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Nonnegative low rank tensor approximations with multidimensional image applications

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Abstract

The main aim of this paper is to develop a new algorithm for computing a nonnegative low rank tensor approximation for nonnegative tensors that arise in many multidimensional imaging applications. Nonnegativity is one of the important properties, as each pixel value refers to a nonzero light intensity in image data acquisitions. Our approach is different from classical nonnegative tensor factorization (NTF), which requires each factorized matrix, and/or tensor, to be nonnegative. In this paper, we determine a nonnegative low Tucker rank tensor to approximate a given nonnegative tensor. We propose an alternating projections algorithm for computing such a nonnegative low rank tensor approximation, which is referred to as NLRT. The convergence of the proposed manifold projection method is established. The experimental results for synthetic data and multidimensional images are presented to demonstrate that the performance of NLRT is better than the state-of-the-art NTF methods.

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Notes

  1. \(\{ \mathcal{Z}^{(k)} \}\) falls in \(\texttt{U}\) means there exists a positive integer \(k_0\) such that \(\{ \mathcal{Z}^{(k)} \}_{k \ge k_0} \subset \texttt{U}\).

  2. To avoid making the entries negative, we first simulate noise with a standard normal distribution and then set the negative noisy value to 0. The SNR in dB is defined as \(\text {SNR}_\text {dB} = 20\log _{10}\frac{\Vert \mathcal {X}_\text {groundtruth}\Vert _F}{\Vert \text {Noise}\Vert _F}\).

  3. Defined as \( \frac{\Vert \mathcal {X}_\text {estimated}-\mathcal {X}_\text {groundtruth}\Vert _F}{\Vert \mathcal {X}_\text {groundtruth}\Vert _F}. \)

  4. Videos are available at http://trace.eas.asu.edu/yuv/ and https://sites.google.com/site/jamiezeminzhang/publications.

  5. Data available at http://www.ehu.eus/ccwintco/index.php?title=Hyperspectral_Remote_Sensing_Scenes.

  6. Data available at https://engineering.purdue.edu/~biehl/MultiSpec/hyperspectral.html.

  7. Data available at https://www.cs.rochester.edu/~jliu/code/TensorCompletion.zip.

  8. Data available at https://aviris.jpl.nasa.gov/data/free_data.html.

  9. Data available at http://openremotesensing.net/knowledgebase/hyperspectral-video/.

  10. https://en.wikipedia.org/wiki/Peak_signal-to-noise_ratio

  11. Data available at https://engineering.purdue.edu/\(\sim \)biehl/MultiSpec/hyperspectral.html.

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Acknowledgements

T.-X. Jiang’s research is supported in part by the National Natural Science Foundation of China under Grant 12001446, the Natural Science Foundation of Sichuan, China under Grant 2022NSFSC1798, and the Fundamental Research Funds for the Central Universities under Grants JBK2202049 and JBK2102001. M. K. Ng’s research is supported in part by Hong Kong Research Grant Council GRF 12300218, 12300519, 17201020, 17300021, C1013-21GF, C7004-21GF and Joint NSFC-RGC N-HKU76921. G.-J. Song’s research is supported in part by the National Natural Science Foundation of China under Grant 12171369 and Key NSF of Shandong Province under Grant ZR2020KA008.

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Authors and Affiliations

  1. School of Computing and Artificial Intelligence, Southwestern University of Finance and Economics, Chengdu, People’s Republic of China

    Tai-Xiang Jiang

  2. Kash Institute of Electronics and Information Industry, Kashi, People’s Republic of China

    Tai-Xiang Jiang

  3. Department of Mathematics, The University of Hong Kong, Pokfulam, Hong Kong, People’s Republic of China

    Michael K. Ng & Junjun Pan

  4. School of Mathematics and Information Sciences, Weifang University, Weifang, 261061, People’s Republic of China

    Guang-Jing Song

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  1. Tai-Xiang Jiang
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Correspondence to Guang-Jing Song.

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Jiang, TX., Ng, M.K., Pan, J. et al. Nonnegative low rank tensor approximations with multidimensional image applications. Numer. Math. 153, 141–170 (2023). https://doi.org/10.1007/s00211-022-01328-6

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