Discrete compressed sensing: lattices and frames (Josiah Park, Georgia Tech)

Lattice valued vector systems have taken an important role in packing, coding, cryptography, and signal processing problems.  In compressed sensing, improvements in sparse recovery methods can be reached with an additional  assumption that the signal of  interest is lattice  valued, as demonstrated by A.  Flinth  and G. Kutyniok. Equiangular  tight  frames are  particular systems  of unit  vectors  with minimal  coherence,  a measure of how well distributed the vectors are, and have provable guarantees for recovery of sparse vectors in standard methods.  The determination whether real equiangular tight frames have integer span on a lattice has been given a characterization within two papers by A. Bottcher, L. Fukshansky, one with S. R. Garcia, H. Maharaj and D. Needell.  Here the corresponding question is considered for the complex case and several families are demonstrated to have either integer span on a lattice or not.  In addition, it is demonstrated that a real Parseval tight frame can have integer span on a lattice if and only if the inner products appearing in the system are rational.  (Collaboration with L. Fukshansky, D. Needell, and Y. Xin)