  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 complex-valued signals, so that
geometrical and measure-theoretic 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.

Functional Analysis;Differential Geometry;Symplectic Geometry