Efficient tensor completion: Low-rank tensor train

TL;DR

This paper introduces new algorithms for tensor completion using tensor train rank, effectively imputing missing data in high-dimensional tensors and outperforming Tucker-based methods in experiments.

Contribution

The paper presents two novel algorithms for tensor completion based on tensor train rank, improving accuracy and efficiency over existing Tucker-based methods.

Findings

Algorithms effectively recover missing entries in synthetic tensors.

Proposed methods outperform Tucker-based algorithms in image tensor completion.

Efficiently handle tensors with low TT rank and Tucker rank.

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.