Tensor completion using geodesics on Segre manifolds

TL;DR

This paper introduces a Riemannian conjugate gradient method for tensor completion that leverages explicit geodesic expressions on Segre manifolds, enabling efficient low-rank tensor approximations with applications in recommender systems and spectroscopy.

Contribution

It provides the first explicit geodesic formulas on Segre manifolds and integrates them into a tensor completion algorithm, demonstrating practical effectiveness.

Findings

Successfully applied to movie rating prediction with high accuracy.

Achieved tensor recovery with less than 10% data in spectroscopy.

Demonstrated efficiency of the Riemannian conjugate gradient approach.

Abstract

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.