Second-Generation Wavelet-inspired Tensor Product with Applications in Hyperspectral Imaging
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| Format: | Preprint |
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2026
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| author | Panchal, Aneesh Behera, Ratikanta |
| author_facet | Panchal, Aneesh Behera, Ratikanta |
| contents | This paper introduces the $w$-product, a novel wavelet-based tensor multiplication scheme leveraging second-generation wavelet transforms to achieve linear transformation complexity while preserving essential algebraic properties. The $w$-product outperforms existing tensor multiplication approaches by enabling fast and numerically stable tensor decompositions by proposing ``$w$-svd'' and its sparse variant ``sp-$w$-svd'', for efficient low-rank approximations with significantly reduced computational costs. Experiments on low-rank hyperspectral image reconstruction demonstrate up to a $92.21$ times speedup compared to state-of-the-art ``$t$-svd'', with comparable PSNR and SSIM metrics. We discuss the Moore-Penrose inverse of tensors based on the $w$-product and examine its essential properties. Numerical examples are provided to support the theoretical results. Then, hyperspectral image deblurring experiments demonstrate up to $27.88$ times speedup with improved image quality. In particular, the $w$-product and the sp-$w$-product exhibit exponentially increasing acceleration with the decomposition level compared to the traditional approach of the $t$-product. This work provides a scalable framework for multidimensional data analysis, with future research directions including adaptive wavelet designs, higher-order tensor extensions, and real-time implementations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_08228 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Second-Generation Wavelet-inspired Tensor Product with Applications in Hyperspectral Imaging Panchal, Aneesh Behera, Ratikanta Numerical Analysis This paper introduces the $w$-product, a novel wavelet-based tensor multiplication scheme leveraging second-generation wavelet transforms to achieve linear transformation complexity while preserving essential algebraic properties. The $w$-product outperforms existing tensor multiplication approaches by enabling fast and numerically stable tensor decompositions by proposing ``$w$-svd'' and its sparse variant ``sp-$w$-svd'', for efficient low-rank approximations with significantly reduced computational costs. Experiments on low-rank hyperspectral image reconstruction demonstrate up to a $92.21$ times speedup compared to state-of-the-art ``$t$-svd'', with comparable PSNR and SSIM metrics. We discuss the Moore-Penrose inverse of tensors based on the $w$-product and examine its essential properties. Numerical examples are provided to support the theoretical results. Then, hyperspectral image deblurring experiments demonstrate up to $27.88$ times speedup with improved image quality. In particular, the $w$-product and the sp-$w$-product exhibit exponentially increasing acceleration with the decomposition level compared to the traditional approach of the $t$-product. This work provides a scalable framework for multidimensional data analysis, with future research directions including adaptive wavelet designs, higher-order tensor extensions, and real-time implementations. |
| title | Second-Generation Wavelet-inspired Tensor Product with Applications in Hyperspectral Imaging |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2601.08228 |