OTLRM: Orthogonal Learning-based Low-Rank Metric for Multi-Dimensional Inverse Problems

TL;DR

This paper introduces a data-driven low-rank tensor model using a learnable orthogonal transform, improving the flexibility and stability of low-rank inverse problem solutions in multi-dimensional data.

Contribution

It proposes a novel orthogonal transform-based low-rank tensor model and a low-rank solver that enhance deep neural network applications for inverse problems.

Findings

Significant improvement in tensor data restoration quality

Effective integration of learnable orthogonal transforms in neural networks

Enhanced stability over traditional SVT-based methods

Abstract

In real-world scenarios, complex data such as multispectral images and multi-frame videos inherently exhibit robust low-rank property. This property is vital for multi-dimensional inverse problems, such as tensor completion, spectral imaging reconstruction, and multispectral image denoising. Existing tensor singular value decomposition (t-SVD) definitions rely on hand-designed or pre-given transforms, which lack flexibility for defining tensor nuclear norm (TNN). The TNN-regularized optimization problem is solved by the singular value thresholding (SVT) operator, which leverages the t-SVD framework to obtain the low-rank tensor. However, it is quite complicated to introduce SVT into deep neural networks due to the numerical instability problem in solving the derivatives of the eigenvectors. In this paper, we introduce a novel data-driven generative low-rank t-SVD model based on the learnable orthogonal transform, which can be naturally solved under its representation. Prompted by the linear algebra theorem of the Householder transformation, our learnable orthogonal transform is achieved by constructing an endogenously orthogonal matrix adaptable to neural networks, optimizing it as arbitrary orthogonal matrices. Additionally, we propose a low-rank solver as a generalization of SVT, which utilizes an efficient representation of generative networks to obtain low-rank structures. Extensive experiments highlight its significant restoration enhancements.