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Tensor completion using geodesics on Segre manifolds
Author(s) -
Swijsen Lars,
Van der Veken Joeri,
Vannieuwenhoven Nick
Publication year - 2022
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2446
Subject(s) - mathematics , geodesic , conjugate gradient method , tensor (intrinsic definition) , rank (graph theory) , parametrization (atmospheric modeling) , tangent vector , nonlinear conjugate gradient method , manifold (fluid mechanics) , riemannian manifold , tangent , gradient descent , algorithm , mathematical analysis , pure mathematics , geometry , combinatorics , computer science , artificial intelligence , mechanical engineering , physics , quantum mechanics , artificial neural network , engineering , radiative transfer
We propose a Riemannian conjugate gradient algorithm for approximating incomplete tensors by canonical polyadic decompositions of low rank. Our main contribution consists of an explicit expression for an almost everywhere complete set of geodesics of the Segre manifold of rank‐1 tensors. These are leveraged in our Riemannian optimization algorithm over a geometrically convenient parametrization of rank‐ r $$ r $$ tensors to move in the direction of a tangent vector over ther $$ r $$ ‐fold product of Segre manifolds. We apply our method to movie rating predictions in a recommender system for the MovieLens dataset, and for identifying pure fluorophores via fluorescent spectroscopy with missing data. In this last application, we can recover the tensor decomposition from only6 . 5 % $$ 6.5\% $$ of the data. In our numerical experiments, the proposed Riemannian conjugate gradient algorithm was competitive with a state‐of‐the‐art quasi‐Newton method with truncated conjugate gradient inner solves from Tensorlab in terms of accuracy and could reduce the computation time by half.