We provide the first positive result on the nonsmooth optimization landscape
of robust principal component analysis, to the best of our knowledge. It is the
object of several conjectures and remains mostly uncharted territory. We
identify a necessary and sufficient condition for the absence of spurious local
minima in the rank-one case. Our proof exploits the subdifferential regularity
of the objective function in order to eliminate the existence quantifier from
the first-order optimality condition known as Fermat's rule.Comment: 23 pages, 5 figure