Geometric structures in tensor representations
Geometric structures in tensor representations
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张量表示中的几何结构
DOI：
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发表时间：
2013
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通讯作者：
A. Nouy
中科院分区：
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作者：
A. Falcó;W. Hackbusch;A. Nouy
在本文中，我们介绍了一个张量子空间为基础的格式表示的张量在拓扑张量空间。要做到这一点，我们使用的最小子空间，使我们能够描述张量表示的根树。通过使用树结构和相关的最小子空间的维数，我们引入了一个基于树的格式，有界或固定的树为基础的秩的张量集。这个类包含Tucker格式和Hierarchical Tucker格式（包括Tensor Train格式）。特别地，所考虑的拓扑张量空间的任何张量都允许在具有有界树基秩的树基格式中的张量集合中的最佳逼近。此外，我们还证明了具有固定树秩的树格式张量集是一个解析Banach流形。这种局部图表表示的流形往往是至关重要的高维时间相关的偏微分方程和最小化问题的算法治疗。然而，在我们的框架中，在给定张量的切（Banach）空间不是自然环境张量Banach空间中的补子空间。因此，我们研究自然包含映射作为Banach流形之间的态射的微分。它允许我们讨论的Dirac-Frenkel变分原理的拓扑张量空间的框架。2010 AMS主题分类：15 A69，46 B28，46 A32。
In this paper we introduce a tensor subspace based format for the representation of a tensor in a topological tensor space. To do this we use a property of minimal subspaces which allow us to describe the tensor representation by means of a rooted tree. By using the tree structure and the dimensions of the associated minimal subspaces, we introduce the set of tensors in a tree based format with either bounded or fixed tree based rank. This class contains the Tucker format and the Hierarchical Tucker format (including the Tensor Train format). In particular, any tensor of the topological tensor space under consideration admits best approximations in the set of tensors in the tree based format with bounded tree based rank. Moreover, we show that the set of tensors in the tree based format with fixed tree based rank is an analytical Banach manifold. This local chart representation of the manifold is often crucial for an algorithmic treatment of high-dimensional time-dependent PDEs and minimisation problems. However, in our framework, the tangent (Banach) space at a given tensor is not a complemented subspace in the natural ambient tensor Banach space. Therefore, we study the differential of the natural inclusion map as a morphism between Banach manifolds. It allows us to discuss the Dirac-Frenkel variational principle in the framework of topological tensor spaces. 2010 AMS Subject Classifications: 15A69, 46B28, 46A32.