Connectivity and Irreducibility of Algebraic Varieties of Finite Unit Norm Tight Frames

We affirm the conjectures in [K. Dykema and N. Strawn, Int. J. Pure Appl. Math., 28 (2006), pp. 217--256] by demonstrating the connectivity of spaces of finite unit norm tight frames (FUNTFs). Our central technique involves explicit continuous lifts of paths from the polytope of eigensteps (see [J. Cahill et al., Appl. Comput. Harmon. Anal., 35 (2013), pp. 52--73]) to spaces of FUNTFs. After demonstrating this connectivity result, we refine our analysis to show that the set of nonsingular points on these spaces is also connected, and we use this to show that spaces of FUNTFs are irreducible in the algebro-geometric sense. This last result allows us to show that generic FUNTFs are full spark, and hence the full spark FUNTFs are dense in the space of FUNTFs. This resolves an important theoretical question regarding the application of FUNTFs in the field of compressed sensing.