Framelet Representation of Tensor Nuclear Norm for Third-Order Tensor Completion

TL;DR

This paper introduces a novel framelet-based tensor nuclear norm for third-order tensor completion, leveraging framelet sparsity to improve recovery performance on multi-dimensional data.

Contribution

It proposes a new convex tensor completion model using framelet representation, enhancing the tensor nuclear norm approach with basis redundancy for better accuracy.

Findings

Outperforms existing methods on videos, multispectral images, and MRI data.

Utilizes sparse framelet representation for improved tensor completion.

Achieves better recovery results when tensor slices are highly correlated.

Abstract

The main aim of this paper is to develop a framelet representation of the tensor nuclear norm for third-order tensor completion. In the literature, the tensor nuclear norm can be computed by using tensor singular value decomposition based on the discrete Fourier transform matrix, and tensor completion can be performed by the minimization of the tensor nuclear norm which is the relaxation of the sum of matrix ranks from all Fourier transformed matrix frontal slices. These Fourier transformed matrix frontal slices are obtained by applying the discrete Fourier transform on the tubes of the original tensor. In this paper, we propose to employ the framelet representation of each tube so that a framelet transformed tensor can be constructed. Because of framelet basis redundancy, the representation of each tube is sparsely represented. When the matrix slices of the original tensor are highly correlated, we expect the corresponding sum of matrix ranks from all framelet transformed matrix frontal slices would be small, and the resulting tensor completion can be performed much better. The proposed minimization model is convex and global minimizers can be obtained. Numerical results on several types of multi-dimensional data (videos, multispectral images, and magnetic resonance imaging data) have tested and shown that the proposed method outperformed the other testing methods.