text: 95,34,1453038307546476552,7984662,Clayton Shonkwiler,"New paper: Toric Symplectic Geometry and Full Spark Frames, written with Tom Needham. <LINK> (1/7) Here’s one way of approaching the basic problem: let d &lt; N and consider d × N complex matrices where we pre-specify the singular values of the matrix and the norms of the columns. Such conditions are common in various signal processing applications. (2/7) Now, for something like compressed sensing, you would really like it to be true that every d × d submatrix is invertible. So the question: given a random matrix with the prescribed data, what is the probability that every d × d minor is invertible? (3/7) Sometimes your data is incompatible, so there are no such matrices. This isn’t the only condition, but to have such matrices the sum of the squares of the singular values must equal the sum of the squares of the column norms. (4/7) It can also happen that the space of matrices with the prescribed data is non-empty, but *all* matrices in the space have some singular d × d minor, and hence the probability is zero. (5/7) Our main theorem characterizes exactly when these two possibilities happen and shows that in all other cases the probability is equal to 1. Along the way we determine when these spaces are manifolds and characterize the local structure of singularities when they're not. (6/7) The main idea is to show that these spaces are closely related to certain highly structured and extremely symmetric manifolds called toric symplectic manifolds (or toric varieties, if you’re more of an algebraic geometer). (7/7)",https://arxiv.org/abs/2110.11295,"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. ",Toric Symplectic Geometry and Full Spark Frames,7,"['New paper: Toric Symplectic Geometry and Full Spark Frames, written with Tom Needham.\n\n<LINK>\n\n(1/7)', 'Here’s one way of approaching the basic problem: let d &lt; N and consider d × N complex matrices where we pre-specify the singular values of the matrix and the norms of the columns. Such conditions are common in various signal processing applications.\n\n(2/7)', 'Now, for something like compressed sensing, you would really like it to be true that every d × d submatrix is invertible. So the question: given a random matrix with the prescribed data, what is the probability that every d × d minor is invertible?\n\n(3/7)', 'Sometimes your data is incompatible, so there are no such matrices. This isn’t the only condition, but to have such matrices the sum of the squares of the singular values must equal the sum of the squares of the column norms.\n\n(4/7)', 'It can also happen that the space of matrices with the prescribed data is non-empty, but *all* matrices in the space have some singular d × d minor, and hence the probability is zero.\n\n(5/7)', ""Our main theorem characterizes exactly when these two possibilities happen and shows that in all other cases the probability is equal to 1. Along the way we determine when these spaces are manifolds and characterize the local structure of singularities when they're not.\n\n(6/7)"", 'The main idea is to show that these spaces are closely related to certain highly structured and extremely symmetric manifolds called toric symplectic manifolds (or toric varieties, if you’re more of an algebraic geometer).\n\n(7/7)']",21,10,1540