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arXiv · 2005.01299

The Multi-Symplectic Lanczos Algorithm and Its Applications to Color Image Processing

Zhigang Jia·Xuan Liu·Mei-Xiang Zhao
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Abstract

Low-rank approximations of original samples are playing more and more an important role in many recently proposed mathematical models from data science. A natural and initial requirement is that these representations inherit original structures or properties. With this aim, we propose a new multi-symplectic method based on the Lanzcos bidiagonalization to compute the partial singular triplets of JRS-symmetric matrices. These singular triplets can be used to reconstruct optimal low-rank approximations while preserving the intrinsic multi-symmetry. The augmented Ritz and harmonic Ritz vectors are used to perform implicit restarting to obtain a satisfactory bidiagonal matrix for calculating the $k$ largest or smallest singular triplets, respectively. We also apply the new multi-symplectic Lanczos algorithms to color face recognition and color video compressing and reconstruction. Numerical experiments indicate their superiority over the state-of-the-art algorithms.

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Zhigang Jia, Xuan Liu, Mei-Xiang Zhao. 2020-05-04. The Multi-Symplectic Lanczos Algorithm and Its Applications to Color Image Processing. https://arxiv.org/abs/2005.01299

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