We study the least-squares (LS) functional of the canonical polyadic (CP)
tensor decomposition. Our approach is based on the elimination of one factor
matrix which results in a reduced functional. The reduced functional is
reformulated into a projection framework and into a Rayleigh quotient. An
analysis of this functional leads to several conclusions: new sufficient
conditions for the existence of minimizers of the LS functional, the existence
of a critical point in the rank-one case, a heuristic explanation of "swamping"
and computable bounds on the minimal value of the LS functional. The latter
result leads to a simple algorithm -- the Centroid Projection algorithm -- to
compute suboptimal solutions of tensor decompositions. These suboptimal
solutions are applied to iterative CP algorithms as initial guesses, yielding a
method called centroid projection for canonical polyadic (CPCP) decomposition
which provides a significant speedup in our numerical experiments compared to
the standard methods.