具有可逆线性变换的$p$阶张量积积
$p$-order Tensor Products with Invertible Linear Transforms
摘要：
本文研究了张量问题。文中介绍了三种典型的张量分解。在这些分解中，t-SVD是近十年来提出的。不同的秩定义来自张量分解。本文在研究高阶张量积和可逆变换下的张量积的基础上，引入了一种高阶可逆变换下的张量积，这是迄今为止最一般的情形。此外，还证明了一些性质。由于低秩恢复的最优化模型经常使用核范数，本文试图推广核范数，并证明它与多秩张量的关系。该定理为今后高阶张量的低秩恢复铺平了道路。
英文摘要：
This paper studies the issues about tensors. Three typical kinds of tensor decomposition are mentioned. Among these decompositions, the t-SVD is proposed in this decade. Different definitions of rank derive from tensor decompositions. Based on the research about higher order tensor t-product and tensor products with invertible transform, this paper introduces a product performing higher order tensor products with invertible transform, which is the most generalized case so far. Also, a few properties are proven. Because the optimization model of low-rank recovery often uses the nuclear norm, the paper tries to generalize the nuclear norm and proves its relation to multi-rank of tensors. The theorem paves the way for low-rank recovery of higher order tensors in the future.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
