Low-Rank Tensor Completion using Tensor Train Decomposition via Riemannian Optimization on the Quotient Geometry
For researchers in tensor completion and optimization, this work provides a more efficient geometric approach to TT-based completion, though improvements are incremental over existing methods.
The paper introduces a novel Riemannian optimization framework for low-rank tensor completion in Tensor Train format, achieving computational efficiency by reducing per-block unknowns from quadratic to near-half scaling in TT-ranks, with reconstruction accuracy comparable to state-of-the-art methods.
Owing to the effectiveness of Tensor Train (TT) decomposition in managing high-order tensors, low-rank tensor completion within the TT-format has emerged as a prominent research focus. In this paper, we leverage the left-orthogonal property of the TT-decomposition to construct a novel quotient manifold and introduce a family of admissible Riemannian metrics. Within this geometric framework, we propose a new approach to constructing retractions compatible with the quotient structure, realized via two novel retractions based on recursive polar and QR decompositions that respect the recursive orthogonalization structure of the TT format. We then derive Riemannian gradient descent and conjugate gradient methods to solve the tensor completion problem. Theoretically, our approach streamlines the horizontal projection by reducing the number of unknowns per block from a quadratic dependence on the TT-ranks to a near-half scaling, thereby enhancing computational efficiency over conventional quotient-based methods. Numerical experiments demonstrate that the proposed algorithms achieve reconstruction accuracy comparable to state-of-the-art TT-based geometric methods.