Tensor completion using geodesics on Segre manifolds
This work addresses tensor completion for applications like recommender systems and spectroscopy, but it is incremental as it builds on existing Riemannian optimization methods with a specific geometric improvement.
The paper tackles the problem of low-rank tensor completion by proposing a Riemannian conjugate gradient method that uses explicit geodesics on Segre manifolds for retractions, achieving recovery from less than 10% of data in a fluorophore identification application.
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.