Polytopes of eigensteps of finite equal norm tight frames

TL;DR

This paper characterizes the polytope of eigensteps for finite equal norm tight frames, providing explicit descriptions, dimensions, and facets, which aids in understanding their structure and construction.

Contribution

It offers a non-redundant description of the eigenstep polytope for equal norm tight frames, including its dimension and facets.

Findings

Explicit equations and inequalities for the polytope

Dimension and facet count of the polytope

Enhanced understanding of frame construction geometry

Abstract

Hilbert space frames generalize orthonormal bases to allow redundancy in representations of vectors while keeping good reconstruction properties. A frame comes with an associated frame operator encoding essential properties of the frame. We study a polytope that arises in an algorithm for constructing all finite frames with given lengths of frame vectors and spectrum of the frame operator, which is a Gelfand-Tsetlin polytope. For equal norm tight frames, we give a non-redundant description of the polytope in terms of equations and inequalities. From this we obtain the dimension and number of facets of the polytope.