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

2025-01-23

AI Summary

This work addresses the problem of completing high-order low-rank tensors from partial observations—arising in quantum state tomography and hyperspectral imaging. We propose Preconditioned Riemannian Gradient Descent (PRGD), the first preconditioned Riemannian optimization algorithm tailored for the tensor train (TT) format. Theoretically, PRGD achieves linear convergence, overcoming the slow convergence and high computational cost inherent in conventional TT completion methods. By integrating Riemannian optimization, TT decomposition, and adaptive preconditioning on the non-convex low-rank TT manifold, PRGD accelerates convergence by over two orders of magnitude on synthetic data. In practical applications—hyperspectral image completion and quantum state tomography—it substantially reduces iteration counts and improves real-world efficiency. This work establishes a new paradigm for structured tensor completion that simultaneously delivers rigorous theoretical guarantees and superior empirical performance.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Privacy-enhancing technologies

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.

Problem

• Low-Rank Tensor Completion
• Quantum Computing
• Image Processing

Innovation

• Preprocessing Techniques
• Low-Rank Tensor Completion
• Quantum Computing Applications