  The subdifferential of convex functions of the singular spectrum of real
matrices has been widely studied in matrix analysis, optimization and automatic
control theory. Convex optimization over spaces of tensors is now gaining much
interest due to its potential applications in signal processing, statistics and
engineering. The goal of this paper is to present an extension of the approach
by Lewis \cite{lewis1995convex} for the analysis of the subdifferential of
certain convex functions of the spectrum of symmetric tensors. We give a
complete characterization of the subdifferential of Schatten-type tensor norms
for symmetric tensors. Some partial results in this direction are also given
for Orthogonally Decomposable tensors.
