Toric Symplectic Geometry and Full Spark Frames
Abstract
The collection of $d \times N$ complex matrices with prescribed column norms and prescribed (nonzero) singular values forms a compact algebraic variety, which we refer to as a frame space. Elements of frame spaces  i.e., frames  are used to give robust representations of complexvalued signals, so that geometrical and measuretheoretic properties of frame spaces are of interest to the signal processing community. This paper is concerned with the following question: what is the probability that a frame drawn uniformly at random from a given frame space has the property that any subset of $d$ of its columns gives a basis for $\mathbb{C}^d$? We show that the probability is one, generalizing recent work of Cahill, Mixon and Strawn. To prove this, we first show that frame spaces are related to highly structured objects called toric symplectic manifolds. This relationship elucidates the geometric meaning of eigensteps  certain spectral invariants of a frame  and should be a more broadly applicable tool for studying probabilistic questions about the structure of frame spaces. As another application of our symplectic perspective, we completely characterize the norm and spectral data for which the corresponding frame space has singularities, answering some open questions in the frame theory literature.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 DOI:
 10.48550/arXiv.2110.11295
 arXiv:
 arXiv:2110.11295
 Bibcode:
 2021arXiv211011295N
 Keywords:

 Mathematics  Functional Analysis;
 Mathematics  Differential Geometry;
 Mathematics  Symplectic Geometry
 EPrint:
 Added an example on the structure of singularities, plus other minor improvements. 30 pages