Tuesday, December 3, 2019 | 2:30pm | Yost 306

Speaker: Alexander Strang, PhD Candidate (CWRU, Department of Mathematics, Applied Mathematics and Statistics)

Title: Variance Minimization via Delaunay Triangulation

Abstract:We consider the problem of minimizing the variance of a distribution supported on a finite set of points in $\mathbb{R}^n$ given the expected value of the distribution. This is a natural problem as it produces the distribution with the least uncertainty in $X$, in the $L_2$ sense, given the support and the mean.   We show that under an appropriate choice of norm on the covariance, (i) the optimization problem is convex  and (ii) its solution is given by evaluating the tent functions associated with a Delaunay triangulation of the support at the mean. Moreover when the Delaunay triangulation is not unique the space of solutions is the space of convex combinations of solutions associated with each possible triangulation. In some applications there are many possible triangulations. For integer lattices and subsets of integer lattices the space of solutions is characterized by restricting the support of the solution to the unit hypercube containing the mean. On the lattice I present a particular solution which simultaneously minimizes the total variance, maximal projected variance, and Frobenius norm of the covariance.