  We consider mean-field vector spin glasses with self-overlap correction. The
limit of free energy is known to be the Parisi formula, which is an infimum
over matrix-valued paths. We decompose such a path into a Lipschitz
matrix-valued path and the quantile function of a one-dimensional probability
measure. For such a pair, we associate a Parisi PDE generalized for vector
spins. Under mild conditions, we rewrite the Parisi formula in terms of
solutions of the PDE. Moreover, for each fixed Lipschitz path, the Parisi
functional is strictly convex over probability measures.
