Low-Rank Approximation of Tensors

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 CURdecomposition are most suitable.For d-mode tensors T ∈ ⊗ d i=1 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 (r1, . . . , r d )-approximation, which maximizes the 2 norm of the projection of T on ⊗ d i=1 Ui, where Ui is an ri-dimensional subspace R n i . One of the most common methods is the alternating maximization method (AMM). It is obtained by maximizing on one subspace Ui, 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 (r1, . . . , r d )-approximation. We compare numerically different approximation methods.2000 Mathematics Subject Classification. 14M15, 15A18, 15A69, 65H10, 65K10.

On best rank-2 and rank-(2,2,2) approximations of order-3 tensors

“…In this paper, we prove a new criterion for existence of a best rank-2 approximation for real order-3 tensors via (unconstrained) best rank-(2,2,2) approximations as recently suggested in [31]. The 2 × 2 × 2 core tensor of a best rank-(2,2,2) approximation has rank 2 or 3.…”

“…Algorithms for solving problem (2.2) have been proposed in [35,36,37,38,31]. In the higher-order power method of [36] each iteration rotates mass to the target subtensor by using the singular value decomposition (SVD) of the columns of one of the three matrix unfoldings of the rotated tensor corresponding to the subtensor.…”

Remarks on the Symmetric Rank of Symmetric Tensors

“…See [14,16]. The results of [15] yield that T ⋆ is unique for T outside of a semialgebraic set of dimension less than the real dimension of ⊗ d F n .…”

Probabilistic bounds on best rank-one approximation ratio

“…It is often necessary to retain only some key properties of a data set, that corresponds to an approximation of a tensor by another one with a simpler structure. There are several different models, based on tensor decompositions, that are used for this purpose, see [13,9] and references therein. An important special case is an approximation of a given "data"-tensor with a rank-one tensor, see [8].…”