English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

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 containsthe description of the interactions between the variables corresponding to eachof the entries. The rank of a tensor extends to arrays with more than twoentries the notion of rank of a matrix, bearing in mind that there are severalapproaches to build such an extension. When the rank is one, the variables areseparated, and when it is low, the variables are weakly coupled. Manycalculations are simpler on tensors of low rank. Furthermore, approximating agiven tensor by a low-rank tensor makes it possible to compute somecharacteristics of a table, such as the partition function when it is a jointlaw. In this note, we present in detail an integrated and progressive approachto approximate a given tensor by a tensor of lower rank, through a systematicuse of tensor algebra. The notion of tensor is rigorously defined, thenelementary but useful operations on tensors are presented. After recallingseveral different notions for extending the rank to tensors, we show how theseelementary operations can be combined to build best low rank approximationalgorithms. The last chapter is devoted to applying this approach to tensorsconstructed as the discretisation of a multivariate function, to show that on aCartesian grid, the rank of such tensors is expected to be low.

Tensor Ranks for the Pedestrian for Dimension Reduction and Disentangling Interactions