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

arXiv: https://arxiv.org/abs/2412.11165 · HTML: https://mlsanity.com/paper/2412.11165/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 53 references that are in the map.
- First posted on arXiv 15 Dec 2024.

## Builds on (most cited first, 18 of 18)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Tensor Robust Principal Component Analysis with A New Tensor Nuclear Norm](https://mlsanity.com/paper/1804.03728/) (2018)
- [Rank Minimization for Snapshot Compressive Imaging](https://mlsanity.com/paper/1807.07837/) (2018)
- [Exact tensor completion using t-SVD](https://mlsanity.com/paper/1502.04689/) (2015)
- [HDNet: High-resolution Dual-domain Learning for Spectral Compressive Imaging](https://mlsanity.com/paper/2203.02149/) (2022)
- [Degradation-Aware Unfolding Half-Shuffle Transformer for Spectral Compressive Imaging](https://mlsanity.com/paper/2205.10102/) (2022)
- [Deep Gaussian Scale Mixture Prior for Spectral Compressive Imaging](https://mlsanity.com/paper/2103.07152/) (2021)
- [Novel methods for multilinear data completion and de-noising based on tensor-SVD](https://mlsanity.com/paper/1407.1785/) (2014)
- [Deep Hyperspectral Prior: Denoising, Inpainting, Super-Resolution](https://mlsanity.com/paper/1902.00301/) (2019)
- [Self-supervised Neural Networks for Spectral Snapshot Compressive Imaging](https://mlsanity.com/paper/2108.12654/) (2021)
- [DDS2M: Self-Supervised Denoising Diffusion Spatio-Spectral Model for Hyperspectral Image Restoration](https://mlsanity.com/paper/2303.06682/) (2023)
- [Training Deep Networks with Structured Layers by Matrix Backpropagation](https://mlsanity.com/paper/1509.07838/) (2015)
- [Dictionary Learning with Low-rank Coding Coefficients for Tensor Completion](https://mlsanity.com/paper/2009.12507/) (2020)
- [Tensor Q-Rank: New Data Dependent Definition of Tensor Rank](https://mlsanity.com/paper/1910.12016/) (2019)
- [Hyperspectral Image Denoising via Global Spatial-Spectral Total Variation Regularized Nonconvex Local Low-Rank Tensor Approximation](https://mlsanity.com/paper/2006.00235/) (2020)
- [Improving Covariance Conditioning of the SVD Meta-layer by Orthogonality](https://mlsanity.com/paper/2207.02119/) (2022)
- [Transformed Low-Rank Parameterization Can Help Robust Generalization for Tensor Neural Networks](https://mlsanity.com/paper/2303.00196/) (2023)

## Built on by (most cited first, 5 of 5)

- [Deep Delta Learning](https://mlsanity.com/paper/2601.00417/) (2026)
- [HSI-VAR: Rethinking Hyperspectral Restoration through Spatial-Spectral Visual Autoregression](https://mlsanity.com/paper/2602.00749/) (2026)
- [Deep LoRA-Unfolding Networks for Image Restoration](https://mlsanity.com/paper/2602.18697/) (2026)
- [Vision-Language Controlled Deep Unfolding for Joint Medical Image Restoration and Segmentation](https://mlsanity.com/paper/2601.23103/) (2026)
- [ExpoMotion: A Large-Scale Benchmark and A Householder Projection Network for Multi-Exposure Fusion](https://mlsanity.com/paper/2607.03110/) (2026)

## Abstract (arXiv)

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.