Fast and Provable Low-Rank High-Order Tensor Completion via Scaled Gradient Descent
Abstract: This work studies the low-rank high-order tensor completion problem, which aims to exactly recover a low-rank order-$d$ ($d \geq 4$) tensor from partially observed entries. Recently, tensor Singular Value Decomposition (t-SVD)-based low-rank tensor completion has gained considerable attention due to its ability to capture the low-rank structure of multidimensional data. However, existing approaches often rely on the computationally expensive tensor nuclear norm (TNN), thereby limiting their scalability for real-world tensors. Leveraging the low-rank structure under the t-SVD decomposition, we propose an efficient algorithm that directly estimates the high-order tensor factors---starting from a spectral initialization---via scaled gradient descent (ScaledGD). Theoretically, we rigorously establish the recovery guarantees of the proposed algorithm under mild assumptions, demonstrating that it achieves linear convergence to the true low-rank tensor at a constant rate that is independent of the condition number. Numerical experiments on both synthetic and real-world data verify our results and demonstrate the superiority of our method.
Submission Length: Regular submission (no more than 12 pages of main content)
Assigned Action Editor: ~Jie_Shen6
Submission Number: 4158
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