论文标题
基于树的张量格式的几何形状
Geometry of tree-based tensor formats in tensor Banach spaces
论文作者
论文摘要
在论文中，“关于张量的Banach空间的Dirac-Frenkel变异原理”，我们提供了tucker格式的张量的几何描述，并在Tensor Banach空间中具有固定的多线性（或Tucker）等级，以扩展Dirac-Frenkel-Frenkel变性原理，以扩展拓扑框架的框架。本注释的目的是将这些结果扩展到更一般的张量格式。更确切地说，我们提供了基于树（或分层）格式（也称为树张量网络）的张量的新几何描述，它们是tucker格式的张量的相互作用，它们与尺寸集的不同分区相关。基于树格式的张量的张量的几何描述与塔克格式的张量的一种张量兼容。
In the paper `On the Dirac-Frenkel Variational Principle on Tensor Banach Spaces', we provided a geometrical description of manifolds of tensors in Tucker format with fixed multilinear (or Tucker) rank in tensor Banach spaces, that allowed to extend the Dirac-Frenkel variational principle in the framework of topological tensor spaces. The purpose of this note is to extend these results to more general tensor formats. More precisely, we provide a new geometrical description of manifolds of tensors in tree-based (or hierarchical) format, also known as tree tensor networks, which are intersections of manifolds of tensors in Tucker format associated with different partitions of the set of dimensions. The proposed geometrical description of tensors in tree-based format is compatible with the one of manifolds of tensors in Tucker format.
