A Walk From Free Probability to Matrix Discrepancy III: Higher Rank Kadison-Singer and Spectrally Thin Trees

Let $A_1,\ldots,A_N$ be positive semidefinite matrices of rank at most $r$, with $\sum_iA_i=I$ and $\|A_i\|\le\varepsilon$. We prove that the original matrices admit signs with discrepancy $O(\sqrt\varepsilon\log(2r))$, independently of their dimension and number which is a significantly stronger re