# 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.

Synopsis:

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.