High-order data are modeled using matrices whose entries are numerical arrays of a fixed size. These arrays, called t-scalars, form a commutative ring under the convolution product. Matrices with elements in the ring of t-scalars are referred to as t-matrices. The t-matrices can be scaled, added and multiplied in the usual way. There are t-matrix generalizations of positive matrices, orthogonal matrices and Hermitian symmetric matrices. With the t-matrix model, it is possible to generalize many well-known matrix algorithms. In particular, the t-matrices are used to generalize the singular value decomposition (SVD), high-order SVD (HOSVD), principal component analysis (PCA), two-dimensional PCA (2DPCA) and Grassmannian component analysis (GCA). The generalized t-matrix algorithms, namely TSVD, THOSVD, TPCA, T2DPCA and TGCA, are applied to low-rank approximation, reconstruction and supervised classification of images. Experiments show that the t-matrix algorithms compare favorably with standard matrix algorithms.

中文翻译：

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通过张量代数的广义视觉信息分析

高阶数据使用其条目为固定大小的数字数组的矩阵建模。这些称为t标量的数组在卷积积下形成一个交换环。t阶为环形的元素的矩阵称为t矩阵。可以按通常的方式对t矩阵进行缩放，相加和相乘。正矩阵，正交矩阵和Hermitian对称矩阵都有t矩阵的概括。使用t矩阵模型，可以归纳许多众所周知的矩阵算法。特别地，t矩阵用于概括奇异值分解（SVD），高阶SVD（HOSVD），主成分分析（PCA），二维PCA（2DPCA）和格拉斯曼成分分析（GCA）。广义的t矩阵算法，即TSVD，THOSVD，TPCA，T2DPCA和TGCA，用于图像的低秩逼近，重建和监督分类。实验表明，t矩阵算法与标准矩阵算法相比具有优势。