Mathematics Colloquium

Best Approximation by Algebraic and Semi-Algebraic Sets

Speaker: Shmuel Friedland, University of Illinois, Chicago

Location: Warren Weaver Hall 1302

Date: Monday, May 12, 2014, 3:45 p.m.


Many approximation problems can be stated as finding a best approximant of a point \(x \in \mathbb{R}^{n}\) from a given set closed \(S \subset \mathbb{R}^{n}\). In most of the interesting cases \(S\) is either algebraic, (approximation by low rank tensors), or semialgebraic, (approximation by separable quantum states). There are two major problems: characterize the set of points \(E\) in \(\mathbb{R}^{n}\) for which a best approximant is not unique, and count (estimate) the number of critical points of the Euclidean distance of a generic \(x\) to a real irreducible variety \(S\).

In this talk we first show that for \(S\) semi-algebraic \(E\) is a semi-algebraic set of dimension less than \(n\). Next we discuss the case where \(E\) is a real irreducible variety. By considering the complex variety \(E_{C}\) we will show that the number of critical point of generic \(x\) in \(\mathbb{C}^{n}\) is a degree of certain rational map. (This gives an upper estimate of the number of critical points.) Next we will discuss the case where \(S\) is the Segre variety of tensors of rank one. We compute the degree of the above map using Chern classes.

Our talk is based on the two recent papers:

  1. S. Friedland and G. Ottaviani, The number of singular vector tuples and uniqueness of best rank one approximation of tensors, to appear in Foundations of Computational Mathematics, arXiv:1210.8316.
  2. S. Friedland and M. Stawiska, Best approximation on semi-algebraic sets and k-border rank approximation of symmetric tensors, arXiv:1311.1561.