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Fast and Provable Tensor-Train Format Tensor Completion via Precondtioned Riemannian Gradient Descent

arXiv 2501.13385 ↗First posted 23 Jan 2025cs.LG cs.NA math.NAcited by 0

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Abstract (arXiv)

Low-rank tensor completion aims to recover a tensor from partially observed entries, and it is widely applicable in fields such as quantum computing and image processing. Due to the significant advantages of the tensor train (TT) format in handling structured high-order tensors, this paper investigates the low-rank tensor completion problem based on the TT-format. We proposed a preconditioned Riemannian gradient descent algorithm (PRGD) to solve low TT-rank tensor completion and establish its linear convergence. Experimental results on both simulated and real datasets demonstrate the effectiveness of the PRGD algorithm. On the simulated dataset, the PRGD algorithm reduced the computation time by two orders of magnitude compared to existing classical algorithms. In practical applications such as hyperspectral image completion and quantum state tomography, the PRGD algorithm significantly reduced the number of iterations, thereby substantially reducing the computational time.

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References · 19

  1. 2014Guaranteed Matrix Completion via Non-convex Factorizationcited by 81
  2. 2016Efficient tensor completion for color image and video recovery: Low-rank tensor traincited by 61
  3. 2019Inference and Uncertainty Quantification for Noisy Matrix Completioncited by 39
  4. 2021Generalized Low-rank plus Sparse Tensor Estimation by Fast Riemannian Optimizationcited by 24
  5. 2016Guarantees of Riemannian Optimization for Low Rank Matrix Completioncited by 19
  6. 2019Statistical Inferences of Linear Forms for Noisy Matrix Completioncited by 18
  7. 2015Structured Matrix Completion with Applications to Genomic Data Integrationcited by 17
  8. 2021Provable Tensor-Train Format Tensor Completion by Riemannian Optimizationcited by 17
  9. 2018Fast and Accurate Tensor Completion with Total Variation Regularized Tensor Trainscited by 14
  10. 2018High-dimension Tensor Completion via Gradient-based Optimization Under Tensor-train Formatcited by 12
  11. 2020Optimal High-order Tensor SVD via Tensor-Train Orthogonal Iterationcited by 11
  12. 2016Tensor Completion by Alternating Minimization under the Tensor Train (TT) Modelcited by 9
7 more
  1. 2021Implicit Regularization and Entrywise Convergence of Riemannian Optimization for Low Tucker-Rank Tensor Completioncited by 9
  2. 2023Quantum State Tomography for Matrix Product Density Operatorscited by 9
  3. 2023A Preconditioned Riemannian Gradient Descent Algorithm for Low-Rank Matrix Recoverycited by 7
  4. 2017On Tensor Train Rank Minimization: Statistical Efficiency and Scalable Algorithmcited by 4
  5. 2021New Riemannian preconditioned algorithms for tensor completion via polyadic decompositioncited by 4
  6. 2022Tensor Completion via Tensor Train Based Low-Rank Quotient Geometry under a Preconditioned Metriccited by 3
  7. 2024A Single-Mode Quasi Riemannian Gradient Descent Algorithm for Low-Rank Tensor Recoverycited by 1

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