Best approximation on semi-algebraic sets and k-border rank approximation of symmetric tensors

Abstract

In the first part of this paper we study a best approximation of a vector in Euclidean space R n with respect to a closed semi-algebraic set C and a given semi-algebraic norm. Assuming that the given norm and its dual norm are differentiable we show that a best approximation is unique outside a hypersurface. We then study the case where C is an irreducible variety and the approximation is with respect to the Euclidean norm. We show that for a general point in x ∈ R n the number of critical points of the distance function of x to C is bounded above by a degree of a related dominant map. If C induces a smooth projective variety V P ⊂ P (C n ) then this degree is the top Chern number of a corresponding vector bundle on V P . We then study the problem when a best k(≥ 2)-border rank approximation of a symmetric tensor is symmetric. We show that under certain dimensional conditions there exists an open semialgebraic set of symmetric tensors for which a best k-border rank is unique and symmetric.

Some approximation problems in semi-algebraic geometry

“…Clearly, π is a polynomial map. It is a dominating map [7,11]. This is a simple consequence of Lemma 4.3.…”

“…) is an irreducible variety of dimension n [7,11]. (A short justification that Σ g (C C ) is irreducible is given in the proof of Lemma 5.1.)…”

“…While we do not present any explicit computations, we should note that our methods have applications in approximation of matrices and tensors. The relevant results can be found in [10] and [11].…”

On the complexity of finding tensor ranks

“…Also if F has at least d elements then each S ∈ S d F n is a sum of rank-one symmetric tensors [13, Proposition 7.2]. (It is shown in [13,Proposition 7.1] that for a fixed finite field F and n 2 there exist symmetric tensors which are not sum of rank-one symmetric tensors for sufficiently large d.) In the following passage, we assume that |F| d. We define srank S, the symmetric rank of S ∈ S d F n \ {0}, as the minimal number in the decomposition of S as a sum of rank-one symmetric tensors. For matrices over a field of characteristic = 2 the symmetric rank of S ∈ S 2 F n is equal to the (standard) rank of S, whereas for d 3 there are examples of 3-symmetric tensors whose symmetric rank is greater than their tensor rank [26].…”

“…A rank-one symmetric tensor in S d G n is of the form [22]. Since every algebraically closed field has an infinite number of elements it follows that every symmetric tensor has a Waring decomposition [13]. The minimal number of rank-one symmetric tensors in the decomposition of S is called a symmetric rank and is denoted as srank S. Clearly, rank S srank S. It is shown in [10,29] that in certain cases one has equality rank S = srank S. However, even for d = 3 one can have an inequality rank S < srank S [26].…”