# Normalized Iterative Hard Thresholding for Tensor Recovery

arXiv: https://arxiv.org/abs/2507.04228 · HTML: https://mlsanity.com/paper/2507.04228/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 79 references that are in the map.
- First posted on arXiv 6 Jul 2025.

## Builds on (most cited first, 19 of 19)

- [Tensor Ring Decomposition](https://mlsanity.com/paper/1606.05535/) (2016)
- [Efficient tensor completion for color image and video recovery: Low-rank tensor train](https://mlsanity.com/paper/1606.01500/) (2016)
- [Efficient Low Rank Tensor Ring Completion](https://mlsanity.com/paper/1707.08184/) (2017)
- [On the Fenchel Duality between Strong Convexity and Lipschitz Continuous Gradient](https://mlsanity.com/paper/1803.06573/) (2018)
- [Barzilai-Borwein Step Size for Stochastic Gradient Descent](https://mlsanity.com/paper/1605.04131/) (2016)
- [Weighted Low-rank Tensor Recovery for Hyperspectral Image Restoration](https://mlsanity.com/paper/1709.00192/) (2017)
- [Dictionary Learning with Low-rank Coding Coefficients for Tensor Completion](https://mlsanity.com/paper/2009.12507/) (2020)
- [Low-tubal-rank Tensor Completion using Alternating Minimization](https://mlsanity.com/paper/1610.01690/) (2016)
- [Linked Component Analysis from Matrices to High Order Tensors: Applications to Biomedical Data](https://mlsanity.com/paper/1508.07416/) (2015)
- [Linear Convergence of Stochastic Iterative Greedy Algorithms with Sparse Constraints](https://mlsanity.com/paper/1407.0088/) (2014)
- [Accelerating Variance-Reduced Stochastic Gradient Methods](https://mlsanity.com/paper/1910.09494/) (2019)
- [Iterative Hard Thresholding for Low CP-rank Tensor Models](https://mlsanity.com/paper/1908.08479/) (2019)
- [Binary Iterative Hard Thresholding Converges with Optimal Number of Measurements for 1-Bit Compressed Sensing](https://mlsanity.com/paper/2207.03427/) (2022)
- [Unsupervised HDR Image and Video Tone Mapping via Contrastive Learning](https://mlsanity.com/paper/2303.07327/) (2023)
- [Stochastic Greedy Algorithms For Multiple Measurement Vectors](https://mlsanity.com/paper/1711.01521/) (2017)
- [Multichannel sparse recovery of complex-valued signals using Huber's criterion](https://mlsanity.com/paper/1504.04184/) (2015)
- [Discrete Optimization Methods for Group Model Selection in Compressed Sensing](https://mlsanity.com/paper/1904.01542/) (2019)
- [Stochastic Variance Reduced Gradient for affine rank minimization problem](https://mlsanity.com/paper/2211.02802/) (2022)
- [Iterative Singular Tube Hard Thresholding Algorithms for Tensor Recovery](https://mlsanity.com/paper/2304.04860/) (2023)

## Abstract (arXiv)

Low-rank recovery builds upon ideas from the theory of compressive sensing, which predicts that sparse signals can be accurately reconstructed from incomplete measurements. Iterative thresholding-type algorithms-particularly the normalized iterative hard thresholding (NIHT) method-have been widely used in compressed sensing (CS) and applied to matrix recovery tasks. In this paper, we propose a tensor extension of NIHT, referred to as TNIHT, for the recovery of low-rank tensors under two widely used tensor decomposition models. This extension enables the effective reconstruction of high-order low-rank tensors from a limited number of linear measurements by leveraging the inherent low-dimensional structure of multi-way data. Specifically, we consider both the CANDECOMP/PARAFAC (CP) rank and the Tucker rank to characterize tensor low-rankness within the TNIHT framework. At the same time, we establish a convergence theorem for the proposed TNIHT method under the tensor restricted isometry property (TRIP), providing theoretical support for its recovery guarantees. Finally, we evaluate the performance of TNIHT through numerical experiments on synthetic, image, and video data, and compare it with several state-of-the-art algorithms.