  Conjugate partial-symmetric (CPS) tensors are the high-order generalization
of Hermitian matrices. As the role played by Hermitian matrices in matrix
theory and quadratic optimization, CPS tensors have shown growing interest
recently in tensor theory and optimization, particularly in many
application-driven complex polynomial optimization problems. In this paper, we
study CPS tensors with a focus on ranks, rank-one decompositions and
approximations, as well as their applications. The analysis is conducted along
side with a more general class of complex tensors called partial-symmetric
tensors. We prove constructively that any CPS tensor can be decomposed into a
sum of rank-one CPS tensors, which provides an alternative definition of CPS
tensors via linear combinations of rank-one CPS tensors. Three types of ranks
for CPS tensors are defined and shown to be different in general. This leads to
the invalidity of the conjugate version of Comon's conjecture. We then study
rank-one approximations and matricizations of CPS tensors. By carefully
unfolding CPS tensors to Hermitian matrices, rank-one equivalence can be
preserved. This enables us to develop new convex optimization models and
algorithms to compute best rank-one approximation of CPS tensors. Numerical
experiments from various data are performed to justify the capability of our
methods.
