On the subdifferential of symmetric convex functions of the spectrum for symmetric and orthogonally decomposable tensors

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.