Jameson Cahill

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.

Fusion Frames and the Restricted Isometry Property

We show that RIP frames, tight frames satisfying the restricted isometry property, give rise to nearly tight fusion frames, which are nearly orthogonal and, hence, are nearly equi-isoclinic. We also show how to replace parts of the RIP frame with orthonormal sets while maintaining the restricted isometry property.

Algebraic Geometry and Finite Frames

Interesting families of finite frames often admit characterizations in terms of algebraic constraints, and thus it is not entirely surprising that powerful results in finite frame theory can be obtained by utilizing tools from algebraic geometry. In this chapter, our goal is to demonstrate the power of these techniques. First, we demonstrate that algebro-geometric ideas can be used to explicitly construct local coordinate systems that reflect intuitive degrees of freedom within spaces of finite unit norm tight frames (and more general spaces), and that optimal frames can be characterized by useful algebraic conditions. In particular, we construct locally well-defined real-analytic coordinate systems on spaces of finite unit norm tight frames, and we demonstrate that many types of optimal Parseval frames are dense and that further optimality can be discovered through embeddings that naturally arise in algebraic geometry.

Dimension invariance of finite frames of translates and Gabor frames

A dimension invariance property for finite frames of translates and Gabor frames is discussed. Under appropriate support conditions among the frame and dual frame generating functions, we show that a pair of dual frames evaluated in a given space remains a valid dual set if they are naturally embedded in the underlying space of almost arbitrarily enlarged dimension. Consequently, the evaluation of duals in a very large dimensional space is now easily accessible by merely working in a space of some much smaller dimension. A number of uniform and non-uniform schemes are studied. To satisfy the support conditions, a method of finding valid alternate dual functions with small support via a known parametric dual frame formula is discussed. Oftentimes it is convenient to have truncated approximate duals that satisfy the support conditions. Stability studies of the dimension invariance principle via such approximate duals are also presented.

Grassmannians in frame theory

The Grassmannian is the set of k dimensional subspace of an n dimensional vector space. This set naturally parametrizes isomorphism classes of frames. We will present several results which exploit this fact. In particular, we will use the Plucker embedding of the Grassmannian to identify a class of frames which is optimal for the maximal number of erasures.

Connectivity of spaces of finite unit-norm tight frames

We show that the spaces of finite unit norm tight frames are connected, which verifies a conjecture first appearing in Dykema and Strawn (2006). Our central technique involves continuous liftings of paths from the polytope of eigensteps (see Cahill et al., 2012), or Gelfand-Tsetlin patterns, 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 result to show that spaces of FUNTFs are irreducible in the algebro-geometric sense.

Using projections for phase retrieval

The mathematical study of phase retrieval was started in 2006 in a landmark paper of Balan, Casazza and Edidin. This quickly became a heavily studied topic with implications for many areas of research in both applied mathematics and engineering. Recently there have been developments in a new area of study pertaining to phase retrieval given by projections. We give an extensive overview of the papers regarding projection phase retrieval.

Non-orthogonal fusion frames

Fusion frames have become a major tool in the implementation of distributed systems. The effectiveness of fusion frame applications in distributed systems is reflected in the efficiency of the end fusion process. This requires the inversion of the fusion frame operator which is difficult or impossible in practice. What we want is for the fusion frame operator to be the identity. But in most applications, especially to sensor networks, this almost never occurs. We will solve this problem by introducing the notion of non-orthogonal fusion frames which have the property that in most cases we can turn a family of subspaces of a Hilbert space into a non-orthogonal fusion frame which has a fusion frame operator which is the identity.