Best rank-k approximations for tensors: generalizing Eckart–Young

Abstract

Given a tensor f in a Euclidean tensor space, we are interested in the critical points of the distance function from f to the set of tensors of rank at most k, which we call the critical rank-at-most-k tensors for f . When f is a matrix, the critical rank-one matrices for f correspond to the singular pairs of f . The critical rank-one tensors for f lie in a linear subspace H f , the critical space of f . Our main result is that, for any k, the critical rank-at-most-k tensors for a sufficiently general f also lie in the critical space H f . This is the part of Eckart-Young Theorem that generalizes from matrices to tensors. Moreover, we show that when the tensor format satisfies the triangle inequalities, the critical space H f is spanned by the complex critical rank-one tensors. Since f itself belongs to H f , we deduce that also f itself is a linear combination of its critical rank-one tensors.2000 Mathematics Subject Classification. 15A69, 15A18, 14M17, 14P05.

Nonlinear algebra and applications

“…The geometry of the CPD is well-studied in nonlinear algebra. One result worth mentioning is [76], where the authors show cases in which the best rank-r approximation of a generic tensor lies in the space spanned by its critical rank-1 approximations. For applications this implies that we can precondition the problem of computing best rank-r approximations by first computing critical rank-1 approximations.…”

High order singular value decomposition for plant diversity estimation

“…It succeeds for the so called orthogonally decomposable tensors [9]. Another type of generalization of the concept of "best rank-r approximation of a tensor" proposed in [17] works for general tensors. Despite the possible non-existence of the best approximation of a tensor under the construction of [13], this technique turns out to be very convenient from the computational point of view and it is possible to prove that the outcome is a "quasi-optimal solution".…”

“…There are various techniques to approximate tensors by taking advantage of the HOSVD, those that will be implemented in this paper are the so called Truncated HOSVD (T-HOSVD) and Sequentially Truncated HOSVD (ST-HOSVD). Even if only for some special cases HOSVD provides optimal results [9,17,41], it is possible to present an estimate of the tensor approximation errors. Indeed the core of the present paper will be: (i) the application of T-HOSVD and ST-HOSVD techniques, (ii) their modern versions and (iii) the associated errors.…”

Skew-symmetric tensor decomposition

“…The problem of decomposing a structured tensor in terms of its structured rank has been extensively studied in the last decades ( [2,3,10,8,12,13,17,18,28,36,39,40]). Most of the well-known results are for symmetric tensors, for tensors without any symmetry and for tensors with partial symmetries.…”