arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2001.11708cs.CVcs.LGmath.ACmath.RA
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基于张量代数的广义视觉信息分析

Generalized Visual Information Analysis via Tensorial Algebra

Liang Liao, Stephen John Maybank

摘要:

高阶数据使用矩阵建模,矩阵的条目是固定大小的数值数组。这些数组称为t标量,在卷积积下形成一个交换环。元素在t-标量环中的矩阵称为t-矩阵。t矩阵可以按通常的方式进行缩放、相加和相乘。正矩阵、正交矩阵和厄米特对称矩阵的t-矩阵推广。利用t矩阵模型,可以推广许多著名的矩阵算法。特别地,t矩阵被用于推广奇异值分解(SVD)、高阶奇异值分解(HOSVD)、主成分分析(PCA)、二维主成分分析(二维的)和格拉斯曼成分分析(GCA)。将广义t矩阵算法TSVD、THOSVD、TPCA、T2 DPCA和TGCA应用于图像的低秩近似、重建和监督分类。实验结果表明,t矩阵算法与标准矩阵算法相比,具有更好的性能。

英文摘要:

Higher order data is 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 SVD (Singular Value Decomposition), HOSVD (High Order SVD), PCA (Principal Component Analysis), 2DPCA (Two Dimensional PCA) and GCA (Grassmannian Component Analysis). 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.

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