论文标题
张量缩小尺寸和解开互动的行人排名
Tensor Ranks for the Pedestrian for Dimension Reduction and Disentangling Interactions
论文作者
论文摘要
张量是一个多路阵列，除了数据集外，还可以表示联合定律或多元函数的表达。因此，它包含了与每个条目相对应的变量之间的相互作用的描述。张量的等级扩展到具有两个以上条目的数组，矩阵等级的概念要记住，有几种构建此类扩展的方法。当等级为一个时，变量将分开，而当级别较低时，变量会弱耦合。对于低等级的张量，许多计算更简单。此外，通过低级张量近似给定的张量，可以计算表格的某些特征，例如当它是联合定律时的分区函数。在本说明中，我们通过系统地使用张量代数，详细介绍了一种较低等级的张量近似张量的集成和渐进方法。张量的概念是严格定义的，然后提出了基本但对张量的有用操作。 在回忆了几个不同的概念以将等级扩展到张量之后，我们展示了如何将这些基本操作组合起来以构建最佳的低级近似算法。最后一章致力于将这种方法应用于构建的多元函数离散化的张量，以表明在笛卡尔网格上，预计此类张量的等级将很低。
A tensor is a multi-way array that can represent, in addition to a data set, the expression of a joint law or a multivariate function. As such it contains the description of the interactions between the variables corresponding to each of the entries. The rank of a tensor extends to arrays with more than two entries the notion of rank of a matrix, bearing in mind that there are several approaches to build such an extension. When the rank is one, the variables are separated, and when it is low, the variables are weakly coupled. Many calculations are simpler on tensors of low rank. Furthermore, approximating a given tensor by a low-rank tensor makes it possible to compute some characteristics of a table, such as the partition function when it is a joint law. In this note, we present in detail an integrated and progressive approach to approximate a given tensor by a tensor of lower rank, through a systematic use of tensor algebra. The notion of tensor is rigorously defined, then elementary but useful operations on tensors are presented. After recalling several different notions for extending the rank to tensors, we show how these elementary operations can be combined to build best low rank approximation algorithms. The last chapter is devoted to applying this approach to tensors constructed as the discretisation of a multivariate function, to show that on a Cartesian grid, the rank of such tensors is expected to be low.
