A preconditioned Riemannian gradient descent method that weights gradient entries by row norms of the TT-unfolded gradient achieves linear convergence for low tensor-train rank completion and is reported to be orders of magnitude faster than standard RGD.
A Single-Mode Quasi Riemannian Gradient Descent Algorithm for Low-Rank Tensor Recovery
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abstract
This paper focuses on recovering a low-rank tensor from its incomplete measurements. We propose a novel algorithm termed the Single Mode Quasi Riemannian Gradient Descent (SM-QRGD). By exploiting the benefits of both fixed-rank matrix tangent space projection in Riemannian gradient descent and sequentially truncated high-order singular value decomposition (ST-HOSVD), SM-QRGD achieves a much faster convergence speed than existing state-of-the-art algorithms. Theoretically, we establish the convergence of SM-QRGD through the Tensor Restricted Isometry Property (TRIP) and the geometry of the fixed-rank matrix manifold. Numerically, extensive experiments are conducted, affirming the accuracy and efficacy of the proposed algorithm.
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Fast and Provable Tensor-Train Format Tensor Completion via Precondtioned Riemannian Gradient Descent
A preconditioned Riemannian gradient descent method that weights gradient entries by row norms of the TT-unfolded gradient achieves linear convergence for low tensor-train rank completion and is reported to be orders of magnitude faster than standard RGD.