Codes and Expansions (CodEx) Seminar

Kasso Okoudjou (Tufts University)
The search for universal minimizers of \(p\) frame potentials

Given \(d\geq 2\), \(p\in (0, \infty]\), and \(N\geq 2\), let \[\mu_{p, d, N}=\min \bigg\{\sum_{k, \ell=1}^N |\langle \varphi_k, \varphi_\ell \rangle|^p:\, \{\varphi_{k}\}_{k=1}^N \subset S^{d-1}\bigg\}\] where \(S^{d-1}\) is the unit sphere in \(\mathbb{R}^d\). Among the questions one can ask about this function, the following are of interest to us in this talk:

  • For fixed \(d\) and \(N\), find an explicit formula for the function \(\mu_{p, d, N}\).
  • For fixed \(d\) and \(N\), what are the optimal configurations \(\{\varphi_{k}\}_{k=1}^N \subset S^{d-1}\)?
  • Do these optimal configurations give rise to minimizers of the continuous \(p\)-frame potentials defined by \[\iint_{S^{d-1}\times S^{d-1}}|\langle x, y \rangle|^p d\mu(x) d\mu(y)\] where \(\mu\) runs over the set of all Borel probability measures defined on \(S^{d-1}\)?
Answers to these questions are known in certain cases, e.g., when \(p=2\) Benedetto and Fickus proved that FUNTFs are the optimal configurations. In addition, for certain values of \(p\), the optimal configurations are related to ETFs or Grassmannian frames. But we still don't know much about the function \(\mu_{p, d, N}\), even when \(d=2\).

After reviewing some facts about these questions, I will focus on recent results we obtained when \(d=2\). In this case, we prove that for each \(N\) there exists \(p_N\) such that for all \(p\geq p_N\), Grassmannian frames of \(N\) vectors are the optimal configurations. In the particular case when \(N=4\) we are able to completely classify the minimizers of \(\mu_{p, 2, 4}\) for \(p\in (0, \infty]\). But for \(N\geq 5\) a similar classification is far from complete when \(p\in (0, p_N)\). I will finish the talk with a number of open questions about the minimizers of this family of potentials.

This talk is based on joint work with X. Chen, E. Goodman, M. Ehler, V. Gonzales, and S. Kang.