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.
On Tensor Train Rank Minimization: Statistical Efficiency and Scalable Algorithm
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abstract
Tensor train (TT) decomposition provides a space-efficient representation for higher-order tensors. Despite its advantage, we face two crucial limitations when we apply the TT decomposition to machine learning problems: the lack of statistical theory and of scalable algorithms. In this paper, we address the limitations. First, we introduce a convex relaxation of the TT decomposition problem and derive its error bound for the tensor completion task. Next, we develop an alternating optimization method with a randomization technique, in which the time complexity is as efficient as the space complexity is. In experiments, we numerically confirm the derived bounds and empirically demonstrate the performance of our method with a real higher-order tensor.
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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.