Low-Rank Approximation of Tensors

Shmuel Friedland () and Venu Tammali ()
Additional contact information
Shmuel Friedland: University of Illinois at Chicago, Department of Mathematics, Statistics and Computer Science
Venu Tammali: University of Illinois at Chicago, Department of Mathematics, Statistics and Computer Science

Chapter Chapter 14 in Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory, 2015, pp 377-411 from Springer

Abstract: Abstract In many applications such as data compression, imaging or genomic data analysis, it is important to approximate a given tensor by a tensor that is sparsely representable. For matrices, i.e. 2-tensors, such a representation can be obtained via the singular value decomposition, which allows to compute best rank k-approximations. For very big matrices a low rank approximation using SVD is not computationally feasible. In this case different approximations are available. It seems that variants of the CUR-decomposition are most suitable. For d-mode tensors $$\mathcal{T} \in \otimes _{i=1}^{d}\mathbb{R}^{n_{i}}$$ , with d > 2, many generalizations of the singular value decomposition have been proposed to obtain low tensor rank decompositions. The most appropriate approximation seems to be best (r 1, …, r d )-approximation, which maximizes the ℓ 2 norm of the projection of $$\mathcal{T}$$ on ⊗ i = 1 d U i , where U i is an r i -dimensional subspace $$\mathbb{R}^{n_{i}}$$ . One of the most common methods is the alternating maximization method (AMM). It is obtained by maximizing on one subspace U i , while keeping all other fixed, and alternating the procedure repeatedly for i = 1, …, d. Usually, AMM will converge to a local best approximation. This approximation is a fixed point of a corresponding map on Grassmannians. We suggest a Newton method for finding the corresponding fixed point. We also discuss variants of CUR-approximation method for tensors. The first part of the paper is a survey on low rank approximation of tensors. The second new part of this paper is a new Newton method for best (r 1, …, r d )-approximation. We compare numerically different approximation methods.

Keywords: Orthonormal Basis; Singular Value Decomposition; Newton Method; Maximization Problem; Singular Vector (search for similar items in EconPapers)
Date: 2015
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-15260-8_14

Ordering information: This item can be ordered from
http://www.springer.com/9783319152608

DOI: 10.1007/978-3-319-15260-8_14

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().