Riemannian preconditioned algorithms for tensor completion via tensor ring decomposition

Bin Gao (), Renfeng Peng () and Ya-xiang Yuan ()
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Bin Gao: Academy of Mathematics and Systems Science, Chinese Academy of Sciences
Renfeng Peng: Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and University of Chinese Academy of Sciences
Ya-xiang Yuan: Academy of Mathematics and Systems Science, Chinese Academy of Sciences

Computational Optimization and Applications, 2024, vol. 88, issue 2, No 2, 443-468

Abstract: Abstract We propose Riemannian preconditioned algorithms for the tensor completion problem via tensor ring decomposition. A new Riemannian metric is developed on the product space of the mode-2 unfolding matrices of the core tensors in tensor ring decomposition. The construction of this metric aims to approximate the Hessian of the cost function by its diagonal blocks, paving the way for various Riemannian optimization methods. Specifically, we propose the Riemannian gradient descent and Riemannian conjugate gradient algorithms. We prove that both algorithms globally converge to a stationary point. In the implementation, we exploit the tensor structure and adopt an economical procedure to avoid large matrix formulation and computation in gradients, which significantly reduces the computational cost. Numerical experiments on various synthetic and real-world datasets—movie ratings, hyperspectral images, and high-dimensional functions—suggest that the proposed algorithms have better or favorably comparable performance to other candidates.

Keywords: Tensor completion; Tensor ring decomposition; Riemannian optimization; Preconditioned gradient (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10589-024-00559-7

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