论文标题
张量子stiefel歧管的计算：初步研究
Computation over Tensor Stiefel Manifold: A Preliminary Study
论文作者
论文摘要
令$*$表示两个三阶张量之间的T产品。这项工作的目的是研究集合$ st（n，p，l）的基本计算：= \ {\ nathcal q \ in \ mathbb r^{n \ times p \ times p \ times p \ times p \ times l} \ mathcal q^{\ top Q^{\ top} $ n \ times p \ times l $和$ \ mathcal i $（$ n \ geq p $）是身份张量。首先证实了$ st（n，p，l）$赋予通常的frobenius norm形成riemannian歧管，该歧管在这项工作中被称为（第三阶）\ emph {tensor stiefel歧管}。然后，我们在$ st（n，p，l）$上得出了切线空间，里曼尼亚渐变和里曼尼亚的黑森。此外，还提供了基于T-QR，T极极分解，Cayley Transform和T-Expentential以及矢量传输的各种缩回的公式。预计本研究中得出的公式类似于其矩阵对应物，可以用作分析张张式stiefel歧管优化问题的基础，并为其设计Riemannian算法。
Let $*$ denote the t-product between two third-order tensors. The purpose of this work is to study fundamental computation over the set $St(n,p,l):= \{\mathcal Q\in \mathbb R^{n\times p\times l} \mid \mathcal Q^{\top}* \mathcal Q = \mathcal I \}$, where $\mathcal Q$ is a third-order tensor of size $n\times p \times l$ and $\mathcal I$ ($n\geq p$) is the identity tensor. It is first verified that $St(n,p,l)$ endowed with the usual Frobenius norm forms a Riemannian manifold, which is termed as the (third-order) \emph{tensor Stiefel manifold} in this work. We then derive the tangent space, Riemannian gradient, and Riemannian Hessian on $St(n,p,l)$. In addition, formulas of various retractions based on t-QR, t-polar decomposition, Cayley transform, and t-exponential, as well as vector transports, are presented. It is expected that analogous to their matrix counterparts, the formulas derived in this study may serve as building blocks for analyzing optimization problems over the tensor Stiefel manifold and designing Riemannian algorithms for them.
