] can be approximated in operator norm to
order \epsilon^{1/8} by a matrix of the form \sum_{i \in S} u_iu_i^*.
