Minimizing the p-frame potential on unit balls

High Dimensional Seminar
Wednesday, October 10, 2018 - 12:55pm
1 hour (actually 50 minutes)
Skiles 006
Georgia institute of Technology
It has been known that when an equiangular tight frame (ETF) of size |Φ|=N exists, Φ ⊂ Fd&nbsp;(real or complex), for p > 2 the p-frame potential ∑i ≠ j&nbsp;| < φj, φk&nbsp;> |p&nbsp;achieves its minimum value on an ETF over all N sized collections of vectors. We are interested in minimizing a related quantity: 1/ N2&nbsp;∑i, j=1&nbsp;| < φj, φk&nbsp;> |p&nbsp;. In particular we ask when there exists a configuration of vectors for which this quantity is minimized over all sized subsets of the real or complex sphere of a fixed dimension. Also of interest is the structure of minimizers over all unit vector subsets of Fd&nbsp;of size N. We shall present some results for p in (2, 4) along with numerical results and conjectures. Portions of this talk are based on recent work of D. Bilyk, A. Glazyrin,&nbsp;R. Matzke, and O. Vlasiuk.