  We propose a Riemannian conjugate gradient (CG) optimization method for
finding low rank approximations of incomplete tensors. Our main contribution
consists of an explicit expression of the geodesics on the Segre manifold.
These are exploited in our algorithm to perform the retractions. We apply our
method to movie rating predictions in a recommender system for the MovieLens
dataset, and identification of pure fluorophores via fluorescent spectroscopy
with missing data. In this last application, we recover the tensor
decomposition from less than $10\%$ of the data.
