Efficient tensor completion: Low-rank tensor train

Abstract

This paper proposes a novel formulation of the tensor completion problem to impute missing entries of data represented by tensors. The formulation is introduced in terms of tensor train (TT) rank which can effectively capture global information of tensors thanks to its construction by a well-balanced matricization scheme. Two algorithms are proposed to solve the corresponding tensor completion problem. The first one called simple low-rank tensor completion via tensor train (SiLRTC-TT) is intimately related to minimizing the TT nuclear norm. The second one is based on a multilinear matrix factorization model to approximate the TT rank of the tensor and called tensor completion by parallel matrix factorization via tensor train (TMac-TT). These algorithms are applied to complete both synthetic and real world data tensors. Simulation results of synthetic data show that the proposed algorithms are efficient in estimating missing entries for tensors with either low Tucker rank or TT rank while Tucker-based algorithms are only comparable in the case of low Tucker rank tensors. When applied to recover color images represented by ninth-order tensors augmented from third-order ones, the proposed algorithms outperforms the Tucker-based algorithms.

11 pages, 9 figures

References in corpus (2)

• Consistency of trace norm minimization
• Fixed Point and Bregman Iterative Methods for Matrix Rank Minimization

Cited by in corpus (13)

• Tensor Ring Decomposition
• Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives
• Tensor Completion by Alternating Minimization under the Tensor Train (TT) Model
• Wide Compression: Tensor Ring Nets
• On Tensor Train Rank Minimization: Statistical Efficiency and Scalable Algorithm
• Tensor Completion Algorithms in Big Data Analytics
• High-dimension Tensor Completion via Gradient-based Optimization Under Tensor-train Format
• New Riemannian preconditioned algorithms for tensor completion via polyadic decomposition
• Fast and Accurate Tensor Completion with Total Variation Regularized Tensor Trains
• Efficient Low Rank Tensor Ring Completion
• Lower and Upper Bounds on the VC-Dimension of Tensor Network Models
• Riemannian Conjugate Gradient Descent Method for Third-Order Tensor Completion
• MARS: Masked Automatic Ranks Selection in Tensor Decompositions