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

Fengmiao Bian, Jian-Feng Cai, Xiaoqun Zhang, Yuanwei Zhang

TL;DR

This work tackles the problem of recovering a high-order tensor from partial observations by leveraging the TT-format and formulating a low $TT$-rank tensor completion task. It introduces a preconditioned Riemannian gradient descent (PRGD) that employs a data-driven metric on the tangent space, enabling faster convergence and a robust retraction via TT-SVD. The authors prove linear convergence under good initialization and sufficient samples, with a contraction factor around 0.3574, and demonstrate substantial empirical speedups over prior Riemannian gradient methods across synthetic data, hyperspectral imaging, and quantum state tomography. The proposed approach significantly enhances scalability for structured high-order tensor problems and offers practical impact in applications requiring efficient tensor completion.

Abstract

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.

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

TL;DR

This work tackles the problem of recovering a high-order tensor from partial observations by leveraging the TT-format and formulating a low -rank tensor completion task. It introduces a preconditioned Riemannian gradient descent (PRGD) that employs a data-driven metric on the tangent space, enabling faster convergence and a robust retraction via TT-SVD. The authors prove linear convergence under good initialization and sufficient samples, with a contraction factor around 0.3574, and demonstrate substantial empirical speedups over prior Riemannian gradient methods across synthetic data, hyperspectral imaging, and quantum state tomography. The proposed approach significantly enhances scalability for structured high-order tensor problems and offers practical impact in applications requiring efficient tensor completion.

Abstract

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.
Paper Structure (22 sections, 14 theorems, 116 equations, 7 figures, 2 algorithms)

This paper contains 22 sections, 14 theorems, 116 equations, 7 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem and Preliminary of TT-format Tensors
  3. Preliminaries
  4. Methodology
  5. Riemannian Gradient Descent Algorithm
  6. Data-Driven Metric and Precondition Riemannian Gradient Descent
  7. Data-Driven Metric
  8. Preconditioned Riemannian Gradient Descent under $\langle\cdot, \cdot\rangle_{\mathscr{W}_l}$
  9. Tangent Space Parameterization and Computation of $\widetilde{\mathscr{P}}_\mathbb{T}$
  10. Recovery Guarantee
  11. Key Lemmas
  12. Proof of Main Theorem
  13. Numerical Experiments
  14. Synthetic Data
  15. Hyperspectral Images
  16. ...and 7 more sections

Key Result

Proposition 3.1

For tensor-train format tensor $\mathcal{T} = \left[T_1, \dots, T_m\right]$ with $T_i\in \mathbb{R}^{r_{i-1}\times d_i\times r_i}$, one has

Figures (7)

  • Figure 1: Illustration of scaling operation on $\mathcal{G}_l$'s elements.
  • Figure 2: Results of CPU-time (in seconds) and the number of iterations for TT-rank tensor completion on simulated data.
  • Figure 3: Results of CPU-time (in seconds) and the number of iterations for TT-rank tensor completion on simulated data.
  • Figure 4: Results of CPU-time (in seconds) and the number of iterations for TT-rank tensor completion on simulated data.
  • Figure 5: Results of relative error and the number of iterations for TT-rank tensor completion on different noise levels.
  • ...and 2 more figures

Theorems & Definitions (25)

  • Proposition 3.1
  • proof
  • Definition 3.2: New left orthogonal decomposition
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Remark 3.6
  • ...and 15 more