Tensors in Statistics

Xuan Bi, Xiwei Tang, Yubai Yuan, Yanqing Zhang, Annie Qu

Research output: Contribution to journalReview articlepeer-review


This article provides an overview of tensors, their properties, and their applications in statistics. Tensors, also known as multidimensional arrays, are generalizations of matrices to higher orders and are useful data representation architectures. We first review basic tensor concepts and decompositions, and then we elaborate traditional and recent applications of tensors in the fields of recommender systems and imaging analysis. We also illustrate tensors for network data and explore the relations among interacting units in a complex network system. Some canonical tensor computational algorithms and available software libraries are provided for various tensor decompositions. Future research directions, including tensors in deep learning, are also discussed.

Original languageEnglish (US)
Pages (from-to)345-368
Number of pages24
JournalAnnual Review of Statistics and Its Application
StatePublished - Mar 7 2021

Bibliographical note

Funding Information:
The authors would like to acknowledge support for this project from the National Science Foundation grants DMS-1821198 and DMS-1952406 and the National Natural Science Foundation of P.R. China (11871420). The authors thank the Editorial Committee, Production Editor, and anonymous reviewers for their suggestions and helpful feedback, which improved the article significantly.

Publisher Copyright:
© 2021 Annual Reviews Inc.. All rights reserved.

Copyright 2021 Elsevier B.V., All rights reserved.


  • high-order networks
  • imaging analyses
  • recommender systems
  • tensor applications
  • tensor properties

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