The collection of $d\times N$ complex matrices with prescribed column norms and singular values forms an algebraic variety, which we refer to as a frame space. Elements of frame spaces—i.e., frames—are used to give robust signal representations, so that geometrical properties of frame spaces are of interest to the signal processing community. This paper is concerned with the question: what is the probability that a frame drawn 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 show that frame spaces are related to highly structured objects called toric symplectic manifolds. As another application, we characterize the norm and spectral data for which the corresponding frame space has singularities, answering an open question in the literature.

的集合$d\times ñ$具有规定列范数和奇异值的复矩阵形成代数簇，我们将其称为框架空间。帧空间的元素（即帧）用于给出稳健的信号表示，因此帧空间的几何特性对信号处理界很感兴趣。本文关注的问题是：从给定帧空间中随机绘制的帧具有其列的任何d子集的属性的概率是多少${\mathbb{C}}^d$? 我们证明概率是 1，概括了 Cahill、Mixon 和 Strawn 最近的工作。为了证明这一点，我们证明了框架空间与称为复曲面辛流形的高度结构化对象有关。作为另一个应用，我们描述了相应框架空间具有奇点的范数和光谱数据，回答了文献中的一个悬而未决的问题。