A Riemannian Rank‐Adaptive Method for Higher‐Order Tensor Completion in the Tensor‐Train Format

Abstract

A new Riemannian rank adaptive method (RRAM) is proposed for the low‐rank tensor completion problem (LRTCP). This problem is formulated as a least‐squares optimization problem on the algebraic variety of tensors of bounded tensor‐train (TT) rank. The RRAM iteratively optimizes over fixed‐rank smooth manifolds using a Riemannian conjugate gradient algorithm from Steinlechner. In between, the rank is increased by computing a descent direction selected in the tangent cone to the variety. A numerical method to estimate the rank increase is proposed. This numerical method is based on a new theoretical result for the low‐rank tensor approximation problem and a definition of an estimated TT‐rank. When the iterate comes close to a lower‐rank set, the RRAM decreases the rank based on the TT‐rounding algorithm from Oseledets and a definition of a numerical rank. It is shown that the TT‐rounding algorithm can be considered an approximate projection onto the lower‐rank set, which satisfies a certain angle condition to ensure that the image is sufficiently close to that of an exact projection. Several numerical experiments illustrate the use of the RRAM and its subroutines in Matlab. In all experiments, the proposed RRAM significantly outperforms the state‐of‐the‐art RRAM for tensor completion in the TT format from Steinlechner in terms of computation time.