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

Abstract

It is well known that a best rank-R approximation of order-3 tensors may not exist for R ≥ 2. A best rank-(R, R, R) approximation always exists, however, and is also a best rank-R approximation when it has rank (at most) R. For R = 2 and real order-3 tensors it is shown that a best rank-2 approximation is also a local minimum of the best rank-(2,2,2) approximation problem. This implies that if all rank-(2,2,2) minima have rank larger than 2, then a best rank-2 approximation does not exist. This provides an easy-to-check criterion for existence of a best rank-2 approximation. The result is illustrated by means of simulations.

Real rank two geometry

“…The following theorem shows that case (b) cannot happen. This was proven for tensors by Stegeman and Friedland [18,Lemma 3.4]. We generalize their result to arbitrary varieties.…”

“…This vanishes on τ (X). For any pair (µ, ν) as in (18), we write f (µ,ν) for the unique preimage of (22) under the map C[x] → C[a, b]. This is well-defined by Proposition 5.2.…”

Tensor decomposition for infectious disease incidence data

“…Characterizing the rank of a tensor is a complex mathematical problem without a simple solution (Alexeev, Forbes, & Tsimerman, 2011; Ballico, Bernardi, Chiantini, & Guardo, 2018; Kolda & Bader, 2009; Stegeman & Friedland, 2017). Therefore, we attempt tensor decomposition beginning by selecting a single component, and increasing the number of components until the algorithm consistently converges.…”

Complexity of model testing for dynamical systems with toric steady states

“…When the constraints are given by polynomial equations these problems may be solved using methods from algebraic geometry, techniques for this have been developed by several authors [13,19]. In mathematics the algebraic geometric ED problem have been studied in the contexts of low rank tensor and low rank matrix approximation, see for example [19,21]. In systems biology, as discussed above, the ED problem has been used to study the model selection problem in [9].…”