  Matrix concentration inequalities give bounds for the spectral-norm deviation
of a random matrix from its expected value. These results have a weak
dimensional dependence that is sometimes, but not always, necessary. This paper
identifies one of the sources of the dimensional term and exploits this insight
to develop sharper matrix concentration inequalities. In particular, this
analysis delivers two refinements of the matrix Khintchine inequality that use
information beyond the matrix variance to reduce or eliminate the dimensional
dependence.
