Parseval frames are spanning sets of R^d that provide redundant encoding of vectors in terms of their expansion coefficients. Especially important for applications are the equal-norm Parseval frames which are known to be most robust in some sense for one erasure. While they have great potential as encoding schemes, equal norm Parseval frames are very difficult to construct. A related notion is that of an ε-nearly equal-norm Parseval frame. These type of frames, on the other hand, can be constructed via probabilistic algorithms. The classical Paulsen Problem asks to determine how far a given ε-nearly equal-norm Parseval frame is from the set of all equal-norm Parseval frames of n vectors in R^d.
Going beyond the classical set-up of frames of vectors in R^d, frames of matrices of arbitrary rank and size are the main tools for applications to distributed sensing, parallel processing, and packet encoding, just to name a few.
In this talk I will describe an approach to Paulsen’s problem based on quiver invariant theory. We will be particularly interested in sigma-critical representations which arise naturally in the context of the Kempf-Ness theorem on closed orbits in Invariant Theory. After introducing all the relevant concepts, I will first present a result that gives necessary and sufficient conditions for the orbit of a quiver representation to contain sigma-critical representation. I will then explain how this result can be used to solve the Paulsen Problem for matrix frames. This is based on joint work with Jasim Ismaeel.
Join via Zoom: https://illinois.zoom.us/j/84912824216?pwd=UnZGcWFrN25IQjR1QnljbW91T0pUZz09
