BREAKING THE CURSE OF DIMENSIONALITY, OR HOW TO USE SVD IN MANY DIMENSIONS
BREAKING THE CURSE OF DIMENSIONALITY, OR HOW TO USE SVD IN MANY DIMENSIONS
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DOI：
10.1137/090748330
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发表时间：
2009-01-01
影响因子：
3.1
通讯作者：
Tyrtyshnikov, E. E.
中科院分区：
文献类型：
--
作者：
Oseledets, I. V.;Tyrtyshnikov, E. E.
对于d维张量，可能大d > 3，分层数据结构，称为Tree-Tucker格式，作为一种替代的规范分解。它具有渐近相同（通常甚至更少）的表示参数和可行的稳定性。该方法涉及到一个递归的结构所描述的树的叶子对应的Tucker分解的三维张量，是基于一个序列的SVD的递归获得的展开矩阵和辅助维度添加到初始的“空间”维度。它示出了如何将这种格式可以应用到多维卷积的问题。给出了令人信服的数值例子。
For d-dimensional tensors with possibly large d > 3, an hierarchical data structure, called the Tree-Tucker format, is presented as an alternative to the canonical decomposition. It has asymptotically the same (and often even smaller) number of representation parameters and viable stability properties. The approach involves a recursive construction described by a tree with the leafs corresponding to the Tucker decompositions of three-dimensional tensors, and is based on a sequence of SVDs for the recursively obtained unfolding matrices and on the auxiliary dimensions added to the initial "spatial" dimensions. It is shown how this format can be applied to the problem of multidimensional convolution. Convincing numerical examples are given.